An application of non-positively curved cubings of alternating links

TL;DR

This paper demonstrates that certain ideal triangulations of hyperbolic alternating link complements, derived from non-positively curved cubings, are non-degenerate, ensuring solutions to hyperbolicity equations and potential applications to the volume conjecture.

Contribution

It proves non-degeneracy of ideal triangulations from non-positively curved cubings of alternating links, facilitating hyperbolic structure analysis and applications to the volume conjecture.

Findings

Ideal triangulations are non-degenerate for hyperbolic alternating links.

Hyperbolicity equations have solutions corresponding to complete structures.

Potential implications for the volume conjecture.

Abstract

By using non-positively curved cubings of prime alternating link exteriors, we prove that certain ideal triangulations of their complements, derived from reduced alternating diagrams, are non-degenerate, in the sense that none of the edges is homotopic relative its endpoints to a peripheral arc. This guarantees that the hyperbolicity equations for those triangulations for hyperbolic alternating links have solutions corresponding to the complete hyperbolic structures. Since the ideal triangulations considered in this paper are often used in the study of the volume conjecture, this result has a potential application to the volume conjecture.