---
date: 2021-11-22
tags: [ web/quick-notes ]
---

# 2021-11-22

Tags:
#web/quick-notes

Refs:
?

## 01:01

- You can define $p\dash$curvature for arithmetic schemes.
  Are there analogs of the [Riemann curvature tensor](Riemann%20curvature%20tensor)? [Ricci curvature](Ricci%20curvature.md)?
  - How to make [Floer homology](Floer%20homology) or [Morse homology](Morse%20homology.md) work for [varieties](varieties) or [schemes](schemes)?
  Not clear how to define things like gradient flows algebraically.

## UGA Topology Seminar, Irving Dai, Equivariant Concordance and Knot Floer Homology

- WIP! Joint with Mallick and Stoffregen. [Equivariant](Equivariant.md) [concordance](concordance) and [knot Floer homology](knot%20Floer%20homology)
- Equivariant [knots](knots.md): pairs $(K, \tau)$ where $K \subseteq S^3$ and $\tau: S^3\selfmap$ an orientation-preserving involution preserving $K$, so $\tau(K) = K$.
- Symmetries of the trefoil:

<!-- Xournal file: /home/zack/figures/2021-11-22_15-08.xoj -->


![](figures/2021-11-22_15-10-22.png)

- The first case is a strong inversion, the second is a 2-periodic involution (given by twisting about a core torus).

- One can assume that $\tau$ is rotation about some axis.

- There is an extension of $\tau$ to $\BB^4$, so define an equivariant slice surface $\Sigma$ if $\tau \Sigma = \Sigma$, and define an equivariant (slice?) genus as the minimal genus among such surfaces $\tilde g_4(K)$

- Study $\tilde g_4(K) - g_4(K)$. 
  [Boyle-Issan](https://arxiv.org/pdf/2101.05413.pdf) show this difference is unbounded for a family of periodic knots.

- Prove a similar theorem: given $(K, \tau)$, define a set of numerical invariants using Floer homology which are 
  - Equivariant concordance invariants
  - Functions of these bound $\tilde g_4(K)$ from below.

- Produced a family of strongly invertible slice knots where $\tilde g_4$ is unbounded.

- Most (small crossing) knots admit a strong inversion.

- Next: how to apply this machinery to seemingly non-equivariant things.

- A slice surface $\Sigma$ is isotopy equivariant iff $\tau_{\BB^4} \isotopic \Sigma$ rel boundary.
  Define isotopy equivariant genus $\tilde{ig}_4(K)$ as the minimal genus of such $\Sigma$.

- Calculating this invariants gives a way of finding non-isotopic surfaces for $K$.

- Recent work: topologically isotopic but not smoothly isotopic surfaces.
  - JMZ: higher genus construction using knot floer homology.
    Needs high genus, won't work for slice discs.
  - H, HS: for slice discs using Khovanov homology.

- Proving topologically isotopic: a known theorem involving equivalence of $\pi_1$.

- Theorem: produced a knot where $\tilde{ig}_4(K) > 0$.

- Does $\BB^4$ actually matter here?
  The answer is no, can take $\ZHB^4$.

- A generalized isotopy equivariant surface is a triple $(W, \tau_W, \Sigma)$ where
  - $W \in \ZHB^4$ with $\bd W = S^3$
  - $\tau_W: W\to W$ extends $\tau$ (in any way!)
  - $\tau_W \Sigma \isotopic \Sigma$ rel $K$.

- Another application: let $\Sigma, \Sigma'$ be two slices surfaces in $\BB^4$ for $K$.
  Interpolate: take \( \Sigma = \Sigma_0 \to \Sigma_1 \to \cdots \to \Sigma_n = \Sigma' \) where each arrow is a stabilization or destabilization or isotopy rel $K$.
  How many arrows are needed?
  Define this as $M_{\st}(\Sigma, \Sigma')$, the stabilization number.
  - Q: given a number of arrows, can this be achieved by picking a suitable genus?

- Theorem: for any $m$, produce a knot $J_m$ with two slice disks with stabilization distance exactly $m$.

- Theorem: if $(K, \tau)$ is strongly invertible slice and $\Sigma$ is any slice disk for $K$, then $M_{\st}(\Sigma, \tau_{\BB^4} \Sigma) \geq \cdots$, some function of the numerical invariants.

  - So this can show non-isotopic, and require many stabilizations to become isotopic.

- These all induce maps on $\CFK(K)$, where the $\tau$ action induces a $\tau$ action on $\CFK(K)$.
  Isotopy equivariant knot cobordisms $K_1\to K_2$ induce $\tau\dash$equivariant maps $\CFK(K_1) \to \CFK(K_2)$ in the sense that this commutes with the two different $\tau$ actions on either side.

- Can use this to find knots that are concordant but not equivariantly concordant by using algebraic restrictions on bigraded $\CFK(K)$

- Doing this with higher order diffeomorphisms: the roadblock is defining $\HF$ mod $p$!

## 19:57

A nice modern intro to homotopy theory: 
<https://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/Bunke/intro-homoto.pdf>

Quotients are colimits:

![](figures/2021-11-22_19-57-59.png)

[Geometric realization](Geometric%20realization.md) as a [coend](coend.md)

![](figures/2021-11-22_19-59-48.png)

[homotopy fiber](homotopy%20fiber.md):

![](figures/2021-11-22_20-01-20.png)

[Homotopy cofiber](Homotopy%20cofiber):

![](figures/2021-11-22_20-02-04.png)

[Spectra](Spectra.md) as a [presentable](presentable) [infty-category](infty-category)

![](figures/2021-11-22_20-05-06.png)