On non-isotopic spanning surfaces for a class of arborescent knots

TL;DR

This paper demonstrates that certain arborescent knots can bound exponentially many non-isotopic minimal genus spanning surfaces, revealing complex surface structures within knot theory.

Contribution

It introduces a sequence of prime arborescent knots with exponentially many non-isotopic minimal genus spanning surfaces, expanding understanding of knot surface diversity.

Findings

A sequence of prime knots with at least 2^{2n-1} non-isotopic minimal spanning surfaces.

Use of methods from Hedden, Juhasz, and Sarkar to construct these surfaces.

Exponential growth in the number of distinct minimal genus surfaces for the knots.

Abstract

We use the methods of Hedden, Juhasz, and Sarkar to exhibit a set of arborescent knots that bound large numbers of non-isotopic minimal genus spanning surfaces. In particular, we describe a sequence of prime knots K_{n} which will bound at least 2^{2n-1} non-isotopic minimal spanning surfaces of genus n.