Knots, i.e. embeddings of spheres

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Contents

1 Introduction

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, \S1, \S3].

2 Examples

Analogously to the Haefliger trefoil knot for k>1 one constructs a smooth embedding t:S^{2k-1}\to\Rr^{3k}, see [Skopenkov2016h, \S5]. For k even this embedding is a generator of E_D^{3k}(S^{2k-1})\cong\Zz; it is not smoothly isotopic to the standard embedding, but is piecewise smoothly isotopic to it [Haefliger1962]. It would be interesting to know if for k odd this embedding is a generator of E_D^{3k}(S^{2k-1})\cong\Zz_2. The last phrase of [Haefliger1962t] suggests that this is true for k=3.

3 Classification

Theorem 3.1 [Levine1965, Corollary in p. 44], [Haefliger1966]. For m-n\ge3 the group E^m_D(S^n) is finite unless n=4k-1 and m\le6k, when E^m_D(S^n) is the sum of \Z and a finite group.

Theorem 3.2 (Haefliger-Milgram). We have the following table for the group E^m_D(S^n); in the table k\ge1:

\displaystyle \begin{array}{c|c|c|c|c|c|c|c} (m,n) &2m\ge3n+4 &(6k,4k-1) &(6k+3,4k+1) &(7,4) &(6k+4,4k+2) &(12k+7,8k+4) & (12k+1,8k+1)\\ \hline E^m_D(S^n)&0 &\Z &\Z_2 &\Z_{12} &0 &\Z_4 &\Z_2\oplus\Z_2 \end{array}

Proof for the first four columns, and for the fifth column, k even, are presented in [Haefliger1966] (or are deduced from that paper using simple calculations, cf. [Skopenkov2005, \S3]). The remaining results were announced in [Milgram1972]; no details of the proofs appeared. Alternative proofs for the case (m,n)=(7,4) are given in [Skopenkov2005], [Crowley&Skopenkov2008].

Theorem 3.3 [Milgram1972, Corollary G]. We have E^m_D(S^n)=0 if and only if either 2m\ge3n+4, or (m,n)=(6k+4,4k+2), or (m,n)=(3k,2k) and k\equiv3,11\mod12, or (m,n)=(3k+2,2k+2) and k\equiv14,22\mod24.

For a description of 2-components of E^m_D(S^n) see [Milgram1972, Theorem F].

For m\ge n+3 the group E^m_D(S^n) has been described as follows, in terms of exact sequences [Haefliger1966], cf. [Levine1965], [Haefliger1966a], [Milgram1972], [Habegger1986].

Theorem 3.4 [Haefliger1966]. For m-n\ge3 there is the following exact sequence of abelian groups:

\displaystyle \ldots \to \pi_{n+1}(SG,SO) \xrightarrow{~u~} E^m_D(S^n) \xrightarrow{~a~} \pi_n(SG_n,SO_n) \xrightarrow{~s~} \pi_n(SG,SO) \xrightarrow{~u~} E^{m-1}_D(S^{n-1})\to \ldots~.

Here SG_n be the space of maps S^{n-1} \to S^{n-1} of degree 1. Restricting an element of SO_n to S^{n-1} \subset \Rr^n identifies SO_n as a subspace of SG_n. Let SG:=SG_1\cup\ldots\cup SG_n\cup\ldots. Analogously define SO. Let s be the stabilization homomorphism. The attaching invariant a and the map u are defined in [Haefliger1966], see also [Skopenkov2005, \S3].

4 Codimension 2 knots

For the best known specific case, i.e. for codimension 2 embeddings of spheres (in particular, for the classical theory of knots in \Rr^3), a complete readily calculable classification (in the sense of Remark 1.2 of [Skopenkov2016c]) is neither known nor expected at the time of writing. However, there is a vast literature on codimension 2 knots. See e.g. the interesting papers [Farber1981], [Farber1983], [Kearton1983], [Farber1984].

On the other hand, if one studies embeddings up to the weaker relation of concordance, then much is known. See e.g. [Levine1969a] and [Ranicki1998].

5 References

  • [Crowley&Skopenkov2008] D. Crowley and A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, II, Intern. J. Math., 22:6 (2011) 731-757. Available at the arXiv:0808.1795.
  • [Farber1981] M. Sh. Farber, Classification of stable fibered knots, Mat. Sb. (N.S.), 115(157):2(6) (1981) 223–262.
  • [Farber1983] M. Sh. Farber, The classification of simple knots, Russian Math. Surveys, 38:5 (1983).

