# 2021-12-06

Tags: #web/quick-notes

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

• Assumptions: $$X\in {\mathsf{sm}}{\mathsf{Mfd}}^4$$, $$b_1({{\partial}}X) = 0$$, so get a nondegenerate quadratic form $$q: H_2(X){ {}^{ \scriptstyle\otimes_{{\mathbb{Z}}}^{2} } } \to {\mathbb{R}}$$.

• Get positive/negative definite eigenspaces of dimensions $$b^+, b^-$$, define $$\operatorname{sig}(X) = b^+ - b^-$$.

• Take $$K\hookrightarrow S^3$$, find a surface $$F \subseteq {\mathbb{B}}^4$$ with $${{\partial}}F = K$$.

• Write $$\Sigma_n(F)$$ for the $$n$$th cyclic branched cover of $${\mathbb{B}}^4$$ branched along $$F$$. Boundary is a $$\mathbb{Q}\kern-0.5pt\operatorname{HS}^3$$, and $$\operatorname{sig}(\Sigma_n(F))$$ is determined by $$K$$. For $$n=2$$, recovers ordinary signature of $$K$$.

• Covering map induces an action $$C_n \curvearrowright H_2(\Sigma_n(F))$$, so take eigenspaces. Write $$C_n = \left\langle{t}\right\rangle$$, take $$\ker(t-\zeta_n)$$ for $$\zeta_n$$ an $$n$$th root of unity (working over $${\mathbb{C}}$$ now).

• Define $$\sigma_K(w)$$ for the signature restructure to $$\ker(t-w)$$ for $$w\coloneqq\zeta_n$$ – the Levine-Tristam, signature.

• Find a rep-theoretic description of $$\sigma_K(w)$$: consider $$\mathop{\mathrm{Hom}}(\pi, G)$$ for $$G\in \mathsf{Lie}{\mathsf{Grp}}$$. We’ll take $$G \coloneqq{\operatorname{SU}}_2$$ and $$\pi \coloneqq\pi_1(K)$$.

• Set $$P\coloneqq\mathop{\mathrm{Hom}}(\pi_1(T), {\operatorname{SU}}_2) \cong \mathop{\mathrm{Hom}}({\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{2} } }, {\operatorname{SU}}_2)/\sim$$ where $${\operatorname{SU}}_2$$ acts by conjugation. Equivalently $$P = \left\{{(\theta_1, \theta_2) \in S^1\times S^1 {~\mathrel{\Big\vert}~}(\theta_1, \theta_2) = (- \theta_1, - \theta_2)}\right\}$$. This yields a pillowcase:

• Define $$\chi^*(K) \coloneqq{\operatorname{Homeo}}(\pi, {\operatorname{SU}}_2)$$ where $$\pi \coloneqq\pi_1(S^3\setminus\nu(K))$$ and we take homeomorphisms that do not have abelian image, modulo conjugation as before.
• For $$\alpha\in [0, 1/2]$$, define $$\chi_\alpha^*(K) \coloneqq\left\{{\phi\in \chi^*(K) {~\mathrel{\Big\vert}~}\phi(\mu) \sim { \begin{bmatrix} {e^{2\pi i \alpha}} & {0} \\ {0} & {e^{-2\pi i \alpha}} \end{bmatrix} } }\right\}$$ 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 $${{\partial}}\nu(K)$$.
• 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 $$\chi_\alpha^*(K) = r^{-1}(C_\alpha)$$ where $$C_\alpha \coloneqq\left\{{(\alpha, t) {~\mathrel{\Big\vert}~}t\in [0, 1/2]}\right\} \subseteq P$$ 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: $$# \chi_\alpha^*(K) \leq {1\over 2}{\left\lvert {\sigma_K(e^{4\pi i \alpha})} \right\rvert}$$.
• Say $$K$$ is $${\operatorname{SU}}_2{\hbox{-}}$$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 $${\operatorname{SU}}_2{\hbox{-}}$$basic knots?
• Theorem: if $$K$$ is $${\operatorname{SU}}_2{\hbox{-}}$$basic, $$S:K\to K'$$ a concordance, then any $$\phi\in \chi^*(K)$$ extends to $$\tilde\phi\in \mathop{\mathrm{Hom}}_{\mathsf{Grp}}(\pi_1(S^4\times I \setminus S), {\operatorname{SU}}_2)$$.
• Proof uses instanton Floer homology, Yang-Mills gauge theory.
• Definition of ribbon concordance: $$S:K\to K'$$ is ribbon if, noting $$S \subseteq I \times S^3$$, 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 $${\operatorname{SU}}_2$$ 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 $${\operatorname{SU}}_2$$ basic knots:
• Cables $$C_{pqk + 1, k}(T_{p, q})$$..
• Twisted torus knots $$T(3, 3n-1, 2, 1)$$. Interestingly, these are all $$L{\hbox{-}}$$space knots.
• Question: can $${\operatorname{SU}}_2$$ basic knots be classified using Dehn surgery.