The unoriented band unknotting numbers of torus knots

TL;DR

This paper proves that for torus knots, the minimal number of unoriented band surgeries needed to unknot them equals a specific calculable invariant called the pinch number, using knot Floer homology techniques.

Contribution

It establishes the equality between the unoriented band unknotting number and the pinch number for torus knots, linking geometric and algebraic invariants.

Findings

Unoriented band unknotting number equals pinch number for torus knots.

Pinch number can be computed from torus knot parameters.

Uses torsion order of unoriented knot Floer homology in proof.

Abstract

The unoriented band unknotting number of a knot is the minimum number of oriented or non-oriented band surgeries that turn the knot into the unknot. Batson introduced a certain non-oriented band surgery for a torus knot. The minimum number of these operations required to turn a torus knot into the unknot is called the pinch number, and it can be easily calculated from the parameters of the torus knot. In this paper, we show that the unoriented band unknotting number and the pinch number coincide for torus knots. In the proof, we use the torsion order of the unoriented knot Floer homology.