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The knot-state microsupport conjecture

Let KK be a knot in S3S^3, let EKE_K be its exterior, and let Σ\Sigma be its peripheral torus. Let EE be the phase space used to quantize the character variety of Σ\Sigma, with projection π:E→E/R\pi:E\to E/R, and let r:M(EK)→M(Σ)r:{\mathcal{M}}(E_K)\to {\mathcal{M}}(\Sigma) be restriction. For the knot-state sequence (Zk(EK))k(Z_k(E_K))_k, write MS⁡(Zk(EK))\operatorname{MS}(Z_k(E_K)) for its microsupport.

Knot-state microsupport conjecture.

π(MS⁡(Zk(EK)))⊂r(M(EK)).\pi\bigl(\operatorname{MS}(Z_k(E_K))\bigr)\subset r({\mathcal{M}}(E_K)).

A stronger version asserts equality, but the stated inclusion is the conjecture. It is proved for the trivial knot and, in the weaker pointwise formulation, for the figure-eight knot; the general statement remains open. The equality is known for the trivial knot, while the paper reports no counterexample to equality in general.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Knot-state microsupport conjecture

    Let KK be a knot with exterior EKE_K and peripheral torus Σ\Sigma. Let Zk(EK)Z_k(E_K) be its family of knot states, and let r:M(EK)→M(Σ)r:{\mathcal M}(E_K)\to{\mathcal M}(\Sigma) be the restriction map between the corresponding character moduli spaces. The microsupport of an admissible family is the set outside which the family is rapidly decreasing in every power of kk. Microsupport conjecture. The microsupport of (Zk(EK))(Z_k(E_K)) is contained in r(M(EK))r({\mathcal M}(E_K)). The text states that this conjecture was proved for the figure-eight knot and for torus knots, but does not give a general proof.

    source: Laurent Charles and Julien Marche, “Knot state asymptotics II, Witten conjecture and irreducible representations”, arXiv:1107.1646 (2011).

Sources & referencesView supporting material

Primary source

Laurent Charles and Julien Marche, “Knot state asymptotics I, AJ Conjecture and abelian representations”, arXiv:1107.1645 (2011).

character-varietiesknot-theorymicrolocal-analysis

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