Prime knots with arc index up to 11 and an upper bound of arc index for non-alternating knots

TL;DR

This paper classifies all prime knots with an arc index up to 11 and establishes an upper bound of arc index based on crossing number for non-alternating knots, aiding knot complexity understanding.

Contribution

It identifies all prime knots with arc index up to 11 and proves an upper bound for arc index in non-alternating knots, advancing knot classification methods.

Findings

All prime knots with arc index ≤ 11 identified.

Crossing number bounds arc index for non-alternating knots.

Arc index determined for prime knots up to twelve crossings.

Abstract

Every knot can be embedded in the union of finitely many half planes with a common boundary line in such a way that the portion of the knot in each half plane is a properly embedded arc. The minimal number of such half planes is called the arc index of the knot. We have identified all prime knots with arc index up to 11. We also proved that the crossing number is an upperbound of arc index for non-alternating knots. As a result the arc index is determined for prime knots up to twelve crossings.