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.

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