, $\S]{Skopenkov2016c}. == Examples == ; Analogously to [[3-manifolds_in_6-space#Examples|the Haefliger trefoil knot]] for $k>1$ one constructs a smooth embedding $t:S^{2k-1}\to\Rr^{3k}$, see \cite[$\S]{Skopenkov2016h}. For $k$ even this embedding is a generator of $E_D^{3k}(S^{2k-1})\cong\Zz$; it is not ''smoothly'' isotopic to the standard embedding, but is ''piecewise smoothly'' isotopic to it \cite{Haefliger1962}. It would be interesting to know if for $k$ odd this embedding is a generator of $E_D^{3k}(S^{2k-1})\cong\Zz_2$. The last phrase of \cite{Haefliger1962t} suggests that this is true for $k=3$. == Classification == ; {{beginthm|Theorem|\cite[Corollary in p. 44]{Levine1965}, \cite{Haefliger1966}}}\label{t:leha} For $m-n\ge3$ the group $E^m_D(S^n)$ is finite unless $n=4k-1$ and $m\le6k$, when $E^m_D(S^n)$ is the sum of $\Z$ and a finite group. {{endthm}} {{beginthm|Theorem|(Haefliger-Milgram)}}\label{t:hami} We have the following table for the group $E^m_D(S^n)$; in the table $k\ge1$: $$\begin{array}{c|c|c|c|c|c|c|c} (m,n) &2m\ge3n+4 &(6k,4k-1) &(6k+3,4k+1) &(7,4) &(6k+4,4k+2) &(12k+7,8k+4) & (12k+1,8k+1)\ \hline E^m_D(S^n)&0 &\Z &\Z_2 &\Z_{12} &0 &\Z_4 &\Z_2\oplus\Z_2 \end{array}$$ {{endthm}} Proof for the first four columns, and for the fifth column, $k$ even, are presented in \cite{Haefliger1966} (or are deduced from that paper using simple calculations, cf. \cite[$\S]{Skopenkov2005}). The remaining results were announced in \cite{Milgram1972}; no details of the proofs appeared. Alternative proofs for the case $(m,n)=(7,4)$ are given in \cite{Skopenkov2005}, \cite{Crowley&Skopenkov2008}. {{beginthm|Theorem|\cite[Corollary G]{Milgram1972}}}\label{t:mi} We have $E^m_D(S^n)=0$ if and only if either m\ge3n+4$, or $(m,n)=(6k+4,4k+2)$, or $(m,n)=(3k,2k)$ and $k\equiv3,11\mod12$, or $(m,n)=(3k+2,2k+2)$ and $k\equiv14,22\mod24$. {{endthm}} For a description of 2-components of $E^m_D(S^n)$ see \cite[Theorem F]{Milgram1972}. For $m\ge n+3$ the group $E^m_D(S^n)$ has been described as follows, in terms of exact sequences \cite{Haefliger1966}, cf. \cite{Levine1965}, \cite{Haefliger1966a}, \cite{Milgram1972}, \cite{Habegger1986}. {{beginthm|Theorem|\cite{Haefliger1966}}}\label{t:knots} For $m-n\ge3$ there is the following exact sequence of abelian groups: $$ \ldots \to \pi_{n+1}(SG,SO) \xrightarrow{~u~} E^m_D(S^n) \xrightarrow{~a~} \pi_n(SG_n,SO_n) \xrightarrow{~s~} \pi_n(SG,SO) \xrightarrow{~u~} E^{m-1}_D(S^{n-1})\to \ldots~.$$ Here $SG_n$ be the space of maps $S^{n-1} \to S^{n-1}$ of degree \S1, \S3].

2 Examples

Analogously to the Haefliger trefoil knot for k>1 one constructs a smooth embedding t:S^{2k-1}\to\Rr^{3k}, see [Skopenkov2016h, \S5]. For k even this embedding is a generator of E_D^{3k}(S^{2k-1})\cong\Zz; it is not smoothly isotopic to the standard embedding, but is piecewise smoothly isotopic to it [Haefliger1962]. It would be interesting to know if for k odd this embedding is a generator of E_D^{3k}(S^{2k-1})\cong\Zz_2. The last phrase of [Haefliger1962t] suggests that this is true for k=3.

3 Classification

Theorem 3.1 [Levine1965, Corollary in p. 44], [Haefliger1966]. For m-n\ge3 the group E^m_D(S^n) is finite unless n=4k-1 and m\le6k, when E^m_D(S^n) is the sum of \Z and a finite group.

