Prime knots whose arc index is smaller than the crossing number

TL;DR

This paper investigates conditions under which prime knots have an arc index smaller than their crossing number, providing explicit minimal grid diagrams for certain nonalternating knots with 13 and 14 crossings.

Contribution

It identifies cases where the arc index is strictly less than the crossing number and constructs minimal grid diagrams for specific prime nonalternating knots.

Findings

Arc index of alternating knots equals crossing number plus two.

Arc index of prime nonalternating knots is at most the crossing number.

Constructed minimal grid diagrams for some 13- and 14-crossing prime nonalternating knots.

Abstract

It is known that the arc index of alternating knots is the minimal crossing number plus two and the arc index of prime nonalternating knots is less than or equal to the minimal crossing number. We study some cases when the arc index is strictly less than the minimal crossing number. We also give minimal grid diagrams of some prime nonalternating knots with 13 crossings and 14 crossings whose arc index is the minimal crossing number minus one.