Lambda lengths in The figure eight knot complement

TL;DR

This paper characterizes lambda lengths and inter-cusp distances in the figure eight knot complement's hyperbolic structure, revealing they correspond to Eisenstein integers and their norms, respectively.

Contribution

It identifies the set of lambda lengths as Eisenstein integers and relates inter-cusp distances to their norms, providing a precise algebraic description.

Findings

Lambda lengths are Eisenstein integers.

Inter-cusp distances are norms of Eisenstein integers.

The correspondence uses spinor and spin-decorated horosphere techniques.

Abstract

In the complete hyperbolic structure on the complement of the figure eight knot, we determine the set of lambda lengths from the maximal cusp to itself. Using the correspondence between spinors and spin-decorated horospheres, we show that these lambda lengths are precisely the Eisenstein integers, up to multiplication by a unit. We also show that the inter-cusp distances from the maximal cusp to itself are precisely the norms of Eisenstein integers.