On an Inequality Between Unknotting Number and Crossing Number of Links

Abstract

It is well-known that for any link L, twice the unknotting number of L is less than or equal to the crossing number of L. Taniyama characterized the links which satisfy the equality. We characterize the links where twice the unknotting number is equal to the crossing number minus one. As a corollary, we show that for any link L with twice the unknotting number of L is greater than or equal to the crossing number of L minus two, every minimal diagram of L realizes the unknotting number.

The Warping Degree of a Knot Diagram

“…Hanaki and Kanadome characterized the link diagrams D which satisfy u(D) = (c(D) − 1)/2 as follows [4]: Let D be a knot diagram with u(D) = (c(D) − 2)/2. They showed in [1] that for any crossing point p of D, one of the components of D p is a reduced alternating diagram of a (2, p)-torus knot and the other component of D p has no self-crossings, where D p is the diagram obtained from D by smoothing at p. In addition, they showed that any minimal crossing diagram D of a knot K with u(K) = (c(K) − 2)/2 satisfies the above condition.…”

“…We give an example of Lemma 3.3. The equality holds if and only if D has property C. 4 The linking warping degree and linking number…”

“…Hanaki and Kanadome characterized the link diagrams D which satisfy u(D) = (c(D) − 1)/2 as follows [4]: By adding to (4), we have the following corollary: Corollary 6.1. For a knot diagram E, we have…”