Infinitely many knots with non-integral trace

TL;DR

This paper demonstrates the existence of infinitely many hyperbolic knots with complements containing elements whose traces are algebraic non-integers, revealing new algebraic properties of knot groups.

Contribution

It proves the existence of infinitely many hyperbolic knots with non-integral trace elements in their fundamental groups, a novel algebraic property.

Findings

Infinitely many hyperbolic knots with non-integral trace elements.

Existence of non-homeomorphic knot complements with algebraic non-integer traces.

New insights into the algebraic structure of knot groups.

Abstract

We prove that there are infinitely many non-homeomorphic hyperbolic knot complements $S^3\setminus K_i = \mathbb{H}^3/\Gamma_i$ for which $\Gamma_i$ contains elements whose trace is an algebraic non-integer.