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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 \(q: H_2(X){ {}^{ \scriptstyle\otimes_{{\mathbf{Z}}}^{2} } } \to {\mathbf{R}}\).

  • 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 \(\mathbf{Q}\kern-0.5pt\operatorname{HS}^3\), 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 \({\mathbf{C}}\) 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 \(P\coloneqq\mathop{\mathrm{Hom}}(\pi_1(T), {\operatorname{SU}}_2) \cong \mathop{\mathrm{Hom}}({\mathbf{Z}}{ {}^{ \scriptscriptstyle\times^{2} } }, {\operatorname{SU}}_2)/\sim\) 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.