Blanchfield form

From Manifold Atlas

(Difference between revisions)

Jump to: navigation, search

Latest revision as of 18:07, 2 April 2013

This page has not been refereed. The information given here might be incomplete or provisional.

[edit] 1 Background: equivariant intersection forms

In 1939 Reidemeister [Reidemeister1939] defined an equivariant, sesquilinear intersection form on the homology of a covering space $ {{Stub}} == Introduction == <wikitex>; After Poincar\'{e} and Lefschetz, a closed oriented manifold $N^{n}$ has a bilinear intersection form defined on its homology: $$I_N \colon H_k(N;\mathbb{Z}) \times H_{n-k}(N;\mathbb{Z}) \to \mathbb{Z}; ([p],[q]) \mapsto \langle p, q \rangle$$ such that $$I_N(x,y) = (-)^{k(n-k)}I_N(y,x).$$ Given a ${k}$--chain $p \in C_{k}(N;\mathbb{Z})$ and an $(n-k)$--chain $q \in C_{n-k}(N;\mathbb{Z})$ which is transverse to $q$, the signed count of the intersections between $p$ and $q$ gives an intersection number $\langle\, p \, , \, q\, \rangle \in \mathbb{Z}$. By bilinearity, the intersection form vanishes on the torsion part of the homology. The analogue of the intersection pairing for the torsion part of the homology of a closed manifold $N^n$ is the bilinear $\Q/\mathbb{Z}$--valued linking form, which is due to Seifert: $$L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \Q/\mathbb{Z}$$ such that $$L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x).$$ Given $[x] \in TH_\ell(N;\mathbb{Z})$ and $[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$ represented by cycles $x \in C_\ell(N;\mathbb{Z})$, let $w \in C_{\ell+1}(N;\mathbb{Z})$ be such that $\partial w = sy$, for some $s \in \mathbb{Z}$. Then we define: $$L_N([x],[y]) := \langle x, w \rangle/s \in \Q/\mathbb{Z}.$$ The resulting element is independent of the choices of $x,y,w$ and $s$. As an example, let $N = \mathbb{RP}^3$, so that $\ell=1$ and $n=3$. Now $H_1(\mathbb{RP}^3;\mathbb{Z}) \cong \mathbb{Z}_2$. Let $\theta \in H_1(\mathbb{RP}^3;\mathbb{Z})$ be the non--trivial element. To compute the linking $L_{\mathbb{RP}^3}(\theta,\theta)$, consider $\mathbb{RP}^3$ modelled as $D^3/\sim$, with antipodal points on $\partial D^2$ identified, and choose two representative \widetilde{M}$ of an $m$ -dimensional closed manifold $M$ whose deck transformation group $G$ is abelian.

$\displaystyle \begin{array}{rcl} I_{\widetilde{M}} \colon H_k(\widetilde{M};\mathbb{Z}) \times H_{m-k}(\widetilde{M};\mathbb{Z}) &\to & \mathbb{Z}[G]; \\ ([p],[q]) & \mapsto & \sum_{g \in G} \langle g \cdot p, q \rangle g^{-1}, \end{array}$ $\displaystyle \begin{array}{rcl} I_{\widetilde{M}} \colon H_k(\widetilde{M};\mathbb{Z}) \times H_{m-k}(\widetilde{M};\mathbb{Z}) &\to & \mathbb{Z}[G]; \\ ([p],[q]) & \mapsto & \sum_{g \in G} \langle g \cdot p, q \rangle g^{-1}, \end{array}$

where

$\displaystyle \langle \cdot , \cdot \rangle \colon C_k(\widetilde{M};\mathbb{Z}) \times C_{m-k}(\widetilde{M};\mathbb{Z}) \to \mathbb{Z}$ $\displaystyle \langle \cdot , \cdot \rangle \colon C_k(\widetilde{M};\mathbb{Z}) \times C_{m-k}(\widetilde{M};\mathbb{Z}) \to \mathbb{Z}$

counts transverse intersections between chains algebraically.

The intersections of each possible $G$ -translate of $p$ and $q$ are counted, and indexed according to the deck transformation which produced that intersection number.

[edit] 2 Definition of the Blanchfield form

In his 1954 Princeton PhD thesis R. C. Blanchfield [Blanchfield1957] made the corresponding generalisation for linking forms. Let $X^{m}$ be a compact manifold, now possibly with non-empty boundary, with a surjective homomorphism $\pi_1(X) \to \Gamma$ , for some free abelian group $\Gamma$ . Let $\mathbb{Z}[\Gamma]$ be the group ring of $\Gamma$ and let $\mathbb{Q}(\Gamma)$ be its field of fractions.

The $\mathbb{Z}[\Gamma]$ $\mathbb{Z}[\Gamma]$ -torsion submodule of a $\mathbb{Z}[\Gamma]$ $\mathbb{Z}[\Gamma]$ module $P$ $P$ is the submodule

$\displaystyle TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}[\Gamma]\}.$ $\displaystyle TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}[\Gamma]\}.$

The Blanchfield form is a sesquilinear $\mathbb{Q}(\Gamma)/\mathbb{Z}[\Gamma]$ --valued pairing which is defined on the $\mathbb{Z}[\Gamma]$ -torsion submodules of the homology of the $\Gamma$ --cover $\widetilde{X}$ of $X$ :

$\displaystyle \mathop{\mathrm{Bl}} \colon TH_\ell(\widetilde{X};\mathbb{Z}) \times TH_{m-\ell-1}(\widetilde{X},\partial\widetilde{X};\mathbb{Z}) \to \mathbb{Q}(\Gamma)/\mathbb{Z}[\Gamma].$ $\displaystyle \mathop{\mathrm{Bl}} \colon TH_\ell(\widetilde{X};\mathbb{Z}) \times TH_{m-\ell-1}(\widetilde{X},\partial\widetilde{X};\mathbb{Z}) \to \mathbb{Q}(\Gamma)/\mathbb{Z}[\Gamma].$

Given

$\displaystyle [x] \in TH_\ell(\widetilde{X};\mathbb{Z}) \cong TH_\ell(X;\mathbb{Z}[\Gamma])$ $\displaystyle [x] \in TH_\ell(\widetilde{X};\mathbb{Z}) \cong TH_\ell(X;\mathbb{Z}[\Gamma])$

and

$\displaystyle [y] \in TH_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z}) \cong TH_{m-\ell-1}(X,\partial X;\mathbb{Z}[\Gamma])$ $\displaystyle [y] \in TH_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z}) \cong TH_{m-\ell-1}(X,\partial X;\mathbb{Z}[\Gamma])$

represented by cycles $x \in C_\ell(\widetilde{X};\mathbb{Z})$ $x \in C_\ell(\widetilde{X};\mathbb{Z})$ and $y \in C_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z})$ $y \in C_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z})$ , let $w \in C_{m-\ell}(\widetilde{X},\partial\widetilde{X};\mathbb{Z})$ $w \in C_{m-\ell}(\widetilde{X},\partial\widetilde{X};\mathbb{Z})$ be such that $\partial w = \Delta y$ $\partial w = \Delta y$ , for some $\Delta \in \mathbb{Z}[\Gamma]$ $\Delta \in \mathbb{Z}[\Gamma]$ . Then we define:

$\displaystyle \mathop{\mathrm{Bl}}([x],[y]) := \sum_{g \in \Gamma} \langle g \cdot x, w \rangle g^{-1}/\Delta \in \mathbb{Q}(\Gamma)/\mathbb{Z}[\Gamma],$ $\displaystyle \mathop{\mathrm{Bl}}([x],[y]) := \sum_{g \in \Gamma} \langle g \cdot x, w \rangle g^{-1}/\Delta \in \mathbb{Q}(\Gamma)/\mathbb{Z}[\Gamma],$

where $\Gamma$ acts on $C_\ell(\widetilde{X};\mathbb{Z})$ by the action induced from the deck transformation.

The Blanchfield form can also be defined via homology, using a Bockstein homomorphism and a cup product. This is entirely analogous to the treatment the situation with the linking form. Just change $\mathbb{Z}$ to $\mathbb{Z}[\mathbb{Z}]$ and $\mathbb{Q}$ to $\mathbb{Q}(\mathbb{Z})$ . In the case of knot exteriors, a homological definition is also given below.

