Dunfield–Friedl–Jackson conjecture for hyperbolic knots

Let KK be a hyperbolic knot in S3S^3, let EK=S3ν(K)E_K=S^3\setminus\nu(K) be its exterior, and let x(K)x(K) denote its Thurston norm, equivalently

x(K)=2genus(K)1.x(K)=2\,\mathrm{genus}(K)-1.

Let ρgeom\rho^{\mathrm{geom}} be a lift of the holonomy representation of EKE_K to SL2(C)\mathrm{SL}_2(\mathbb C), and let τ2,ρgeom(t)\tau_{2,\rho^{\mathrm{geom}}}(t) be the associated \mathbbmC2\mathbbm C^2-torsion polynomial. Dunfield–Friedl–Jackson conjecture.

x(K)=12degτ2,ρgeom(t),x(K)=\frac{1}{2}\deg\tau_{2,\rho^{\mathrm{geom}}}(t),

and the exterior EKE_K fibers over the circle if and only if τ2,ρgeom(t)\tau_{2,\rho^{\mathrm{geom}}}(t) is monic. The degree equality would sharpen the known torsion lower bound for the Thurston norm, while the monicity criterion predicts that the torsion polynomial detects fiberedness. It is known computationally for all hyperbolic knots with at most 15 crossings, but remains a conjecture in general.

Sources & referencesView supporting material

Primary source

Nathan M. Dunfield, Stavros Garoufalidis and Seokbeom Yoon, “1-loop equals torsion for fibered 3-manifolds”, arXiv:2304.00469 (2024).

Additional references

9 papers in this index state this conjecture (2013–2023). The statement above is taken from the most recent of them; the others are arXiv:2203.03239, arXiv:2108.08645, arXiv:1606.06360, arXiv:1511.00400, arXiv:1510.03216, arXiv:1412.4483, arXiv:1406.4626, arXiv:1301.1447.

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