Narrow Results

We give some relationships of the Jones and Q polynomials between two links which are related by a band surgery. Then we consider two applications: The first one is to an evaluation of the ribbon-fusion number, the least fusion number of a ribbon knot. The second one is to DNA knot theory, helping us to understand the action of the Xer site-specific recombination at psi site.

An oriented 2-component link is called band-trivializable, if it can be unknotted by a single band surgery. We consider whether a given 2-component link is band-trivializable or not. Then we can completely determine the band-trivializability for the prime links with up to 9 crossings. We use the signature, the Jones and Q polynomials, and the Arf invariant. Since a band-trivializable link has 4-ball genus zero, we also give a table for the 4-ball genus of the prime links with up to 9 crossings. Furthermore, we give an additional answer to the problem of whether a (2n + 1)-crossing 2-bridge knot is related to a (2, 2n) torus link or not by a band surgery for n = 3, 4, which comes from the study of a DNA site-specific recombination.

In this paper, we consider the nullification numbers $n_i(L)$ of the 249 prime knots up to 10 crossings to an unlink with $i$ components. This extends an existing table of knot nullification numbers up to nine crossings in [C. Ernst, A. Montemayor and A. Stasiak, Nullification of small knots, Prog. Theor. Phys. Suppl.191 (2011) 66–77].

We give two criteria of links concerning a band surgery: The first one is a condition on the determinants of links which are related by a band surgery using Nakanishi’s criterion on knots with Gordian distance one. The second one is a criterion on knots with $H(2)$-Gordian distance two by using a special value of the Jones polynomial, where an $H(2)$-move is a band surgery preserving a component number. Then, we give an improved table of $H(2)$-Gordian distances between knots with up to seven crossings, where we add Zeković’s result.

In the paper, we study the incoherent nullification number $\bar{n}(L)$ of knots and links. We establish an upper bound on the incoherent nullification number of torus knots and links and conjecture that this upper bound is the actual incoherent nullification number of this family. Finally, we establish the actual incoherent nullification number of particular subfamilies of torus knots and links.

We give a lower bound of the crosscap number of alternating knots using band surgery. This lower bound is equal to the minimum number of the first Betti number, i.e. one minus the maximal Euler characteristic, over all (orientable and non-orientable) surfaces spanning a given alternating knot.

In this paper, by applying the results in [4], we shall compute lower bounds for the unknotting numbers of the knots obtained from certain links, for example, by band surgery.