15:07
Tags: #knots #concordance #geometric_topology
\(K_1, K_2\) 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? #idle_thoughts
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See torus knot
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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).