A family of pseudo-Anosov braids with large conjugacy invariant sets

TL;DR

This paper constructs a family of pseudo-Anosov braids demonstrating that the conjugacy invariant sets grow exponentially with the braid index, indicating the conjugacy problem remains computationally hard as the braid complexity increases.

Contribution

It introduces a new family of pseudo-Anosov braids with quantifiable growth in conjugacy invariant sets, highlighting limitations of current algorithms for the conjugacy problem.

Findings

Conjugacy invariant sets grow exponentially with braid index

Growth is linear in the braid length

Conjugacy problem remains exponential in complexity

Abstract

We show that there is a family of pseudo-Anosov braids independently parameterized by the braid index and the (canonical) length whose smallest conjugacy invariant sets grow exponentially in the braid index and linearly in the length and conclude that the conjugacy problem remains exponential in the braid index under the current knowledge.