On the semi-threading of knot diagrams with minimal overpasses

TL;DR

This paper introduces semi-threading circles for knot diagrams, linking minimal overpasses to braid index, and proves they are equal for torus knots, providing new insights into knot complexity.

Contribution

It defines semi-threading circles for minimal knot diagrams and establishes their relation to braid index, especially showing equality for torus knots.

Findings

Braid index is at least the minimal number of overpasses for any nontrivial knot.

For torus knots, the braid index equals the minimal overpasses.

Semi-threading circles serve as a bridge between diagram overpasses and braid representations.

Abstract

Given a knot diagram $D$, we construct a semi-threading circle for it which can be an axis of $D$ as a closed braid depending on knot diagrams. In particular, we consider semi-threading circles for minimal diagrams of a knot with respect to overpasses which give us some information related to the braid index. By this notion, we show that, for every nontrivial knot $K$, the braid index $b(K)$ of $K$ is not less than the minimum number $l(K)$ of overpasses of diagrams. Moreover, they are the same for a torus knot.