Signed mosaic graphs and alternating mosaic number of knots

Abstract

Lomonaco and Kauffman introduced knot mosaics in their work on quantum knots. This definition is intended to represent an actual physical quantum system. A knot n-mosaic is an n×n matrix of 11 kinds of specific mosaic tiles representing a knot or a link. In this paper, we consider the alternating mosaic number of an alternating knot K which is defined as the smallest integer n for which K is representable as a reduced alternating knot n-mosaic. We define a signed mosaic graph and a diagonal grid graph and construct Hamiltonian cycles derived from the diagonal grid graphs. Using the cycles, we completely determine the alternating mosaic number of torus knots of type (2,q) for q≥2, which grows in an order of q1/2.