---
date: 2021-12-06
tags: [ web/quick-notes ]
---

# 2021-12-06

Tags:
#web/quick-notes

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## Ali Daemi: Signature functions and basic knots (15:04)

- Assumptions: $X\in \smooth\Mfd^4$, $b_1(\bd X) = 0$, so get a nondegenerate quadratic form $q: H_2(X)\tensorpower{\ZZ}{2} \to \RR$.
- Get positive/negative definite eigenspaces of dimensions $b^+, b^-$, define $\signature(X) = b^+ - b^-$.
- Take $K\embeds S^3$, find a surface $F \subseteq \BB^4$ with $\bd F = K$.
- Write $\Sigma_n(F)$ for the $n$th cyclic branched cover of $\BB^4$ branched along $F$.
  Boundary is a $\QHS^3$, and $\signature(\Sigma_n(F))$ is determined by $K$.
  For $n=2$, recovers ordinary signature of $K$.
- Covering map induces an action $C_n \actson H_2(\Sigma_n(F))$, so take eigenspaces.
  Write $C_n = \gens{t}$, take $\ker(t-\zeta_n)$ for $\zeta_n$ an $n$th root of unity (working over $\CC$ now).
- Define $\sigma_K(w)$ for the signature restructure to $\ker(t-w)$ for $w\da \zeta_n$ -- the *Levine-Tristam, signature*.
- Find a rep-theoretic description of $\sigma_K(w)$: consider $\Hom(\pi, G)$ for $G\in \Lie\Grp$.
  We'll take $G \da \SU_2$ and $\pi \da \pi_1(K)$.

- Set $P\da \Hom(\pi_1(T), \SU_2) \cong \Hom(\ZZ\cartpower{2}, \SU_2)/\sim$ where $\SU_2$ acts by conjugation.
  Equivalently $P = \ts{(\theta_1, \theta_2) \in S^1\times S^1 \st (\theta_1, \theta_2) = (- \theta_1, - \theta_2)}$.
  This yields a pillowcase:

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- Define $\chi^*(K) \da \Homeo(\pi, \SU_2)$ where $\pi \da \pi_1(S^3\sm \nu(K))$ and we take homeomorphisms that do not have abelian image, modulo conjugation as before.
- For $\alpha\in [0, 1/2]$, define $\chi_\alpha^*(K) \da \ts{\phi\in \chi^*(K) \st \phi(\mu) \sim \matt{e^{2\pi i \alpha}}{0}{0}{e^{-2\pi i \alpha}} }$ for $\mu$ a meridian of $K$.
- Morally: $\sigma_K(e^{2\pi i \alpha})$ is a signed count of $\chi_\alpha^*(K)$.
- There is a map $r: \chi^*(K) \to P$ given by restriction to $\bd \nu(K)$.
- Generically $\chi^*(K)$ is a 1-dimensional variety with boundary, and its image under $r$ is the bottom line of the pillowcase.
  We can also write $\chi_\alpha^*(K) = r\inv(C_\alpha)$ where $C_\alpha \da \ts{(\alpha, t) \st t\in [0, 1/2]} \subseteq P$ is a vertical line.
- Need to remove a finite set $S_k$, the $\alpha$ for which $e^{4\pi i \alpha}$ is a root of the Alexander polynomial.
- Get a lower bound for the number of elements in the character variety: $\# \chi_\alpha^*(K) \leq {1\over 2}\abs{\sigma_K(e^{4\pi i \alpha})}$.
- Say $K$ is $\SU_2\dash$basic if $\chi_\alpha^*(K)$ is as small as possible, so equality in this inequality, plus some transversality conditions.
  - Examples: $T_{p, q}$.
  - Also the pretzel knot $P(-2,3,7)$

- Question: can we classify all $\SU_2\dash$basic knots?
- **Theorem**: if $K$ is $\SU_2\dash$basic, $S:K\to K'$ a concordance, then any $\phi\in \chi^*(K)$ extends to $\tilde\phi\in \Hom_\Grp(\pi_1(S^4\cross I \sm S), \SU_2)$.
  - Proof uses instanton Floer homology, Yang-Mills gauge theory.
- Definition of ribbon concordance: $S:K\to K'$ is *ribbon* if, noting $S \subseteq I \cross S^3$, the projection $S\to I$ is Morse without any critical points of index 2.
- **Theorem**: let $S:K\to K'$ be a ribbon concordance, then the same kind of lift exists.
- Question: if $K$ is $\SU_2$ basic and $S:K\to K'$ is a concordance, can $S$ be exchanged for $S'$ a ribbon concordance?
  - Slice-ribbon conjecture for $K=U$ implies that the answer is yes. 
    The theorem says that a negative answer wouldn't be useful in disproving this conjecture.

- Other examples of $\SU_2$ basic knots:
  - Cables $C_{pqk + 1, k}(T_{p, q})$..
  - Twisted torus knots $T(3, 3n-1, 2, 1)$.
    Interestingly, these are  all $L\dash$space knots.

- Question: can $\SU_2$ basic knots be classified using Dehn surgery.


## 20:31

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