On wrapping number, adequacy and the crossing number of satellite knots

TL;DR

This paper establishes a new lower bound for the minimal crossing number of satellite knots based on the wrapping number and the crossing number of the companion knot, with specific results for adequate and alternating companions.

Contribution

It provides the tightest known lower bound relating satellite knot crossing number to the companion's crossing number and wrapping number, improving previous bounds.

Findings

New lower bound: c(Sat(P,C)) > (M^2/2) * c(C)

Bound proven for adequate companion knots

Refined bound for alternating companion knots

Abstract

In this work we establish the tightest lower bound up-to-date for the minimal crossing number of a satellite knot based on the minimal crossing number of the companion used to build the satellite. If $M$ is the wrapping number of the pattern knot, we essentially show that $c(Sat(P,C))>\frac{M^2}{2}c(C)$. The existence of this bound will be proven when the companion knot is adequate, and it will be further tuned in the case of the companion being alternating.