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 H(2)-move is an unknotting operation of a knot, which is performed by adding a half-twisted band. We define the H(2)-Gordian distance of two knots to be the minimum number of H(2)-moves needed to transform one into the other. We give several methods to estimate the H(2)-Gordian distance of knots. Then we give a table of H(2)-Gordian distances of knots with up to 7 crossings.

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.

We consider an $H(2)$-move between two torus links of type $(2,2n)$, $T_{2n}$. We give some necessary conditions for two links $T_{2n}$ and $T_{2m}$ which are related by a single $H(2)$-move. In particular, we show that $T_{2n}$ is obtained by a single $H(2)$-move from the Hopf link if and only if $n=\pm 1$, $\pm 3$, and $T_{2n}$ is obtained by a single $H(2)$-move from the trivial 2-component link if and only if $n=0$, $\pm 2$,