Weights of essential surfaces in 2-bridge knot complements

TL;DR

This paper explicitly determines the structure of Serre trees for essential surfaces in 2-bridge knot complements and derives formulas for counting associated ideal points, advancing understanding of 3-manifold invariants.

Contribution

It provides a direct method to compute Serre trees and ideal points for all 2-bridge knots from their diagrams, improving previous abstract techniques.

Findings

Explicit structure of Serre trees for essential surfaces

Formula for counting ideal points associated with incompressible surfaces

Application to all 2-bridge knots from their diagrams

Abstract

Understanding ideal points in the character varieties of knot complements has led to a number of important invariants for 3-manifolds. Ohtsuki (1994) counted the ideal points for character varieties of 2-bridge knot complements, and he made his techniques more concrete in an ensuing paper (1996). Drawing on these ideas, for all 2-bridge knots $K$, we explicitly determine the structure of a Serre tree for each essential surface in the knot complement directly from the knot diagram. Using these trees, we derive a formula for the number of ideal points associated to each incompressible surface.