Garside theory and subsurfaces: some examples in braid groups

TL;DR

This paper investigates the size of the sliding circuit set in braid groups, proposing a polynomial bound related to braid length and strands, supported by constructed examples illustrating the geometric properties involved.

Contribution

It introduces a conjectured polynomial bound for the sliding circuit set size in rigid braids and provides explicit examples that support this bound and reveal geometric properties.

Findings

Proposes a polynomial bound of C·L^{N-2} for the sliding circuit set size.

Constructs example braids that realize the potential worst case.

Shows that large sliding circuit sets relate to geometric properties like subsurfaces and subsurface projections.

Abstract

Garside-theoretical solutions to the conjugacy problem in braid groups depend on the determination of a characteristic subset of the conjugacy class of any given braid, e.g. the sliding circuit set. It is conjectured that, among rigid braids with a fixed number of strands, the size of this set is bounded by a polynomial in the length of the braids. In this paper we suggest a more precise bound: for rigid braids with $N$ strands and of Garside length $L$, the sliding circuit set should have at most $C\cdot L^{N-2}$ elements, for some constant $C$. We construct a family of braids which realise this potential worst case. Our example braids suggest that having a large sliding circuit set is a geometric property of braids, as our examples have multiple subsurfaces with large subsurface projection; thus they are "almost reducible" in multiple ways, and act on the curve graph with small translation distance.