Multiple points of view: The simultaneous crossing number for knots with doubly transvergent diagrams

Abstract

The simultaneous crossing number is a new knot invariant which is defined for strongly invertible knots having diagrams with two orthogonal transvergent axes of strong inversions. Because the composition of the two inversions gives a cyclic period of order 2 with an axis orthogonal to the two axes of strong inversion, knot diagrams with this property have three characteristic orthogonal directions. We define the simultaneous crossing number, $\operatorname{sim}(K)$, as the minimum of the sum of the numbers of crossings of projections in the 3 directions, where the minimum is taken over all embeddings of $K$ satisfying the symmetry condition. Dividing the simultaneous crossing number by the usual crossing number, $\operatorname{cr}(K)$, of a knot gives a number $\ge 3$, because each of the 3 diagrams is a knot diagram of the knot in question. We show that $\liminf_{\operatorname{cr}(K) \to \infty} \operatorname{sim}(K)/\operatorname{cr}(K) \le 8$, when the minimum over all knots and the limit over increasing crossing numbers is considered.