On two crossing numbers of algebraic knots under Hopf fibration

TL;DR

This paper investigates two different crossing numbers for algebraic knots under Hopf fibration, demonstrating their potential disparity and providing bounds for specific knot families.

Contribution

It establishes that the algebraic crossing number and the topological crossing number can differ arbitrarily and offers upper bounds for certain knot families.

Findings

$C_{alg}(K)-h(K)$ can be arbitrarily large

Upper bounds for $h$ on torus knots $T(2,n)$ and twist knots

Comparison of two crossing number notions for algebraic knots

Abstract

We answer a question posed by Fielder in [1] concerning two notions of crossing number for algebraic knots $K$ under Hopf fibration, one topological, denoted $h(K)$, the other coming from the realization of such knots around complex singularities, denoted $C_{alg}(K)$. We show that $C_{alg}(K)-h(K)$ can be arbitrarily large. We also give an upper bound for $h$ of some families of knots such as torus knots $T(2,n)$, twist knots and their mirror images.