Taizo Kanenobu

Band surgery on knots and links, III

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 Zekovic’s result.

Two filtrations of ribbon 2-knots

Abstract We have constructed a ‘Vassiliev-like’ filtration on the free abelian group generated by the set of ribbon 2-knots in 4-space in two ways: one is from a ribbon 2-disk, and the other from a projection of a ribbon 2-knot onto a generic 3-space whose singular set consists of only double points. Each filtration determines a notion of finite type invariants for ribbon 2-knots. We prove that the two filtrations are the same, and thus, the two finite type invariants are coincident.

H(2)-GORDIAN DISTANCE OF KNOTS

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.

VASSILIEV KNOT INVARIANTS OF ORDER 6

We give a basis for the space of the Vassiliev knot invariants of order 6, which implies a sixth order Vassiliev knot invariant is determined by its HOMFLY and Kauffman polynomials. Furthermore, we show that there is a seventh order invariant which cannot be obtained from these polynomials of a knot.

Genus and Kauffman polynomial of a 2-bridge knot

In [6, Theorem 6], arbitrarily many 2-bridge knots sharing the same Jones polynomial are constructed. The construction is as follows (See Example 2.): Two 2-bridge knots 1022 and 10^ [15, Table] are obtained as symmetric skew unions [11] of 52. They have the same Jones polynomial [4] but have distinct genera, and so have distinct Alexander polynomials [3,14]. From these knots, we get four 2-bridge knots as symmetric skew unions. Continuing this construction, we have 2 distinct 2-bridge knots with the same Jones polynomial for any positive integer N. The question is whether the Alexander polynomials, or genera, of these 2-bridge knots are mutually distinct or not. In [7, Theorem 5], we constructed a pair of 2-bridge knots which have the same Kaufϊman polynomial and so have the same Jones polynomial, but have distinct Alexander polynomials. In fact, in the set of all the 2-bridge knots through 22 crossings, there are 239 pairs sharing the same Kauίfman polynomial, among which 58 pairs also share the same homfly polynomial and the rest have distinct Alexander polynomials [9]. Also in [8], we constructed arbitrarily many skein equivalent 2-bridge knots with the same Kauffman polynomial, and so they have the same homfly, Jones and Alexander polynomials. We refer [13] for the definition of the skein equivalence and the homfly polynomial, and [10] for the Kauffman and L polynomials and the writhe. In this paper, we prove:

RIBBON TORUS KNOTS PRESENTED BY VIRTUAL KNOTS WITH UP TO FOUR CROSSINGS

A ribbon torus knot embedded in the 4-space is presented by a welded virtual knot through the tube operation due to Shin Satoh. We make an attempt of classification of ribbon torus knots presented by virtual knots with up to four crossings, where we use the list of virtual knots enumerated by Jeremy Green. We mainly investigate the groups of virtual knots.

The homfly and the Kauffman bracket polynomials for the generalized mutant of a link

Abstract Conways mutation of a link is achieved by flipping a 2-strand tangle. Two mutant links share the same polynomial invariants. Anstee et al. generalized a mutation by flipping a many-string tangle which has rotational symmetry. We give another generalization of mutation: We consider a link constructed with 3-strand tangles T 1 , T 2 ,…, T n and a 2 n -strand tangle S . Under some conditions, by permuting T 1 , T 2 ,…, T n or flipping S , the homfly or the Kauffman bracket polynomial do not change.

Polynomial invariants of 2-bridge knots through 22 crossings

We calculate the homfly, Kauffman, Jones, Q, and Conway polynomials of 2-bridge knots through 22 crossings and list all the pairs sharing the same polynomial invariants.

AN UNKNOTTING OPERATION ON RIBBON 2-KNOTS

We define a local move on a ribbon 2-knot diagram, called an HC-move. We show that it is an unknotting operation for a ribbon 2-knot, and that the application of a single HC-move to a ribbon 2-knot changes the second derivative at t=1 of its normalized Alexander polynomial by either ±2 or 0. This result is applied to the calculation of the HC-unknotting numbers of ribbon 2-knots. We also consider a relation with a 1-handle unknotting operation.

WEAK UNKNOTTING NUMBER OF A COMPOSITE 2-KNOT

The weak unknotting number of a 2-knot K in a 4-sphere is the minimum number of elements g1, g2,…, gn of the knot group πK of K such that πK becomes the infinite cyclic group by adding the relations gix=xgi, where x is a meridian. We give an example of the nonadditivity of the weak unknotting number under connected sum of 2-knots.