Trisection genus of knot traces

Abstract

We classify knot traces with trisection genus at most 2. We give infinitely many torus knots whose traces have trisection genus 3, and infinitely many hyperbolic knots whose traces have trisection genus 4. We also show that there exist infinite families of knots whose traces have arbitrarily large trisection genus. In addition, we determine or give sharp bounds for the trisection genus of the traces of several well-known knots, such as the figure-eight knot, the $(p,pq+1)$-torus knots, and the $(-2,3,2n+1)$-pretzel knots.

25 pages, 27 figures, Main results improved, exposition improved