The arc-faithful diagram conjecture for hyperbolic knots

Let KK be a hyperbolic knot, and call a diagram of KK arc-faithful when its associated octahedral coloring has the required non-pinched property and holonomy equal to the complete hyperbolic structure. Arc-faithful diagram conjecture. Every hyperbolic knot has an arc-faithful diagram. The conjecture is motivated by computational evidence: every hyperbolic knot with at most 1212 crossings has an arc-faithful diagram, and such diagrams are relevant to a general proof of the Volume Conjecture.

Sources & referencesView supporting material

Primary source

Calvin McPhail-Snyder, “Octahedral coordinates from the Wirtinger presentation”, arXiv:2404.19155 (2025).

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