Volume is bounded by determinant for alternating hyperbolic links

Let KK be an alternating hyperbolic link, let vol(K)\operatorname{vol}(K) denote its hyperbolic volume, and let det(K)\det(K) denote its determinant. Alternating volume–determinant conjecture.

vol(K)<2πlogdet(K).\operatorname{vol}(K)<2\pi\log\det(K).

The inequality had been verified for alternating knots through 16 crossings and for the specified families of weaving knots and links. The source also says the constant is sharp, but the general conjecture remains open.

Sources & referencesView supporting material

Primary source

Abhijit Champanerkar, Ilya Kofman and Jessica S. Purcell, “Density spectra for knots”, arXiv:1506.05841 (2015).

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