[edit] 3 Example of knot exteriors

We now turn to an example. We will focus on the case of knots in $S^3$ . For a knot $K \subset S^3$ , let $X_K$ denote its exterior, which is the complement of a regular neighbourhood of $K$ : $X_K:= S^3 \setminus \nu K$ . Now $\ell=1$ , $m=3$ and the abelianisation gives a homomorphism $\pi_1(X_K) \to H_1(X_K;\mathbb{Z}) \xrightarrow{\simeq} \mathbb{Z}$ . The Blanchfield form can in this case be defined without relative homology, on $H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \cong H_1(\widetilde{X}_K;\mathbb{Z})$ . The form

$\displaystyle \mathop{\mathrm{Bl}} \colon H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \times H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to \mathbb{Q}(\mathbb{Z})/\mathbb{Z}[\mathbb{Z}]$ $\displaystyle \mathop{\mathrm{Bl}} \colon H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \times H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to \mathbb{Q}(\mathbb{Z})/\mathbb{Z}[\mathbb{Z}]$

is non--singular, sesquilinear and Hermitian. Note that $H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ is entirely $\mathbb{Z}[\mathbb{Z}]$ -torsion, so $H_1(X_K;\mathbb{Z}[\mathbb{Z}]) = TH_1(X_K;\mathbb{Z}[\mathbb{Z}])$ . The adjoint of this form is given by the following sequence of homomorphisms:

$\displaystyle \begin{aligned} &H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(X_K,\partial X_K;\mathbb{Z}[\mathbb{Z}]) \to H^2(X_K;\mathbb{Z}[\mathbb{Z}]) \\ & \to H^1(X_K;\mathbb{Q}(\mathbb{Z})/\mathbb{Q}[\mathbb{Z}]) \to \Hom_{\mathbb{Z}[\mathbb{Z}]}(H_1(X_K;\mathbb{Z}[\mathbb{Z}]),\mathbb{Q}(\mathbb{Z})/\mathbb{Q}[\mathbb{Z}]), \end{aligned}$ $\displaystyle \begin{aligned} &H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(X_K,\partial X_K;\mathbb{Z}[\mathbb{Z}]) \to H^2(X_K;\mathbb{Z}[\mathbb{Z}]) \\ & \to H^1(X_K;\mathbb{Q}(\mathbb{Z})/\mathbb{Q}[\mathbb{Z}]) \to \Hom_{\mathbb{Z}[\mathbb{Z}]}(H_1(X_K;\mathbb{Z}[\mathbb{Z}]),\mathbb{Q}(\mathbb{Z})/\mathbb{Q}[\mathbb{Z}]), \end{aligned}$

which arise from the long exact sequence of a pair, equivariant Poincaré-Lefschetz duality, a Bockstein homomorphism, and universal coefficients. Showing that these maps are isomorphisms proves that $\mathop{\mathrm{Bl}}$ is non-singular. A good exercise is to trace through this sequence of isomorphisms to check that it really does coincide with the definition of the Blanchfield form given above.

A formula for the Blanchfield pairing of a knot in terms of a Seifert matrix for it is given in [Kearton1975]. The algebraic relationship between Seifert matrices and the Blanchfield pairing was studied in [Trotter1973],[Trotter1978] and [Ranicki2003]. In particular, every Blanchfield form is determined by a Seifert matrix, and two Seifert matrices are $S$ -equivalent (resp. cobordant) if and only if they determine isomorphic (resp. cobordant) Blanchfield forms.

[edit] 4 Applications to knot concordance

One of the Blanchfield form's main applications is in knot concordance, a notion first defined in [Fox&Milnor1966]. Two knots $K,J$ $K,J$ are concordant if there is an annulus embedded in $S^3 \times I$ $S^3 \times I$ whose boundary is $K \times \{0\} \cup -J \times \{1\}$ $K \times \{0\} \cup -J \times \{1\}$ . A knot which is concordant to the unknot is called a slice knot; equivalently a slice knot bounds an embedded disk in $D^4$ $D^4$ . We say that a Blanchfield form is metabolic if there is a submodule $P \subset H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ $P \subset H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ which is self-orthogonal with respect to $\mathop{\mathrm{Bl}}$ $\mathop{\mathrm{Bl}}$ , called a metaboliser. The Blanchfield form of a slice knot is metabolic, so that the Blanchfield form provides an obstruction to concordance [Kearton1975], which is equivalent to Levine's Seifert form obstruction [Levine1969a], but which is more intrinsic, since for a given knot there are potentially many Seifert surfaces but only one knot exterior. The proof that the Blanchfield form of a slice knot is metabolic rests on the observation that, if $A \subset D^4$ $A \subset D^4$ is a slice disk for $K$ $K$ , the Blanchfield form vanishes on the kernel of the inclusion induced map

$\displaystyle H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(D^4\setminus \nu A;\mathbb{Z}[\mathbb{Z}]).$ $\displaystyle H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(D^4\setminus \nu A;\mathbb{Z}[\mathbb{Z}]).$

For high-dimensional knots, $K \colon S^{2k-1} \subset S^{2k+1}$ , where $k \geq 3$ , the Blanchfield form is metabolic if and only if $K$ is slice [Levine1969a], [Kearton1975]. Levine [Levine1977] classified the modules which can arise as the homology of high-dimensional knots: the key property that a knot module must satisfy is Blanchfield duality. For a comprehensive account of the algebraic theory of high-dimensional knots, such as how the Blanchfield form can be used to compute the high-dimensional knot cobordism group, see [Ranicki1998].

For classical knots in the $3$ $3$ -sphere, there are many non-slice knots with metabolic Blanchfield form, the first of which were found in [Casson&Gordon1986]. Cochran, Orr and Teichner [Cochran&Orr&Teichner2003] defined an infinite filtration of the knot concordance group, each of whose associated graded groups has infinite rank [Cochran&Orr&Teichner2004], [Cochran&Teichner2007], [Cochran&Harvey&Leidy2009]. Their obstructions are obtained by defining representations into progressively more solvable groups. The Blanchfield form, and so-called higher order Blanchfield forms, play a crucial role in controlling the representations which extend from the knot exterior across a potential slice disc exterior

Tex syntax error

$W$ , whose existence one wishes to deny. Let $M_K$ $M_K$ be the closed 3-manifold obtained from zero-framed surgery on $S^3$ $S^3$ along $K$ $K$ . Note that $\partial W = M_K$ $\partial W = M_K$ . Then the kernel

$\displaystyle P:= \ker(H_1(M_K;\mathbb{Q}[\mathbb{Z}]) \to H_1(W;\mathbb{Q}[\mathbb{Z}]))$ $\displaystyle P:= \ker(H_1(M_K;\mathbb{Q}[\mathbb{Z}]) \to H_1(W;\mathbb{Q}[\mathbb{Z}]))$

is a metaboliser for the rational Blanchfield form of $M_K$ $M_K$ . Therefore, given $p \in P$ $p \in P$ , the representation

$\displaystyle H_1(M_K;\mathbb{Q}[\mathbb{Z}]) \to \mathbb{Q}(\mathbb{Z})/\mathbb{Q}[\mathbb{Z}];\; x \mapsto \mathop{\mathrm{Bl}}(p,x)$ $\displaystyle H_1(M_K;\mathbb{Q}[\mathbb{Z}]) \to \mathbb{Q}(\mathbb{Z})/\mathbb{Q}[\mathbb{Z}];\; x \mapsto \mathop{\mathrm{Bl}}(p,x)$

extends over

Tex syntax error

$W$ .

We give a special case of the results of [Cochran&Orr&Teichner2003] below, which shows the use of the Blanchfield form in an archetypal obstruction theorem for knot concordance problems. Given a closed 3-manifold $M$ and a representation $\phi \colon \pi_1(M) \to G$ there is defined a real number $\rho^{(2)}(M,\phi)$ called the Cheeger-Gromov-Von Neumann $\rho$ -invariant of $(M,\phi)$ , which can be computed using $L^{(2)}$ -signatures of 4-manifolds with boundary $M_K$ .

Let $K$ be a slice knot. Then there exists a metaboliser $P = P^{\bot}$ for the rational Blanchfield form of $K$ such that for each $p \in P$ there is a representation $\phi_p \colon \pi_1(M_K) \to G$ for which $\rho^{(2)}(M_K,\phi_p)=0$ .

Finally, it is somewhat remarkable that the classical Blanchfield form continues to have new and interesting applications. For example, Borodzik and Friedl [Borodzik&Friedl2012] recently used the minimal size of a square matrix which presents the Blanchfield form of a given knot to compute many previously unknown unknotting numbers of low crossing number knots. See the linking form page, for the definition of a presentation of a linking form.

