An Index-Type Invariant of Knot Diagrams Giving Bounds for Unknotting Framed Unknots

TL;DR

This paper introduces the Self-Crossing Index (SCI), a new knot diagram invariant, and uses it to establish bounds for unknotting framed unknots, revealing complexities in framed Reidemeister moves and exploring relations with existing invariants.

Contribution

The paper presents the SCI invariant and demonstrates its effectiveness in bounding unknotting processes for framed unknots, also analyzing its relation to other known invariants.

Findings

SCI provides bounds for unknotting framed unknots.

Unknotting framed unknots can be more complex than regular unknots.

SCI's change under Reidemeister moves depends only on move orientation.

Abstract

We introduce a new knot diagram invariant called the Self-Crossing Index (SCI). Using SCI, we provide bounds for unknotting two families of framed unknots. For one of these families, unknotting using framed Reidemeister moves is significantly harder than unknotting using regular Reidemeister moves. We also investigate the relation between SCI and Arnold's curve invariant St, as well as the relation with Hass and Nowik's invariant, which generalizes cowrithe. In particular, the change of SCI under {\Omega}3 moves depends only on the forward/backward character of the move, similar to how the change of St or cowrithe depends only on the positive/negative quality of the move.