Theorem 3.2 (Haefliger-Milgram). We have the following table for the group E^m_D(S^n); in the table k\ge1:

\displaystyle \begin{array}{c|c|c|c|c|c|c|c} (m,n) &2m\ge3n+4 &(6k,4k-1) &(6k+3,4k+1) &(7,4) &(6k+4,4k+2) &(12k+7,8k+4) & (12k+1,8k+1)\\ \hline E^m_D(S^n)&0 &\Z &\Z_2 &\Z_{12} &0 &\Z_4 &\Z_2\oplus\Z_2 \end{array}

Proof for the first four columns, and for the fifth column, k even, are presented in [Haefliger1966] (or are deduced from that paper using simple calculations, cf. [Skopenkov2005, \S3]). The remaining results were announced in [Milgram1972]; no details of the proofs appeared. Alternative proofs for the case (m,n)=(7,4) are given in [Skopenkov2005], [Crowley&Skopenkov2008].

Theorem 3.3 [Milgram1972, Corollary G]. We have E^m_D(S^n)=0 if and only if either 2m\ge3n+4, or (m,n)=(6k+4,4k+2), or (m,n)=(3k,2k) and k\equiv3,11\mod12, or (m,n)=(3k+2,2k+2) and k\equiv14,22\mod24.

For a description of 2-components of E^m_D(S^n) see [Milgram1972, Theorem F].

For m\ge n+3 the group E^m_D(S^n) has been described as follows, in terms of exact sequences [Haefliger1966], cf. [Levine1965], [Haefliger1966a], [Milgram1972], [Habegger1986].

Theorem 3.4 [Haefliger1966]. For m-n\ge3 there is the following exact sequence of abelian groups:

\displaystyle \ldots \to \pi_{n+1}(SG,SO) \xrightarrow{~u~} E^m_D(S^n) \xrightarrow{~a~} \pi_n(SG_n,SO_n) \xrightarrow{~s~} \pi_n(SG,SO) \xrightarrow{~u~} E^{m-1}_D(S^{n-1})\to \ldots~.

Here SG_n be the space of maps S^{n-1} \to S^{n-1} of degree 1. Restricting an element of SO_n to S^{n-1} \subset \Rr^n identifies SO_n as a subspace of SG_n. Let SG:=SG_1\cup\ldots\cup SG_n\cup\ldots. Analogously define SO. Let s be the stabilization homomorphism. The attaching invariant a and the map u are defined in [Haefliger1966], see also [Skopenkov2005, \S3].

4 Codimension 2 knots

For the best known specific case, i.e. for codimension 2 embeddings of spheres (in particular, for the classical theory of knots in \Rr^3), a complete readily calculable classification (in the sense of Remark 1.2 of [Skopenkov2016c]) is neither known nor expected at the time of writing. However, there is a vast literature on codimension 2 knots. See e.g. the interesting papers [Farber1981], [Farber1983], [Kearton1983], [Farber1984].

On the other hand, if one studies embeddings up to the weaker relation of concordance, then much is known. See e.g. [Levine1969a] and [Ranicki1998].

5 References

  • [Crowley&Skopenkov2008] D. Crowley and A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, II, Intern. J. Math., 22:6 (2011) 731-757. Available at the arXiv:0808.1795.
  • [Farber1981] M. Sh. Farber, Classification of stable fibered knots, Mat. Sb. (N.S.), 115(157):2(6) (1981) 223–262.
  • [Farber1983] M. Sh. Farber, The classification of simple knots, Russian Math. Surveys, 38:5 (1983).

$. Restricting an element of $SO_n$ to $S^{n-1} \subset \Rr^n$ identifies $SO_n$ as a subspace of $SG_n$. Let $SG:=SG_1\cup\ldots\cup SG_n\cup\ldots$. Analogously define $SO$. Let $s$ be the stabilization homomorphism. The attaching invariant $a$ and the map $u$ are defined in \cite{Haefliger1966}, see also \cite[$\S]{Skopenkov2005}. {{endthm}} == Codimension 2 knots == ; For the best known specific case, i.e. for codimension 2 embeddings of spheres (in particular, for the classical theory of knots in $\Rr^3$), a complete readily calculable classification (in the sense of [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|Remark 1.2]] of \cite{Skopenkov2016c}) is neither known nor expected at the time of writing. However, there is a vast literature on codimension 2 knots. See e.g. the interesting papers \cite{Farber1981}, \cite{Farber1983}, \cite{Kearton1983}, \cite{Farber1984}. On the other hand, if one studies embeddings up to the weaker relation of [[Isotopy|''concordance'']], then much is known. See e.g. \cite{Levine1969a} and \cite{Ranicki1998}. == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\S1, \S3].