[edit] 5 References

[Blanchfield1957] R. C. Blanchfield, Intersection theory of manifolds with operators with applications to knot theory, Ann. of Math. (2) 65 (1957), 340–356. MR0085512 (19,53a) Zbl 0080.16601
[Borodzik&Friedl2012] M. Borodzik and S. Friedl, The unknotting number and classical invariants I, (2012). Available at the arXiv:1203.3225.
[Casson&Gordon1986] A. J. Casson and C. M. Gordon, Cobordism of classical knots, with an appendix by P. M. Gilmer. Progr. Math., 62, À la recherche de la topologie perdue, 181–199, Birkhäuser Boston, Boston, MA, 1986. MR900252
[Cochran&Harvey&Leidy2009] T. D. Cochran, S. Harvey and C. Leidy, Knot concordance and higher-order Blanchfield duality, Geom. Topol. 13 (2009), no.3, 1419–1482. MR2496049 (2009m:57006) Zbl 1175.57004
[Cochran&Orr&Teichner2003] T. D. Cochran, K. E. Orr and P. Teichner, Knot concordance, Whitney towers and $L^2$ -signatures, Ann. of Math. (2) 157 (2003), no.2, 433–519. MR1973052 (2004i:57003) Zbl 1044.57001
[Cochran&Orr&Teichner2004] T. D. Cochran, K. E. Orr and P. Teichner, Structure in the classical knot concordance group, Comment. Math. Helv. 79 (2004), no.1, 105–123. MR2031301 (2004k:57005) Zbl 1061.57008
[Cochran&Teichner2007] T. D. Cochran and P. Teichner, Knot concordance and von Neumann $\rho$ -invariants, Duke Math. J. 137 (2007), no.2, 337–379. MR2309149 (2008f:57005) Zbl 1186.57006
[Fox&Milnor1966] R. H. Fox and J. W. Milnor, Singularities of $2$ -spheres in $4$ -space and cobordism of knots, Osaka J. Math. 3 (1966), 257–267. MR0211392 (35 #2273) Zbl 0146.45501
[Kearton1975] C. Kearton, Cobordism of knots and Blanchfield duality, J. London Math. Soc. (2) 10 (1975), no.4, 406–408. MR0385873 (52 #6732) Zbl 0305.57015
[Levine1969a] J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244. MR0246314 (39 #7618) Zbl 0176.22101
[Levine1977] J. Levine, Knot modules. I, Trans. Amer. Math. Soc. 229 (1977), 1–50. MR0461518 (57 #1503) Zbl 0422.57007
[Ranicki1998] A. Ranicki, High-dimensional knot theory, Springer-Verlag, 1998. MR1713074 (2000i:57044) Zbl 1059.19003
[Ranicki2003] A. Ranicki, Blanchfield and Seifert algebra in high-dimensional knot theory, Mosc. Math. J. 3 (2003), no.4, 1333–1367. MR2058802 (2005b:19008) Zbl 1059.19003
[Reidemeister1939] K. Reidemeister, Durchschnitt und Schnitt von Homotopieketten, Monatsh. Math. Phys. 48 (1939), 226–239. MR0000634 (1,105h) Zbl 0021.43104
[Trotter1973] H. F. Trotter, On $S$ -equivalence of Seifert matrices, Invent. Math. 20 (1973), 173–207. MR0645546 (58 #31100) Zbl 0269.15009
[Trotter1978] H. F. Trotter, Knot module and Seifert matrices, Knot theory (Proc. Sem., Plans-sur-Bex, 1977), Springer (1978), 291–299. MR521739 (81c:57019) Zbl 407.57015

$--chains $x$ and $y$ for $\theta$. Let $x$ be the straight line between north and south poles and let $y$ be half of the equator. Now y = \partial w$, where $w \in C_2(\mathbb{RP}^3;\mathbb{Z})$ is the 2--disk whose boundary is the equator. We see that $\langle x,w \rangle = 1$, so that $$L_{\mathbb{RP}^3}(\theta,\theta) = L_{\mathbb{RP}^3}([x],[y]) = \langle x,y \rangle/2 = 1/2.$$ In 1939 Reidemeister {{cite|Reidemeister1939}} defined an equivariant, sesquilinear intersection form on the homology of a covering space $\widetilde{M}$ of an $m$-dimensional closed manifold $M$ whose deck transformation group $G$ is abelian. $$\begin{array}{rcl} I_{\widetilde{M}} \colon H_k(\widetilde{M};\mathbb{Z}) \times H_{m-k}(\widetilde{M};\mathbb{Z}) &\to & \mathbb{Z}[G]; \ ([p],[q]) & \mapsto & \sum_{g \in G} \langle g \cdot p, q \rangle g^{-1}. \end{array}$$ The intersections of each possible $G$-translate of $p$ and $q$ are counted, and indexed according to the deck transformation which produced that intersection number. In his 1954 Princeton PhD thesis R.~C.~Blanchfield {{cite|Blanchfield1957}} made the corresponding generalisation for linking forms. Let $X^{m}$ be a compact manifold, now possibly with non--empty boundary, with a surjective homomorphism $\pi_1(X) \to \Gamma$, for some free abelian group $\Gamma$. Let $\mathbb{Z}[\Gamma]$ be the group ring of $\Gamma$ and let $\Q(\Gamma)$ be its field of fractions. The $\mathbb{Z}[\Gamma]$-torsion submodule of a $\mathbb{Z}[\Gamma]$ module $P$ is the set $$TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}[\Gamma]\}.$$ The Blanchfield form is a sesquilinear $\Q(\Gamma)/\mathbb{Z}[\Gamma]$--valued form which is defined on the $\mathbb{Z}[\Gamma]$-torsion submodules of the homology of the $\Gamma$--cover $\widetilde{X}$ of $X$: $$\mathop{\mathrm{Bl}} \colon TH_\ell(\widetilde{X};\mathbb{Z}) \times TH_{m-\ell-1}(\widetilde{X},\partial\widetilde{X};\mathbb{Z}) \to \Q(\Gamma)/\mathbb{Z}[\Gamma].$$ Given $$[x] \in TH_\ell(\widetilde{X};\mathbb{Z}) \cong TH_\ell(X;\mathbb{Z}[\Gamma])$$ and $$[y] \in TH_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z}) \cong TH_{m-\ell-1}(X,\partial X;\mathbb{Z}[\Gamma])$$ represented by cycles $x \in C_\ell(\widetilde{X};\mathbb{Z})$ and $y \in C_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z})$, let $w \in C_{m-\ell}(\widetilde{X},\partial\widetilde{X};\mathbb{Z})$ be such that $\partial w = \Delta y$, for some $\Delta \in \mathbb{Z}[\Gamma]$. Then we define: $$\mathop{\mathrm{Bl}}([x],[y]) := \sum_{g \in \Gamma} \langle g \cdot x, w \rangle g^{-1}/\Delta \in \Q(\Gamma)/\mathbb{Z}[\Gamma],$$ where $\Gamma$ acts on $C_\ell(\widetilde{X};\mathbb{Z})$ by the action induced from the deck transformation. We now turn to an example. For simplicity we will focus on the case of knots in $S^3$. For a knot $K \subset S^3$, let $X_K$ denote its exterior, which is the complement of a regular neighbourhood of $K$: $X_K:= S^3 \setminus \nu K$. Now $\ell=1$, $m=3$ and the abelianisation gives a homomorphism $\pi_1(X_K) \to H_1(X_K;\mathbb{Z}) \xrightarrow{\simeq} \mathbb{Z}$. The Blanchfield form can in this case be defined without relative homology, on $H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \cong H_1(\widetilde{X}_K;\mathbb{Z})$. The form $$\mathop{\mathrm{Bl}} \colon H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \times H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to \Q(\mathbb{Z})/\mathbb{Z}[\mathbb{Z}]$$ is non--singular, sesquilinear and Hermitian. Note that $H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ is entirely $\mathbb{Z}[\mathbb{Z}]$--torsion, so $H_1(X_K;\mathbb{Z}[\mathbb{Z}]) = TH_1(X_K;\mathbb{Z}[\mathbb{Z}])$. The adjoint of this form is given by the following sequence of homomorphisms: $$\begin{aligned} &H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(X_K,\partial X_K;\mathbb{Z}[\mathbb{Z}]) \to H^2(X_K;\mathbb{Z}[\mathbb{Z}]) \ & \to H^1(X_K;\Q(\mathbb{Z})/\Q[\mathbb{Z}]) \to \Hom_{\mathbb{Z}[\mathbb{Z}]}(H_1(X_K;\mathbb{Z}[\mathbb{Z}]),\Q(\mathbb{Z})/\Q[\mathbb{Z}]), \end{aligned} $$ which arise from the long exact sequence of a pair, equivariant Poincar\'{e}--Lefschetz duality, a Bockstein homomorphism, and universal coefficients. Showing that these maps are isomorphisms proves that \mathop{\mathrm{Bl}}$ is non--singular. A good exercise is to trace through this sequence of isomorphisms to check that it really does coincide with the definition of the Blanchfield form given above. One of the Blanchfield form's main applications is in knot concordance, a notion first defined in \cite{Fox&Milnor1959}. Two knots $K,J$ are concordant if there is an annulus embedded in $S^3 \times I$ whose boundary is $K \times \{0\} \cup -J \times \{1\}$. A knot which is concordant to the unknot is called a slice knot; equivalently a slice knot bounds an embedded disk in $D^4$. We say that a Blanchfield form is metabolic if there is a submodule $P \subset H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ which is self--orthogonal with respect to $\mathop{\mathrm{Bl}}$, called a metaboliser. The Blanchfield form of a slice knot is metabolic, so that the Blanchfield form provides an obstruction to concordance \cite{Kearton}, which is equivalent to Levine's Seifert form obstruction \cite{Levine}, but which is more intrinsic, since for a given knot there are many Seifert surfaces but only one knot exterior. The proof that Blanchfield form of a slice knot is metabolic rests on the observation that, if $A \subset D^4$ is a slice disk for $K$, the Blanchfield form vanishes on the kernel of the map $$H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(D^4\setminus \nu A;\mathbb{Z}[\mathbb{Z}]).$$ For high--dimensional knots, $K \colon S^{2k-1} \subset S^{2k+1}$, where $k \geq 3$, the Blanchfield form is metabolic if and only if $K$ is slice \cite{Levine, Kearton}. Levine \cite{Levine2} classified the modules which can arise as the homology of high--dimensional knots: the key property that a knot module must satisfy is Blanchfield duality. For a comprehensive account of the algebraic theory of high--dimensional knots, such as how the Blanchfield form can be used to compute the high--dimensional knot cobordism group, see \cite{RanickiHighDimKnotTheory}. For classical knots in the $--sphere, there are many non--slice knots with metabolic Blanchfield form, the first of which were found in \cite{CassonGordon}. Cochran, Orr and Teichner \cite{COT} defined an infinite filtration of the knot concordance group, each of whose associated graded groups has infinite rank \cite{COT2, Cochran&Teichner2007, Cochran&Harvey&Leidy2009}. Their obstructions are obtained by defining representations into progressively more solvable groups. The Blanchfield form, and so--called higher order Blanchfield forms, play a crucial r\^{o}le in controlling the representations which extend from the knot exterior across a potential slice disc exterior $W$, whose existence one wishes to deny. Let $M_K$ be the closed 3--manifold obtained from zero--framed surgery on $S^3$ along $K$. Then the kernel \[P:= \ker(H_1(M_K;\Q[\mathbb{Z}]) \to H_1(W;\Q[\mathbb{Z}]))\] is a metaboliser for the rational Blanchfield form of $M_K$. We give a special case of the results of \cite{Cochran&Orr&Teichner2003} below, which shows the use of the Blanchfield form in an archetypal obstruction theorem for knot concordance problems. Given a closed 3--manifold $M$ and a representation $\phi \colon \pi_1(M) \to G$ there is defined a real number $\rho^{(2)}(M,\phi)$ called the Cheeger--Gromov--Von--Neumann $\rho$--invariant of $(M,\phi)$. {{beginthm |Theorem |Cochran-Orr-Teichner}} Let $K$ be a slice knot. Then there exists a metaboliser $P = P^{\bot}$ for the Blanchfield form of $K$ such that for each $p \in P$ there is a representation $\phi_p \colon \pi_1(M_K) \to G$ for which $\rho^{(2)}(M_K,\phi_p)=0$. {{endthm}} It is somewhat remarkable that the classical Blanchfield form continues to have new and interesting applications. For example, Borodzik and Friedl \cite{BorodzikFriedl12I, BorodzikFriedl12II} recently used the minimal size of a matrix which represents the Blanchfield form of a given knot to compute many previously unknown unknotting numbers of low crossing number knots. == References == {{#RefList:}} [[Category:Theory]]\widetilde{M} of an $m$ $m$ -dimensional closed manifold $M$ $M$ whose deck transformation group $G$ $G$ is abelian.

