# 2021-11-22

Tags: #web/quick-notes

Refs: ?

## 01:01

• You can define $$p{\hbox{-}}$$curvature for arithmetic schemes. Are there analogs of the Riemann curvature tensor? Ricci curvature?
• How to make Floer homology or Morse homology work for varieties or 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 concordance and knot Floer homology
• Equivariant knots: pairs $$(K, \tau)$$ where $$K \subseteq S^3$$ and $$\tau: S^3{\circlearrowleft}$$ an orientation-preserving involution preserving $$K$$, so $$\tau(K) = K$$.
• Symmetries of the trefoil:

• 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 $${\mathbb{B}}^4$$, 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 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 $$\tau_{{\mathbb{B}}^4} \isotopic \Sigma$$ 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 $${\mathbb{B}}^4$$ actually matter here? The answer is no, can take $$\operatorname{ZHB}^4$$.

• A generalized isotopy equivariant surface is a triple $$(W, \tau_W, \Sigma)$$ where

• $$W \in \operatorname{ZHB}^4$$ with $${{\partial}}W = S^3$$
• $$\tau_W: W\to W$$ extends $$\tau$$ (in any way!)
• $$\tau_W \Sigma \isotopic \Sigma$$ rel $$K$$.
• Another application: let $$\Sigma, \Sigma'$$ be two slices surfaces in $${\mathbb{B}}^4$$ 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 $$M_{{~\mathrel{\Big\vert}~}}(\Sigma, \Sigma')$$, 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 $$M_{{~\mathrel{\Big\vert}~}}(\Sigma, \tau_{{\mathbb{B}}^4} \Sigma) \geq \cdots$$, 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 $$\tau{\hbox{-}}$$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 $$\operatorname{HF}$$ 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:

Geometric realization as a coend

homotopy fiber:

Homotopy cofiber:

Spectra as a presentable infty-category

#web/quick-notes