Journal of Knot Theory and Its Ramifications

We develop formulae for local measures of writhing and linking density for closed, circular DNA.

We apply skein relations to derive recurrence relations for the HOMFLY polynomials of a family of Brunnian links. Solution of the recurrence relations yields closed-form formulas for the HOMFLY and hence the Jones polynomials.

A smooth knot is non-trivial precisely if it cannot be unlinked from every small deformation of itself. White's formula expresses the linking number of a representative of a knot and a deformation of it as a sum of two, not necessarily integral, terms — a writhe and a twist. An elementary proof is given of this formula. A byproduct of the proof is an easy derivation of the expression for the writhe of the knot in terms of standard invariants of the curve and the writhe of the knot diagram obtained by projection onto a plane.

A double-torus knot is a knot embedded in a genus two Heegaard surface in S³. We consider double-torus knots L such that is connected, and consider fibred knots in various classes.

An upper bound of the superbridge index of the connected sum of two knots is given in terms of the braid index of the summands. Using this upper bound and minimal polygonal presentations, we give an upper bound in terms of the superbridge index and the bridge index of the summands when they are torus knots. In contrast to the fact that the difference between the sum of bridge indices of two knots and the bridge index of their connected sum is always one, the corresponding difference for the superbridge index can be arbitrarily large.

In this note, we will study on equivalence relations for links and we will give a necessary condition for equivalence in terms of the Alexander polynomial and show their differences.

Recently it has been proved by K. Taniyama, S. Yamada and the author that for any natural number n and any knot K, there are infinitely many unknotting number one knots, all of whose Vassiliev invariants of order less than or equal to n coincide with those of K. In this paper we give another proof of this result by using web diagrams.

In this paper we find a method to compute the classical Seifert-Threlfall linking number for rational homology spheres without using 2-chains bounded by the curves in question. By using a Heegaard diagram for the manifold, we describe link isotopy combinatorially using the three traditional Reidemeister moves along with a fourth move which is essentially a Kirby move along the characteristic curves. This result is mathematical folklore which we set in print. We then use this combinatorial description of link isotopy to develop and prove the invariance of linking numbers. Once the linking numbers are in place, matrix invariants such as the Alexander polynomial can be computed.