Dunfield--Friedl--Jackson conjecture for the discrete faithful representation

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Let K⊂S3K\subset S^3 be a hyperbolic oriented knot, and let πK\pi_K denote its knot group. Let α ⁣:πK→SL⁡(2,C)\alpha\colon \pi_K\to \operatorname{SL}(2,\mathbb{C}) be a lift of the discrete and faithful representation. Dunfield--Friedl--Jackson conjecture. Then

deg⁡(τ(K,α)(t))=2(2genus⁡(K)−1).\deg\bigl(\tau(K,\alpha)(t)\bigr)=2(2\operatorname{genus}(K)-1).

Furthermore, KK is fibered if and only if τ(K,α)(t)\tau(K,\alpha)(t) is monic.

This conjecture would specify a single representation detecting both genus and fiberedness, replacing the existential representations in the known general theorems. The source refers to it as open.

References

Primary source

Jérôme Dubois, Stefan Friedl and Wolfgang Lück, “Three flavors of twisted invariants of knots”, arXiv:1410.6924 (2014).

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