$\displaystyle \begin{array}{rcl} I_{\widetilde{M}} \colon H_k(\widetilde{M};\mathbb{Z}) \times H_{m-k}(\widetilde{M};\mathbb{Z}) &\to & \mathbb{Z}[G]; \\ ([p],[q]) & \mapsto & \sum_{g \in G} \langle g \cdot p, q \rangle g^{-1}, \end{array}$ $\displaystyle \begin{array}{rcl} I_{\widetilde{M}} \colon H_k(\widetilde{M};\mathbb{Z}) \times H_{m-k}(\widetilde{M};\mathbb{Z}) &\to & \mathbb{Z}[G]; \\ ([p],[q]) & \mapsto & \sum_{g \in G} \langle g \cdot p, q \rangle g^{-1}, \end{array}$

where

$\displaystyle \langle \cdot , \cdot \rangle \colon C_k(\widetilde{M};\mathbb{Z}) \times C_{m-k}(\widetilde{M};\mathbb{Z}) \to \mathbb{Z}$ $\displaystyle \langle \cdot , \cdot \rangle \colon C_k(\widetilde{M};\mathbb{Z}) \times C_{m-k}(\widetilde{M};\mathbb{Z}) \to \mathbb{Z}$

counts transverse intersections between chains algebraically.

The intersections of each possible $G$ -translate of $p$ and $q$ are counted, and indexed according to the deck transformation which produced that intersection number.

[edit] 2 Definition of the Blanchfield form

In his 1954 Princeton PhD thesis R. C. Blanchfield [Blanchfield1957] made the corresponding generalisation for linking forms. Let $X^{m}$ be a compact manifold, now possibly with non-empty boundary, with a surjective homomorphism $\pi_1(X) \to \Gamma$ , for some free abelian group $\Gamma$ . Let $\mathbb{Z}[\Gamma]$ be the group ring of $\Gamma$ and let $\mathbb{Q}(\Gamma)$ be its field of fractions.

The $\mathbb{Z}[\Gamma]$ $\mathbb{Z}[\Gamma]$ -torsion submodule of a $\mathbb{Z}[\Gamma]$ $\mathbb{Z}[\Gamma]$ module $P$ $P$ is the submodule

$\displaystyle TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}[\Gamma]\}.$ $\displaystyle TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}[\Gamma]\}.$

The Blanchfield form is a sesquilinear $\mathbb{Q}(\Gamma)/\mathbb{Z}[\Gamma]$ --valued pairing which is defined on the $\mathbb{Z}[\Gamma]$ -torsion submodules of the homology of the $\Gamma$ --cover $\widetilde{X}$ of $X$ :

$\displaystyle \mathop{\mathrm{Bl}} \colon TH_\ell(\widetilde{X};\mathbb{Z}) \times TH_{m-\ell-1}(\widetilde{X},\partial\widetilde{X};\mathbb{Z}) \to \mathbb{Q}(\Gamma)/\mathbb{Z}[\Gamma].$ $\displaystyle \mathop{\mathrm{Bl}} \colon TH_\ell(\widetilde{X};\mathbb{Z}) \times TH_{m-\ell-1}(\widetilde{X},\partial\widetilde{X};\mathbb{Z}) \to \mathbb{Q}(\Gamma)/\mathbb{Z}[\Gamma].$

Given

$\displaystyle [x] \in TH_\ell(\widetilde{X};\mathbb{Z}) \cong TH_\ell(X;\mathbb{Z}[\Gamma])$ $\displaystyle [x] \in TH_\ell(\widetilde{X};\mathbb{Z}) \cong TH_\ell(X;\mathbb{Z}[\Gamma])$

and

$\displaystyle [y] \in TH_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z}) \cong TH_{m-\ell-1}(X,\partial X;\mathbb{Z}[\Gamma])$ $\displaystyle [y] \in TH_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z}) \cong TH_{m-\ell-1}(X,\partial X;\mathbb{Z}[\Gamma])$

represented by cycles $x \in C_\ell(\widetilde{X};\mathbb{Z})$ $x \in C_\ell(\widetilde{X};\mathbb{Z})$ and $y \in C_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z})$ $y \in C_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z})$ , let $w \in C_{m-\ell}(\widetilde{X},\partial\widetilde{X};\mathbb{Z})$ $w \in C_{m-\ell}(\widetilde{X},\partial\widetilde{X};\mathbb{Z})$ be such that $\partial w = \Delta y$ $\partial w = \Delta y$ , for some $\Delta \in \mathbb{Z}[\Gamma]$ $\Delta \in \mathbb{Z}[\Gamma]$ . Then we define:

$\displaystyle \mathop{\mathrm{Bl}}([x],[y]) := \sum_{g \in \Gamma} \langle g \cdot x, w \rangle g^{-1}/\Delta \in \mathbb{Q}(\Gamma)/\mathbb{Z}[\Gamma],$ $\displaystyle \mathop{\mathrm{Bl}}([x],[y]) := \sum_{g \in \Gamma} \langle g \cdot x, w \rangle g^{-1}/\Delta \in \mathbb{Q}(\Gamma)/\mathbb{Z}[\Gamma],$

where $\Gamma$ acts on $C_\ell(\widetilde{X};\mathbb{Z})$ by the action induced from the deck transformation.

The Blanchfield form can also be defined via homology, using a Bockstein homomorphism and a cup product. This is entirely analogous to the treatment the situation with the linking form. Just change $\mathbb{Z}$ to $\mathbb{Z}[\mathbb{Z}]$ and $\mathbb{Q}$ to $\mathbb{Q}(\mathbb{Z})$ . In the case of knot exteriors, a homological definition is also given below.

