---
date: 2021-10-18
tags: [ web/quick-notes ]
---

# 2021-10-18

## 15:07

Tags: #geomtop/knots #geomtop

:::{.definition title="Concordance"}
$K_1, K_2$ are smoothly [concordant](concordant) iff there exists a smoothly embedded cylinder $S^1\times I \embeds S^3\times I$ with $\bd(S^1\times I) = K_1 \disjoint -K_2$.
The [concordance group](concordance%20group) $C$ is the abelian group given by knots $K \embeds S^3$ under connect sum, modulo concordance.
:::

:::{.definition title="Homological concordance"}
If $K_i \embeds Y_i \in \ZHS^3$, then the $K_i$ are **homologically concordant** if there is smoothly embedded cylinder $S^1\times I \embeds W$ with $\bd(W, S^1\times I) = (Y_1, K_1) \disjoint (Y_2, K_2)$ with $W$ a *homology cobordism*:

- $W \in \Mfd^4$ compact oriented,
- $\bd W = Y_1 \disjoint Y_2$,
- There are induced isomorphisms $H_*(Y_i; \ZZ) \iso H_*(W; \ZZ)$.

This yields a homological concordance group $\hat{C}_\ZZ$.
:::

:::{.remark}

There is an injection (?) $C_\ZZ \injects \hat{C}_\ZZ$ which is known by Levine not to be surjective.
What can be said about the cokernel?

- Infinitely generated, known using [d invariants](d%20invariants) and reduced [Heegard-Floer homology](Heegard-Floer%20homology.md).
- Contains a $\ZZ\dash$subgroup using [epsilon invariants](epsilon%20invariants) and [tau invariants](tau%20invariants)
- Contains a $\ZZ^\infty$ subgroup, and in fact a summand

See [Seifert fibered space](Seifert%20fibered%20space), [ZHS3](ZHS3).
These are all [homology cobordant](homology%20cobordant) to $S^3$.

Proof uses [CFK](CFK), a $\FF[u, v]\dash$module.

:::

:::{.definition title="Knot-like complexes"}
A **knot-like complex** over $R$ is a complex $C \in \gr_{\ZZ\cartpower{2}} \Ch(R)$ such that 

- $H_*(C/u)/C_{\tors_v} \conf \FF[v]$
- $H_*(C/v)/C_{\tors_u} \conf \FF[u]$
- Some grading conditions.

:::

:::{.remark}
Some examples: the knot Floer complex [CFK](CFK) over a knot, $\CFK_{\FF[u, v]}(K)$.
Theorem: every such complex is *locally equivalent* to a unique standard complex.
Concordant knots produce locally equivalent complexes $\CFK_R(K)$ for $R \da \FF[u] \tensor_\FF \FF[z] / \gens{uv}$.
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:::{.remark}

Set $\cat{C} \da \Emb(S^1, S^3)$, add the monoidal structure $\size$ 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 $\wait \size X$ for fixed choices of $X$?
Or of other interesting functors? #personal/idle-thoughts
:::

:::{.remark}
- See [torus knot](torus%20knot)

- Can do base changes $\CFK_{\FF[u, v]}(M_n, K_n) \tensor_{\FF[u, v]} \mathcal{X} \leadsto \CFK_{\mathcal{X}}(M_n, K_n)$ (may also need to change basis to get standard complex).

:::