Dunfield, Friedl and Jackson's twisted Alexander polynomial conjecture for hyperbolic knots

Let KK be a hyperbolic knot, let g(K)g(K) denote its genus, and let ρholo:π(K)SL(2,C)\rho_{\rm holo}:\pi(K)\to\operatorname{SL}(2,\mathbb{C}) be a lift of the discrete and faithful representation defining the hyperbolic metric. Dunfield, Friedl and Jackson's conjecture. The twisted Alexander polynomial satisfies

ΔK,ρholo=4g(K)2.\Delta_{K,\rho_{\rm holo}}=4g(K)-2.

Furthermore, KK is fibered if and only if ΔK,ρholo\Delta_{K,\rho_{\rm holo}} is monic. This relates the holonomy twisted Alexander polynomial to the genus and proposes that it detects fibredness for hyperbolic knots; the supplied source does not indicate whether the assertion has been resolved.

Sources & referencesView supporting material

Primary source

Haimiao Chen, “Computing twisted Alexander polynomials for Montesinos links”, arXiv:1709.03116 (2021).

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