2021-10-18

15:07

Tags: #subjects/geomtop/knots #subjects/geomtop

\(K_1, K_2\) are smoothly concordant iff there exists a smoothly embedded cylinder \(S^1\times I \hookrightarrow S^3\times I\) with \({{\partial}}(S^1\times I) = K_1 {\textstyle\coprod}-K_2\). The concordance group \(C\) is the abelian group given by knots \(K \hookrightarrow S^3\) under connect sum, modulo concordance.

If \(K_i \hookrightarrow Y_i \in \mathbb{Z}\operatorname{HS}^3\), then the \(K_i\) are homologically concordant if there is smoothly embedded cylinder \(S^1\times I \hookrightarrow W\) with \({{\partial}}(W, S^1\times I) = (Y_1, K_1) {\textstyle\coprod}(Y_2, K_2)\) with \(W\) a homology cobordism:

  • \(W \in {\mathsf{Mfd}}^4\) compact oriented,
  • \({{\partial}}W = Y_1 {\textstyle\coprod}Y_2\),
  • There are induced isomorphisms \(H_*(Y_i; {\mathbb{Z}}) { \, \xrightarrow{\sim}\, }H_*(W; {\mathbb{Z}})\).

This yields a homological concordance group \(\widehat{C}_{\mathbb{Z}}\).

There is an injection (?) \(C_{\mathbb{Z}}\hookrightarrow\widehat{C}_{\mathbb{Z}}\) which is known by Levine not to be surjective. What can be said about the cokernel?

  • Infinitely generated, known using d invariants and reduced Heegard-Floer homology.
  • Contains a \({\mathbb{Z}}{\hbox{-}}\)subgroup using epsilon invariants and tau invariants
  • Contains a \({\mathbb{Z}}^\infty\) subgroup, and in fact a summand

See Seifert fibered space, ZHS3. These are all homology cobordant to \(S^3\).

Proof uses CFK, a \({\mathbb{F}}[u, v]{\hbox{-}}\)module.

A knot-like complex over \(R\) is a complex \(C \in {\mathsf{gr}\,}_{{\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{2} } }} \mathsf{Ch}(R)\) such that

  • \(H_*(C/u)/C_{{\operatorname{tors}}_v} \conf {\mathbb{F}}[v]\)
  • \(H_*(C/v)/C_{{\operatorname{tors}}_u} \conf {\mathbb{F}}[u]\)
  • Some grading conditions.

Some examples: the knot Floer complex CFK over a knot, \(\CFK_{{\mathbb{F}}[u, v]}(K)\). Theorem: every such complex is locally equivalent to a unique standard complex. Concordant knots produce locally equivalent complexes \(\CFK_R(K)\) for \(R \coloneqq{\mathbb{F}}[u] \otimes_{\mathbb{F}}{\mathbb{F}}[z] / \left\langle{uv}\right\rangle\).

Set \(\mathsf{C} \coloneqq{\operatorname{Emb}}(S^1, S^3)\), add the monoidal structure \({\sharp}\) for connect sum. Take “isotopy” category instead of homotopy category? The unit is \(\one = U\), the unknot up to isotopy. What is the stabilization of \({-}{\sharp}X\) for fixed choices of \(X\)? Or of other interesting functors? #personal/idle-thoughts

  • See torus knot

  • Can do base changes \(\CFK_{{\mathbb{F}}[u, v]}(M_n, K_n) \otimes_{{\mathbb{F}}[u, v]} \mathcal{X} \leadsto \CFK_{\mathcal{X}}(M_n, K_n)\) (may also need to change basis to get standard complex).

#web/quick-notes #subjects/geomtop/knots #subjects/geomtop #personal/idle-thoughts