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

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 Ω3 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. Contents1. Introduction 2. Index-type description of Arnold's curve invariants 2.1. Arnold's invariants 2.2. Indices with respect to a curve 2.3. Viro's formulas for J + and J -2.4. Shumakovich's formulas for St 3. The Self-Crossing Index and bounds for unknotting framed knots 3.1. Definition and properties of SCI 3.2. Bounds for unknotting via SCI 4. Comparison with the Hass-Nowik invariant 4.1. HN and its properties 4.2. Forward/backward, positive/negative and ascending/descending Ω3 moves 5. Appendix References