Mutation invariance of the Arc index for some montesinos knots

Abstract

Every tame knot or link can be embeded into a finite union of half-planes which have the z-axis as their common boundary, so that each half plane intersects the knot or link in a simple arc. Such an embedding is called an arc presentation and the minimal number of half-planes among all arc presentations is called the arc index of the knot or link. Moreover, we introduce an application of the arc index to the mosaic knots. Mutation is an operation on a knot diagram which replaces a two string tangle with its image under a $180 ^\circ C$ rotation. Mutation may change the knot types. For the alternating knots or links, mutations do not change the arc index. For nonalternating knots and links, some of the semi-alternating knots or links have the same property. We mainly focus on the problem of mutation invariance of the arc index for non alternating knots which are not semi-alternating. We found families of infinitely many mutant pairs/triples of Montesinos knots with the same arc index.