Planar and spherical stick indices of knots

TL;DR

This paper introduces planar and spherical stick indices as new 2D invariants of knots, providing bounds, constructions, and distinguishing features, including examples where these indices differ for certain knots.

Contribution

It defines and analyzes planar and spherical stick indices, offering bounds, explicit constructions, and demonstrating their ability to distinguish knots like granny and square knots.

Findings

Spherical stick index distinguishes granny and square knots.

Bounds established for these indices in terms of other invariants.

Constructed examples for torus knots and knot compositions.

Abstract

The stick index of a knot is the least number of line segments required to build the knot in space. We define two analogous 2-dimensional invariants, the planar stick index, which is the least number of line segments in the plane to build a projection, and the spherical stick index, which is the least number of great circle arcs to build a projection on the sphere. We find bounds on these quantities in terms of other knot invariants, and give planar stick and spherical stick constructions for torus knots and for compositions of trefoils. In particular, unlike most knot invariants,we show that the spherical stick index distinguishes between the granny and square knots, and that composing a nontrivial knot with a second nontrivial knot need not increase its spherical stick index.