[edit] 3 Example of knot exteriors

We now turn to an example. We will focus on the case of knots in $S^3$ . For a knot $K \subset S^3$ , let $X_K$ denote its exterior, which is the complement of a regular neighbourhood of $K$ : $X_K:= S^3 \setminus \nu K$ . Now $\ell=1$ , $m=3$ and the abelianisation gives a homomorphism $\pi_1(X_K) \to H_1(X_K;\mathbb{Z}) \xrightarrow{\simeq} \mathbb{Z}$ . The Blanchfield form can in this case be defined without relative homology, on $H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \cong H_1(\widetilde{X}_K;\mathbb{Z})$ . The form

$\displaystyle \mathop{\mathrm{Bl}} \colon H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \times H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to \mathbb{Q}(\mathbb{Z})/\mathbb{Z}[\mathbb{Z}]$ $\displaystyle \mathop{\mathrm{Bl}} \colon H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \times H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to \mathbb{Q}(\mathbb{Z})/\mathbb{Z}[\mathbb{Z}]$

is non--singular, sesquilinear and Hermitian. Note that $H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ is entirely $\mathbb{Z}[\mathbb{Z}]$ -torsion, so $H_1(X_K;\mathbb{Z}[\mathbb{Z}]) = TH_1(X_K;\mathbb{Z}[\mathbb{Z}])$ . The adjoint of this form is given by the following sequence of homomorphisms:

$\displaystyle \begin{aligned} &H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(X_K,\partial X_K;\mathbb{Z}[\mathbb{Z}]) \to H^2(X_K;\mathbb{Z}[\mathbb{Z}]) \\ & \to H^1(X_K;\mathbb{Q}(\mathbb{Z})/\mathbb{Q}[\mathbb{Z}]) \to \Hom_{\mathbb{Z}[\mathbb{Z}]}(H_1(X_K;\mathbb{Z}[\mathbb{Z}]),\mathbb{Q}(\mathbb{Z})/\mathbb{Q}[\mathbb{Z}]), \end{aligned}$ $\displaystyle \begin{aligned} &H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(X_K,\partial X_K;\mathbb{Z}[\mathbb{Z}]) \to H^2(X_K;\mathbb{Z}[\mathbb{Z}]) \\ & \to H^1(X_K;\mathbb{Q}(\mathbb{Z})/\mathbb{Q}[\mathbb{Z}]) \to \Hom_{\mathbb{Z}[\mathbb{Z}]}(H_1(X_K;\mathbb{Z}[\mathbb{Z}]),\mathbb{Q}(\mathbb{Z})/\mathbb{Q}[\mathbb{Z}]), \end{aligned}$

which arise from the long exact sequence of a pair, equivariant Poincaré-Lefschetz duality, a Bockstein homomorphism, and universal coefficients. Showing that these maps are isomorphisms proves that $\mathop{\mathrm{Bl}}$ is non-singular. A good exercise is to trace through this sequence of isomorphisms to check that it really does coincide with the definition of the Blanchfield form given above.

A formula for the Blanchfield pairing of a knot in terms of a Seifert matrix for it is given in [Kearton1975]. The algebraic relationship between Seifert matrices and the Blanchfield pairing was studied in [Trotter1973],[Trotter1978] and [Ranicki2003]. In particular, every Blanchfield form is determined by a Seifert matrix, and two Seifert matrices are $S$ -equivalent (resp. cobordant) if and only if they determine isomorphic (resp. cobordant) Blanchfield forms.

[edit] 4 Applications to knot concordance

One of the Blanchfield form's main applications is in knot concordance, a notion first defined in [Fox&Milnor1966]. Two knots $K,J$ $K,J$ are concordant if there is an annulus embedded in $S^3 \times I$ $S^3 \times I$ whose boundary is $K \times \{0\} \cup -J \times \{1\}$ $K \times \{0\} \cup -J \times \{1\}$ . A knot which is concordant to the unknot is called a slice knot; equivalently a slice knot bounds an embedded disk in $D^4$ $D^4$ . We say that a Blanchfield form is metabolic if there is a submodule $P \subset H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ $P \subset H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ which is self-orthogonal with respect to $\mathop{\mathrm{Bl}}$ $\mathop{\mathrm{Bl}}$ , called a metaboliser. The Blanchfield form of a slice knot is metabolic, so that the Blanchfield form provides an obstruction to concordance [Kearton1975], which is equivalent to Levine's Seifert form obstruction [Levine1969a], but which is more intrinsic, since for a given knot there are potentially many Seifert surfaces but only one knot exterior. The proof that the Blanchfield form of a slice knot is metabolic rests on the observation that, if $A \subset D^4$ $A \subset D^4$ is a slice disk for $K$ $K$ , the Blanchfield form vanishes on the kernel of the inclusion induced map

$\displaystyle H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(D^4\setminus \nu A;\mathbb{Z}[\mathbb{Z}]).$ $\displaystyle H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(D^4\setminus \nu A;\mathbb{Z}[\mathbb{Z}]).$

For high-dimensional knots, $K \colon S^{2k-1} \subset S^{2k+1}$ , where $k \geq 3$ , the Blanchfield form is metabolic if and only if $K$ is slice [Levine1969a], [Kearton1975]. Levine [Levine1977] classified the modules which can arise as the homology of high-dimensional knots: the key property that a knot module must satisfy is Blanchfield duality. For a comprehensive account of the algebraic theory of high-dimensional knots, such as how the Blanchfield form can be used to compute the high-dimensional knot cobordism group, see [Ranicki1998].

For classical knots in the $3$ $3$ -sphere, there are many non-slice knots with metabolic Blanchfield form, the first of which were found in [Casson&Gordon1986]. Cochran, Orr and Teichner [Cochran&Orr&Teichner2003] defined an infinite filtration of the knot concordance group, each of whose associated graded groups has infinite rank [Cochran&Orr&Teichner2004], [Cochran&Teichner2007], [Cochran&Harvey&Leidy2009]. Their obstructions are obtained by defining representations into progressively more solvable groups. The Blanchfield form, and so-called higher order Blanchfield forms, play a crucial role in controlling the representations which extend from the knot exterior across a potential slice disc exterior

Tex syntax error

$W$ , whose existence one wishes to deny. Let $M_K$ $M_K$ be the closed 3-manifold obtained from zero-framed surgery on $S^3$ $S^3$ along $K$ $K$ . Note that $\partial W = M_K$ $\partial W = M_K$ . Then the kernel

$\displaystyle P:= \ker(H_1(M_K;\mathbb{Q}[\mathbb{Z}]) \to H_1(W;\mathbb{Q}[\mathbb{Z}]))$ $\displaystyle P:= \ker(H_1(M_K;\mathbb{Q}[\mathbb{Z}]) \to H_1(W;\mathbb{Q}[\mathbb{Z}]))$

is a metaboliser for the rational Blanchfield form of $M_K$ $M_K$ . Therefore, given $p \in P$ $p \in P$ , the representation

$\displaystyle H_1(M_K;\mathbb{Q}[\mathbb{Z}]) \to \mathbb{Q}(\mathbb{Z})/\mathbb{Q}[\mathbb{Z}];\; x \mapsto \mathop{\mathrm{Bl}}(p,x)$ $\displaystyle H_1(M_K;\mathbb{Q}[\mathbb{Z}]) \to \mathbb{Q}(\mathbb{Z})/\mathbb{Q}[\mathbb{Z}];\; x \mapsto \mathop{\mathrm{Bl}}(p,x)$

extends over

Tex syntax error

$W$ .

We give a special case of the results of [Cochran&Orr&Teichner2003] below, which shows the use of the Blanchfield form in an archetypal obstruction theorem for knot concordance problems. Given a closed 3-manifold $M$ and a representation $\phi \colon \pi_1(M) \to G$ there is defined a real number $\rho^{(2)}(M,\phi)$ called the Cheeger-Gromov-Von Neumann $\rho$ -invariant of $(M,\phi)$ , which can be computed using $L^{(2)}$ -signatures of 4-manifolds with boundary $M_K$ .

Let $K$ be a slice knot. Then there exists a metaboliser $P = P^{\bot}$ for the rational Blanchfield form of $K$ such that for each $p \in P$ there is a representation $\phi_p \colon \pi_1(M_K) \to G$ for which $\rho^{(2)}(M_K,\phi_p)=0$ .

Finally, it is somewhat remarkable that the classical Blanchfield form continues to have new and interesting applications. For example, Borodzik and Friedl [Borodzik&Friedl2012] recently used the minimal size of a square matrix which presents the Blanchfield form of a given knot to compute many previously unknown unknotting numbers of low crossing number knots. See the linking form page, for the definition of a presentation of a linking form.

