Narrow Results

We introduce a new numerical invariant of knots and links from the descending diagrams. It is considered to live between the unknotting number and the bridge number. Some fundamental results and an incomplete table of the invariant for knots with 8-crossings or less are given.

The warping sum $e(K)$ of a knot $K$ is the minimal value of the sum of the warping degrees of a minimal diagram of $K$ with both orientations. In this paper, knots $K$ with $e(K)\le 3$ are characterized, and some knots $K$ with $e(K)=4$ are given.

For a knot, the ascending number is the minimum number of crossing changes which are needed to obtain an ascending diagram. We study the ascending number of a knot by analyzing the Conway polynomial. In this paper, we give a sharper lower bound of the ascending number of a knot and newly determine the ascending number for 26 prime knots up to 10 crossings.

For a knot, the ascending number is the minimum number of crossing changes which are needed to obtain an descending diagram. We study the $\mathrm{\Gamma }$-polynomial of knots with a given ascending number. We give a lower bound of the ascending numbers by using $\mathrm{\Gamma }$-polynomials. We estimate the ascending numbers for 65 prime knots up to 10 crossings by using $\mathrm{\Gamma }$-polynomials, Conway polynomials, and the determinants.

For a knot, the ascending number is the minimum number of crossing changes which are needed to obtain an descending diagram. We study relations between the ascending number and geometrical invariants; the crossing number, the genus and the bridge index. The main aim of this paper is to show that there exists a knot K such that $a(K)=2$ and $g(K)=n$, and that there exists a knot K’ such that $a(K^{\prime })\ge n$ and $g(K^{\prime })=1$ for any positive integer n. We also give an upper bound of the ascending number for a 2-bridge knot.