The spiral index of knots

TL;DR

This paper introduces two new knot invariants, the spiral index and the projective superbridge index, which relate to classical invariants and help classify knots with specific curvature-torsion properties.

Contribution

The paper defines the spiral index and projective superbridge index, linking them to classical invariants and applying them to classify knots with a curvature-torsion invariant of 6 pi.

Findings

Introduced the spiral index and projective superbridge index as new knot invariants.

Established relationships between these invariants and classical invariants.

Determined all knots with curvature-torsion invariant equal to 6 pi.

Abstract

Two new invariants that are closely related to Milnor's curvature-torsion invariant are introduced. The first, the spiral index of a knot, captures the minimum number of maxima among all knot projections that are free of inflection points. This invariant is closely related to both the bridge and braid index of the knot. The second, the projective superbridge index, provides a method of counting the greatest number of maxima that occur in a given knot projection. In addition to investigating how these invariants are related to the classical invariants, we utilize them to determine all knots with curvature-torsion invariant equal to 6 pi.