[edit] 5 References

[Blanchfield1957] R. C. Blanchfield, Intersection theory of manifolds with operators with applications to knot theory, Ann. of Math. (2) 65 (1957), 340–356. MR0085512 (19,53a) Zbl 0080.16601
[Borodzik&Friedl2012] M. Borodzik and S. Friedl, The unknotting number and classical invariants I, (2012). Available at the arXiv:1203.3225.
[Casson&Gordon1986] A. J. Casson and C. M. Gordon, Cobordism of classical knots, with an appendix by P. M. Gilmer. Progr. Math., 62, À la recherche de la topologie perdue, 181–199, Birkhäuser Boston, Boston, MA, 1986. MR900252
[Cochran&Harvey&Leidy2009] T. D. Cochran, S. Harvey and C. Leidy, Knot concordance and higher-order Blanchfield duality, Geom. Topol. 13 (2009), no.3, 1419–1482. MR2496049 (2009m:57006) Zbl 1175.57004
[Cochran&Orr&Teichner2003] T. D. Cochran, K. E. Orr and P. Teichner, Knot concordance, Whitney towers and $L^2$ -signatures, Ann. of Math. (2) 157 (2003), no.2, 433–519. MR1973052 (2004i:57003) Zbl 1044.57001
[Cochran&Orr&Teichner2004] T. D. Cochran, K. E. Orr and P. Teichner, Structure in the classical knot concordance group, Comment. Math. Helv. 79 (2004), no.1, 105–123. MR2031301 (2004k:57005) Zbl 1061.57008
[Cochran&Teichner2007] T. D. Cochran and P. Teichner, Knot concordance and von Neumann $\rho$ -invariants, Duke Math. J. 137 (2007), no.2, 337–379. MR2309149 (2008f:57005) Zbl 1186.57006
[Fox&Milnor1966] R. H. Fox and J. W. Milnor, Singularities of $2$ -spheres in $4$ -space and cobordism of knots, Osaka J. Math. 3 (1966), 257–267. MR0211392 (35 #2273) Zbl 0146.45501
[Kearton1975] C. Kearton, Cobordism of knots and Blanchfield duality, J. London Math. Soc. (2) 10 (1975), no.4, 406–408. MR0385873 (52 #6732) Zbl 0305.57015
[Levine1969a] J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244. MR0246314 (39 #7618) Zbl 0176.22101
[Levine1977] J. Levine, Knot modules. I, Trans. Amer. Math. Soc. 229 (1977), 1–50. MR0461518 (57 #1503) Zbl 0422.57007
[Ranicki1998] A. Ranicki, High-dimensional knot theory, Springer-Verlag, 1998. MR1713074 (2000i:57044) Zbl 1059.19003
[Ranicki2003] A. Ranicki, Blanchfield and Seifert algebra in high-dimensional knot theory, Mosc. Math. J. 3 (2003), no.4, 1333–1367. MR2058802 (2005b:19008) Zbl 1059.19003
[Reidemeister1939] K. Reidemeister, Durchschnitt und Schnitt von Homotopieketten, Monatsh. Math. Phys. 48 (1939), 226–239. MR0000634 (1,105h) Zbl 0021.43104
[Trotter1973] H. F. Trotter, On $S$ -equivalence of Seifert matrices, Invent. Math. 20 (1973), 173–207. MR0645546 (58 #31100) Zbl 0269.15009
[Trotter1978] H. F. Trotter, Knot module and Seifert matrices, Knot theory (Proc. Sem., Plans-sur-Bex, 1977), Springer (1978), 291–299. MR521739 (81c:57019) Zbl 407.57015