2 Examples

Analogously to the Haefliger trefoil knot for k>1 one constructs a smooth embedding t:S^{2k-1}\to\Rr^{3k}, see [Skopenkov2016h, \S5]. For k even this embedding is a generator of E_D^{3k}(S^{2k-1})\cong\Zz; it is not smoothly isotopic to the standard embedding, but is piecewise smoothly isotopic to it [Haefliger1962]. It would be interesting to know if for k odd this embedding is a generator of E_D^{3k}(S^{2k-1})\cong\Zz_2. The last phrase of [Haefliger1962t] suggests that this is true for k=3.

3 Classification

Theorem 3.1 [Levine1965, Corollary in p. 44], [Haefliger1966]. For m-n\ge3 the group E^m_D(S^n) is finite unless n=4k-1 and m\le6k, when E^m_D(S^n) is the sum of \Z and a finite group.

Theorem 3.2 (Haefliger-Milgram). We have the following table for the group E^m_D(S^n); in the table k\ge1:

\displaystyle \begin{array}{c|c|c|c|c|c|c|c} (m,n) &2m\ge3n+4 &(6k,4k-1) &(6k+3,4k+1) &(7,4) &(6k+4,4k+2) &(12k+7,8k+4) & (12k+1,8k+1)\\ \hline E^m_D(S^n)&0 &\Z &\Z_2 &\Z_{12} &0 &\Z_4 &\Z_2\oplus\Z_2 \end{array}

Proof for the first four columns, and for the fifth column, k even, are presented in [Haefliger1966] (or are deduced from that paper using simple calculations, cf. [Skopenkov2005, \S3]). The remaining results were announced in [Milgram1972]; no details of the proofs appeared. Alternative proofs for the case (m,n)=(7,4) are given in [Skopenkov2005], [Crowley&Skopenkov2008].

Theorem 3.3 [Milgram1972, Corollary G]. We have E^m_D(S^n)=0 if and only if either 2m\ge3n+4, or (m,n)=(6k+4,4k+2), or (m,n)=(3k,2k) and k\equiv3,11\mod12, or (m,n)=(3k+2,2k+2) and k\equiv14,22\mod24.

For a description of 2-components of E^m_D(S^n) see [Milgram1972, Theorem F].

For m\ge n+3 the group E^m_D(S^n) has been described as follows, in terms of exact sequences [Haefliger1966], cf. [Levine1965], [Haefliger1966a], [Milgram1972], [Habegger1986].

Theorem 3.4 [Haefliger1966]. For m-n\ge3 there is the following exact sequence of abelian groups:

\displaystyle \ldots \to \pi_{n+1}(SG,SO) \xrightarrow{~u~} E^m_D(S^n) \xrightarrow{~a~} \pi_n(SG_n,SO_n) \xrightarrow{~s~} \pi_n(SG,SO) \xrightarrow{~u~} E^{m-1}_D(S^{n-1})\to \ldots~.

Here SG_n be the space of maps S^{n-1} \to S^{n-1} of degree 1. Restricting an element of SO_n to S^{n-1} \subset \Rr^n identifies SO_n as a subspace of SG_n. Let SG:=SG_1\cup\ldots\cup SG_n\cup\ldots. Analogously define SO. Let s be the stabilization homomorphism. The attaching invariant a and the map u are defined in [Haefliger1966], see also [Skopenkov2005, \S3].

4 Codimension 2 knots

For the best known specific case, i.e. for codimension 2 embeddings of spheres (in particular, for the classical theory of knots in \Rr^3), a complete readily calculable classification (in the sense of Remark 1.2 of [Skopenkov2016c]) is neither known nor expected at the time of writing. However, there is a vast literature on codimension 2 knots. See e.g. the interesting papers [Farber1981], [Farber1983], [Kearton1983], [Farber1984].

On the other hand, if one studies embeddings up to the weaker relation of concordance, then much is known. See e.g. [Levine1969a] and [Ranicki1998].

5 References

  • [Crowley&Skopenkov2008] D. Crowley and A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, II, Intern. J. Math., 22:6 (2011) 731-757. Available at the arXiv:0808.1795.
  • [Farber1981] M. Sh. Farber, Classification of stable fibered knots, Mat. Sb. (N.S.), 115(157):2(6) (1981) 223–262.
  • [Farber1983] M. Sh. Farber, The classification of simple knots, Russian Math. Surveys, 38:5 (1983).

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