The rank of a warping matrix

Abstract

The warping matrix has been defined for knot projections and knot diagrams by using warping degrees. In particular, the warping matrix of a knot diagram represents the knot diagram uniquely. In this paper we show that the rank of the warping matrix is one greater than the crossing number. We also discuss the linearly independence of knot diagrams by considering the warping incidence matrix.Theorem 1.1. Let P be an oriented knot projection on S 2 , and M(P ) the warping matrix of P . We have the following equality:where c(P ) is the crossing number of P .Let D be an oriented knot diagram on S 2 , and c(D) the crossing number of D. Let M (D) be the warping matrix of D without signs, which is mentioned concretely in Section 2. We also have the following theorem:The rest of this paper is organized as follows: In Section 2, we review the warping matrix. In Section 3, we prove Theorem 1.1 and 1.2. In Section 4, we define the warping incidence matrix of a knot diagram. In Section 5, we investigate linearly independence for knot diagrams.

Axis systems of link projections

“…For the standard projection P of a twist knot with n = 2m crossings, we have the following two axes with the words For the standard projection P of a twist knot with n = 2m + 1 crossings, we have the two axes It is unknown if there exists a one-to-one correspondence between axis systems and link projections. If exists, then axis system can be used as a representation of link projections and diagrams such as chord diagram, warping matrix ( [5]), warping incidence matrix ( [3]) and so on. a non-simple axis with length n, give y n .…”

Region Polynomial of a Knot Diagram

“…The warping polynomial was dependent on both crossing and orientation of the knot diagram. She also introduced about warping matrix and its rank [12,13]. In this paper, we introduced a new polynomial defined using region in diagram.…”