Blanchfield form

Latest revision as of 18:07, 2 April 2013

Contents

[edit] 1 Background: equivariant intersection forms

[edit] 2 Definition of the Blanchfield form

[edit] 3 Example of knot exteriors

[edit] 4 Applications to knot concordance

[edit] 5 References

[edit] 2 Definition of the Blanchfield form

[edit] 3 Example of knot exteriors

[edit] 4 Applications to knot concordance

[edit] 5 References

Personal tools

Namespaces

Variants

Views

Actions

Search

Navigation

Interaction

Toolbox

@@ Line 1: / Line 1: @@
-<!-- COMMENT:
-To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:
-- For statements like Theorem, Lemma, Definition etc., use e.g.
-{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.
-- For references, use e.g. {{cite|Milnor1958b}}.
-END OF COMMENT
--->
 {{Stub}}
-== Introduction ==
+== Background: equivariant intersection forms==
 <wikitex>;
+In 1939 Reidemeister {{cite|Reidemeister1939}} defined an equivariant, sesquilinear intersection form on the homology of a covering space $\widetilde{M}$ of an $m$-dimensional closed manifold $M$ whose deck transformation group $G$ is abelian.
+$$\begin{array}{rcl} I_{\widetilde{M}} \colon H_k(\widetilde{M};\mathbb{Z}) \times H_{m-k}(\widetilde{M};\mathbb{Z}) &\to & \mathbb{Z}[G]; \\ ([p],[q]) & \mapsto &  \sum_{g \in G} \langle g \cdot p, q \rangle g^{-1}, \end{array}$$
+where $$\langle \cdot , \cdot \rangle \colon C_k(\widetilde{M};\mathbb{Z}) \times C_{m-k}(\widetilde{M};\mathbb{Z}) \to \mathbb{Z}$$ counts transverse intersections between chains algebraically.
-After Poincar\'{e} and Lefschetz, a closed oriented manifold $N^{n}$ has a bilinear intersection form defined on its homology:
-$$I_N \colon H_k(N;\mathbb{Z}) \times H_{n-k}(N;\mathbb{Z}) \to  \mathbb{Z}; ([p],[q]) \mapsto  \langle p, q \rangle$$
-such that
-$$I_N(x,y) = (-)^{k(n-k)}I_N(y,x).$$
-Given a ${k}$--chain $p \in C_{k}(N;\mathbb{Z})$ and an $(n-k)$--chain $q \in C_{n-k}(N;\mathbb{Z})$ which is transverse to $q$, the signed count of the intersections between $p$ and $q$ gives an intersection number $\langle\, p \, , \, q\, \rangle \in \mathbb{Z}$.
-By bilinearity, the intersection form vanishes on the torsion part of the homology.  The analogue of the intersection pairing for the torsion part of the homology of a closed manifold $N^n$ is the bilinear $\Q/\mathbb{Z}$--valued linking form, which is due to Seifert:
-$$L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \Q/\mathbb{Z}$$
-such that
-$$L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x).$$
-Given $[x] \in TH_\ell(N;\mathbb{Z})$ and $[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$ represented by cycles $x \in C_\ell(N;\mathbb{Z})$, let $w \in C_{\ell+1}(N;\mathbb{Z})$ be such that $\partial w = sy$, for some $s \in \mathbb{Z}$.  Then we define:
-$$L_N([x],[y]) := \langle x, w \rangle/s \in \Q/\mathbb{Z}.$$
-The resulting element is independent of the choices of $x,y,w$ and $s$.
-As an example, let $N = \mathbb{RP}^3$, so that $\ell=1$ and $n=3$.  Now $H_1(\mathbb{RP}^3;\mathbb{Z}) \cong \mathbb{Z}_2$.  Let $\theta \in H_1(\mathbb{RP}^3;\mathbb{Z})$ be the non--trivial element.  To compute the linking $L_{\mathbb{RP}^3}(\theta,\theta)$, consider $\mathbb{RP}^3$ modelled as $D^3/\sim$, with antipodal points on $\partial D^2$ identified, and choose two representative $1$--chains $x$ and $y$ for $\theta$.  Let $x$ be the straight line between north and south poles and let $y$ be half of the equator.  Now $2y = \partial w$, where $w \in C_2(\mathbb{RP}^3;\mathbb{Z})$ is the 2--disk whose boundary is the equator.  We see that $\langle x,w \rangle = 1$, so that
-$$L_{\mathbb{RP}^3}(\theta,\theta) = L_{\mathbb{RP}^3}([x],[y]) = \langle x,y \rangle/2 = 1/2.$$
-In 1939 Reidemeister {{cite|Reidemeister1939}} defined an equivariant, sesquilinear intersection form on the homology of a covering space $\widetilde{M}$ of an $m$-dimensional closed manifold $M$ whose deck transformation group $G$ is abelian.
-$$\begin{array}{rcl} I_{\widetilde{M}} \colon H_k(\widetilde{M};\mathbb{Z}) \times H_{m-k}(\widetilde{M};\mathbb{Z}) &\to & \mathbb{Z}[G]; \\ ([p],[q]) & \mapsto &  \sum_{g \in G} \langle g \cdot p, q \rangle g^{-1}. \end{array}$$
 The intersections of each possible $G$-translate of $p$ and $q$ are counted, and indexed according to the deck transformation which produced that intersection number.
+</wikitex>
-In his 1954 Princeton PhD thesis R.~C.~Blanchfield {{cite|Blanchfield1957}} made the corresponding generalisation for linking forms.  Let $X^{m}$ be a compact manifold, now possibly with non--empty boundary, with a surjective homomorphism $\pi_1(X) \to \Gamma$, for some free abelian group $\Gamma$.  Let $\mathbb{Z}[\Gamma]$ be the group ring of $\Gamma$ and let $\Q(\Gamma)$ be its field of fractions.
+==Definition of the Blanchfield form==
-The $\mathbb{Z}[\Gamma]$-torsion submodule of a $\mathbb{Z}[\Gamma]$ module $P$ is the set $$TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}[\Gamma]\}.$$
+<wikitex>;
-The Blanchfield form is a sesquilinear $\Q(\Gamma)/\mathbb{Z}[\Gamma]$--valued form which is defined on the $\mathbb{Z}[\Gamma]$-torsion submodules of the homology of the $\Gamma$--cover $\widetilde{X}$ of $X$:
+In his 1954 Princeton PhD thesis R. C. Blanchfield {{cite|Blanchfield1957}} made the corresponding generalisation for [[Linking form|linking forms]].  Let $X^{m}$ be a compact manifold, now possibly with non-empty boundary, with a surjective homomorphism $\pi_1(X) \to \Gamma$, for some free abelian group $\Gamma$.  Let $\mathbb{Z}[\Gamma]$ be the group ring of $\Gamma$ and let $\mathbb{Q}(\Gamma)$ be its field of fractions.
-$$\mathop{\mathrm{Bl}} \colon TH_\ell(\widetilde{X};\mathbb{Z}) \times TH_{m-\ell-1}(\widetilde{X},\partial\widetilde{X};\mathbb{Z}) \to \Q(\Gamma)/\mathbb{Z}[\Gamma].$$
+The $\mathbb{Z}[\Gamma]$-torsion submodule of a $\mathbb{Z}[\Gamma]$ module $P$ is the submodule $$TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}[\Gamma]\}.$$
+The Blanchfield form is a sesquilinear $\mathbb{Q}(\Gamma)/\mathbb{Z}[\Gamma]$--valued pairing which is defined on the $\mathbb{Z}[\Gamma]$-torsion submodules of the homology of the $\Gamma$--cover $\widetilde{X}$ of $X$:
+$$\mathop{\mathrm{Bl}} \colon TH_\ell(\widetilde{X};\mathbb{Z}) \times TH_{m-\ell-1}(\widetilde{X},\partial\widetilde{X};\mathbb{Z}) \to \mathbb{Q}(\Gamma)/\mathbb{Z}[\Gamma].$$
 Given $$[x] \in TH_\ell(\widetilde{X};\mathbb{Z}) \cong TH_\ell(X;\mathbb{Z}[\Gamma])$$ and $$[y] \in TH_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z}) \cong TH_{m-\ell-1}(X,\partial X;\mathbb{Z}[\Gamma])$$ represented by cycles $x \in C_\ell(\widetilde{X};\mathbb{Z})$ and $y \in C_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z})$, let $w \in C_{m-\ell}(\widetilde{X},\partial\widetilde{X};\mathbb{Z})$ be such that $\partial w = \Delta y$, for some $\Delta \in \mathbb{Z}[\Gamma]$.  Then we define:
-$$\mathop{\mathrm{Bl}}([x],[y]) := \sum_{g \in \Gamma} \langle g \cdot x, w \rangle g^{-1}/\Delta \in \Q(\Gamma)/\mathbb{Z}[\Gamma],$$
+$$\mathop{\mathrm{Bl}}([x],[y]) := \sum_{g \in \Gamma} \langle g \cdot x, w \rangle g^{-1}/\Delta \in \mathbb{Q}(\Gamma)/\mathbb{Z}[\Gamma],$$
 where $\Gamma$ acts on $C_\ell(\widetilde{X};\mathbb{Z})$ by the action induced from the deck transformation.
-We now turn to an example.  For simplicity we will focus on the case of knots in $S^3$.  For a knot $K \subset S^3$, let $X_K$ denote its exterior, which is the complement of a regular neighbourhood of $K$: $X_K:= S^3 \setminus \nu K$.  Now $\ell=1$, $m=3$ and the abelianisation gives a homomorphism $\pi_1(X_K) \to H_1(X_K;\mathbb{Z}) \xrightarrow{\simeq} \mathbb{Z}$.  The Blanchfield form can in this case be defined without relative homology, on $H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \cong H_1(\widetilde{X}_K;\mathbb{Z})$.  The form
+The Blanchfield form can also be defined via homology, using a Bockstein homomorphism and a cup product.  This is entirely analogous to the treatment the situation with the [[Linking form#Definition_via_cohomology|linking form]].  Just change $\mathbb{Z}$ to $\mathbb{Z}[\mathbb{Z}]$ and $\mathbb{Q}$ to $\mathbb{Q}(\mathbb{Z})$.  In the case of knot exteriors, a homological definition is also given below.
-$$\mathop{\mathrm{Bl}} \colon H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \times H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to \Q(\mathbb{Z})/\mathbb{Z}[\mathbb{Z}]$$
+</wikitex>
-is non--singular, sesquilinear and Hermitian.  Note that $H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ is entirely $\mathbb{Z}[\mathbb{Z}]$--torsion, so $H_1(X_K;\mathbb{Z}[\mathbb{Z}]) = TH_1(X_K;\mathbb{Z}[\mathbb{Z}])$.  The adjoint of this form is given by the following sequence of homomorphisms:
+==Example of knot exteriors==
-$$\begin{aligned} &H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(X_K,\partial X_K;\mathbb{Z}[\mathbb{Z}]) \to H^2(X_K;\mathbb{Z}[\mathbb{Z}]) \\ & \to H^1(X_K;\Q(\mathbb{Z})/\Q[\mathbb{Z}]) \to \Hom_{\mathbb{Z}[\mathbb{Z}]}(H_1(X_K;\mathbb{Z}[\mathbb{Z}]),\Q(\mathbb{Z})/\Q[\mathbb{Z}]),
+<wikitex>;
+We now turn to an example.  We will focus on the case of knots in $S^3$.  For a knot $K \subset S^3$, let $X_K$ denote its exterior, which is the complement of a regular neighbourhood of $K$: $X_K:= S^3 \setminus \nu K$.  Now $\ell=1$, $m=3$ and the abelianisation gives a homomorphism $\pi_1(X_K) \to H_1(X_K;\mathbb{Z}) \xrightarrow{\simeq} \mathbb{Z}$.  The Blanchfield form can in this case be defined without relative homology, on $H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \cong H_1(\widetilde{X}_K;\mathbb{Z})$.  The form
+$$\mathop{\mathrm{Bl}} \colon H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \times H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to \mathbb{Q}(\mathbb{Z})/\mathbb{Z}[\mathbb{Z}]$$
+is non--singular, sesquilinear and Hermitian.  Note that $H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ is entirely $\mathbb{Z}[\mathbb{Z}]$-torsion, so $H_1(X_K;\mathbb{Z}[\mathbb{Z}]) = TH_1(X_K;\mathbb{Z}[\mathbb{Z}])$.  The adjoint of this form is given by the following sequence of homomorphisms:
