2021-11-22 – Notes

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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 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 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:

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 \({\mathbb{B}}^4\), 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 \(p\)!