A variation of McShane's identity for 2-bridge links

TL;DR

This paper presents a new variation of McShane's identity tailored for hyperbolic 2-bridge links, linking cusp shapes to complex translation lengths and characterizing end invariants of associated holonomy representations.

Contribution

It introduces a novel variation of McShane's identity for 2-bridge links and explicitly characterizes end invariants of $SL(2, ext{C})$-characters for these structures.

Findings

Derived a variation of McShane's identity for 2-bridge links

Explicitly determined the set of end invariants for relevant holonomy representations

Connected cusp shapes with complex translation lengths of simple loops

Abstract

We give a variation of McShane's identity, which describes the cusp shape of a hyperbolic 2-bridge link in terms of the complex translation lengths of simple loops on the bridge sphere. We also explicitly determine the set of end invariants of $SL(2,\mathbb{C})$-characters of the once-punctured torus corresponding to the holonomy representations of the complete hyperbolic structures of 2-bridge link complements.