+$$\begin{aligned} &H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(X_K,\partial X_K;\mathbb{Z}[\mathbb{Z}]) \to H^2(X_K;\mathbb{Z}[\mathbb{Z}]) \\ & \to H^1(X_K;\mathbb{Q}(\mathbb{Z})/\mathbb{Q}[\mathbb{Z}]) \to \Hom_{\mathbb{Z}[\mathbb{Z}]}(H_1(X_K;\mathbb{Z}[\mathbb{Z}]),\mathbb{Q}(\mathbb{Z})/\mathbb{Q}[\mathbb{Z}]),
 \end{aligned}
 $$
-which arise from the long exact sequence of a pair, equivariant Poincar\'{e}--Lefschetz duality, a Bockstein homomorphism, and universal coefficients.  Showing that these maps are isomorphisms proves that \mathop{\mathrm{Bl}}$ is non--singular.  A good exercise is to trace through this sequence of isomorphisms to check that it really does coincide with the definition of the Blanchfield form given above.
+which arise from the long exact sequence of a pair, equivariant Poincaré-Lefschetz duality, a Bockstein homomorphism, and universal coefficients.  Showing that these maps are isomorphisms proves that $\mathop{\mathrm{Bl}}$ is non-singular.  A good exercise is to trace through this sequence of isomorphisms to check that it really does coincide with the definition of the Blanchfield form given above.
-One of the Blanchfield form's main applications is in knot concordance, a notion first defined in \cite{Fox&Milnor1959}.  Two knots $K,J$ are concordant if there is an annulus embedded in $S^3 \times I$ whose boundary is $K \times \{0\} \cup -J \times \{1\}$.  A knot which is concordant to the unknot is called a slice knot; equivalently a slice knot bounds an embedded disk in $D^4$.  We say that a Blanchfield form is metabolic if there is a submodule $P \subset H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ which is self--orthogonal with respect to $\mathop{\mathrm{Bl}}$, called a metaboliser.  The Blanchfield form of a slice knot is metabolic, so that the Blanchfield form provides an obstruction to concordance \cite{Kearton}, which is equivalent to Levine's Seifert form obstruction \cite{Levine}, but which is more intrinsic, since for a given knot there are many Seifert surfaces but only one knot exterior.  The proof that Blanchfield form of a slice knot is metabolic rests on the observation that, if $A \subset D^4$ is a slice disk for $K$, the Blanchfield form vanishes on the kernel of the map $$H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(D^4\setminus \nu A;\mathbb{Z}[\mathbb{Z}]).$$
+A formula for the Blanchfield pairing of a knot in terms of a Seifert matrix for it is given in \cite{Kearton1975}.  The algebraic relationship between Seifert matrices and the Blanchfield pairing was studied in \cite{Trotter1973},\cite{Trotter1978} and \cite{Ranicki2003}. In particular, every Blanchfield form is determined by a Seifert matrix, and two Seifert matrices are $S$-equivalent (resp. cobordant) if and only if they determine isomorphic (resp. cobordant) Blanchfield forms.
+</wikitex>
-For high--dimensional knots, $K \colon S^{2k-1} \subset S^{2k+1}$, where $k \geq 3$, the Blanchfield form is metabolic if and only if $K$ is slice  \cite{Levine, Kearton}.  Levine \cite{Levine2} classified the modules which can arise as the homology of high--dimensional knots: the key property that a knot module must satisfy is Blanchfield duality.  For a comprehensive account of the algebraic theory of high--dimensional knots, such as how the Blanchfield form can be used to compute the high--dimensional knot cobordism group, see \cite{RanickiHighDimKnotTheory}.
+==Applications to knot concordance==
+<wikitex>;
+One of the Blanchfield form's main applications is in knot concordance, a notion first defined in \cite{Fox&Milnor1966}.  Two knots $K,J$ are concordant if there is an annulus embedded in $S^3 \times I$ whose boundary is $K \times \{0\} \cup -J \times \{1\}$.  A knot which is concordant to the unknot is called a slice knot; equivalently a slice knot bounds an embedded disk in $D^4$.  We say that a Blanchfield form is metabolic if there is a submodule $P \subset H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ which is self-orthogonal with respect to $\mathop{\mathrm{Bl}}$, called a metaboliser.  The Blanchfield form of a slice knot is metabolic, so that the Blanchfield form provides an obstruction to concordance \cite{Kearton1975}, which is equivalent to Levine's Seifert form obstruction \cite{Levine1969a}, but which is more intrinsic, since for a given knot there are potentially many Seifert surfaces but only one knot exterior.  The proof that the Blanchfield form of a slice knot is metabolic rests on the observation that, if $A \subset D^4$ is a slice disk for $K$, the Blanchfield form vanishes on the kernel of the inclusion induced map $$H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(D^4\setminus \nu A;\mathbb{Z}[\mathbb{Z}]).$$
-For classical knots in the $3$--sphere, there are many non--slice knots with metabolic Blanchfield form, the first of which were found in \cite{CassonGordon}.  Cochran, Orr and Teichner \cite{COT} defined an infinite filtration of the knot concordance group, each of whose associated graded groups has infinite rank \cite{COT2, Cochran&Teichner2007, Cochran&Harvey&Leidy2009}.  Their obstructions are obtained by defining representations into progressively more solvable groups.  The Blanchfield form, and so--called higher order Blanchfield forms, play a crucial r\^{o}le in controlling the representations which extend from the knot exterior across a potential slice disc exterior $W$, whose existence one wishes to deny.  Let $M_K$ be the closed 3--manifold obtained from zero--framed surgery on $S^3$ along $K$.  Then the kernel
+For high-dimensional knots, $K \colon S^{2k-1} \subset S^{2k+1}$, where $k \geq 3$, the Blanchfield form is metabolic if and only if $K$ is slice  \cite{Levine1969a}, \cite{Kearton1975}.  Levine \cite{Levine1977} classified the modules which can arise as the homology of high-dimensional knots: the key property that a knot module must satisfy is Blanchfield duality.  For a comprehensive account of the algebraic theory of high-dimensional knots, such as how the Blanchfield form can be used to compute the high-dimensional knot cobordism group, see \cite{Ranicki1998}.
-\[P:= \ker(H_1(M_K;\Q[\mathbb{Z}]) \to H_1(W;\Q[\mathbb{Z}]))\]
-is a metaboliser for the rational Blanchfield form of $M_K$.
-We give a special case of the results of \cite{Cochran&Orr&Teichner2003} below, which shows the use of the Blanchfield form in an archetypal obstruction theorem for knot concordance problems.  Given a closed 3--manifold $M$ and a representation $\phi \colon \pi_1(M) \to G$ there is defined a real number $\rho^{(2)}(M,\phi)$ called the Cheeger--Gromov--Von--Neumann $\rho$--invariant of $(M,\phi)$.
-{{beginthm |Theorem |Cochran-Orr-Teichner}}
-Let $K$ be a slice knot.  Then there exists a metaboliser $P = P^{\bot}$ for the Blanchfield form of $K$ such that for each $p \in P$ there is a representation $\phi_p \colon \pi_1(M_K) \to G$ for which $\rho^{(2)}(M_K,\phi_p)=0$.
-{{endthm}}
-It is somewhat remarkable that the classical Blanchfield form continues to have new and interesting applications.  For example, Borodzik and Friedl \cite{BorodzikFriedl12I, BorodzikFriedl12II} recently used the minimal size of a matrix which represents the Blanchfield form of a given knot to compute many previously unknown unknotting numbers of low crossing number knots.
+For classical knots in the $3$-sphere, there are many non-slice knots with metabolic Blanchfield form, the first of which were found in \cite{Casson&Gordon1986}.  Cochran, Orr and Teichner \cite{Cochran&Orr&Teichner2003} defined an infinite filtration of the knot concordance group, each of whose associated graded groups has infinite rank \cite{Cochran&Orr&Teichner2004}, \cite{Cochran&Teichner2007}, \cite{Cochran&Harvey&Leidy2009}.  Their obstructions are obtained by defining representations into progressively more solvable groups.  The Blanchfield form, and so-called higher order Blanchfield forms, play a crucial role in controlling the representations which extend from the knot exterior across a potential slice disc exterior $W$, whose existence one wishes to deny.  Let $M_K$ be the closed 3-manifold obtained from zero-framed surgery on $S^3$ along $K$.  Note that $\partial W = M_K$.  Then the kernel
+$$P:= \ker(H_1(M_K;\mathbb{Q}[\mathbb{Z}]) \to H_1(W;\mathbb{Q}[\mathbb{Z}]))$$
+is a metaboliser for the rational Blanchfield form of $M_K$.  Therefore, given $p \in P$, the representation $$H_1(M_K;\mathbb{Q}[\mathbb{Z}]) \to \mathbb{Q}(\mathbb{Z})/\mathbb{Q}[\mathbb{Z}];\; x \mapsto \mathop{\mathrm{Bl}}(p,x)$$
+extends over $W$.
+We give a special case of the results of \cite{Cochran&Orr&Teichner2003} below, which shows the use of the Blanchfield form in an archetypal obstruction theorem for knot concordance problems.  Given a closed 3-manifold $M$ and a representation $\phi \colon \pi_1(M) \to G$ there is defined a real number $\rho^{(2)}(M,\phi)$ called the Cheeger-Gromov-Von Neumann $\rho$-invariant of $(M,\phi)$, which can be computed using $L^{(2)}$-signatures of 4-manifolds with boundary $M_K$.
+{{beginthm |Theorem |(Cochran-Orr-Teichner)}}
+Let $K$ be a slice knot.  Then there exists a metaboliser $P = P^{\bot}$ for the rational Blanchfield form of $K$ such that for each $p \in P$ there is a representation $\phi_p \colon \pi_1(M_K) \to G$ for which $\rho^{(2)}(M_K,\phi_p)=0$.
+{{endthm}}
+Finally, it is somewhat remarkable that the classical Blanchfield form continues to have new and interesting applications.  For example, Borodzik and Friedl \cite{Borodzik&Friedl2012} recently used the minimal size of a square matrix which presents the Blanchfield form of a given knot to compute many previously unknown unknotting numbers of low crossing number knots.  See the [[Linking form|linking form page]], for the definition of a [[Linking form#Presentations_of_linking_forms|presentation of a linking form]].
 </wikitex>
@@ Line 79: / Line 58: @@
 {{#RefList:}}
-[[Category:Theory]]
+[[Category:Definitions]]