Journal of Knot Theory and Its Ramifications

We show that for certain unknotting operations λ their unknotting number u_λ is not additive with respect to connected sums. In general the unknotting number of the connected sum of two knots is smaller than or equal to the sum of the unknotting numbers of the summands; we provide examples where the inequality is strict.

Pulling back the weight systems associated with the exceptional Lie algebras and their standard representations by a modification of the universal Vassiliev-Kontsevich invariant yields link invariants; extending them to coloured 3-nets, we derive for each of them a skein relation.

The group BSL(k) of boundary cobordism classes of boundary k-string links is defined. An epimorphism from BSL(k) to a group of cobordism classes of matrices is defined. An action of a certain group of pure braids on BSL(k) provides all possible splittings for a given boundary k-link. A necessary and sufficient condition is given for two elements of BSL(k) to have the same closure as an F(k)-link (i.e., a boundary k-link with one of its splittings), up to F(k)-cobordism.

We prove that genus one, three-bridge knots are pretzel knots.

Contrary to the case of knots, the property that a link has an unlinking number one is reflected in its Conway polynomial. Examples of the determination of the unlinking number are given.

Our main theorem is that the local move between any non equivalent oriented trivial tangle diagrams is an unknotting operation.

We characterize composite double torus knots and links, 2-string composite double torus knots.

Let t be a virtual tangle embedded in a virtual knot or link k. If a prime p divides the determinants of both the numerator and the denominator closure of t, then p divides the determinant of k.

In 1993, Cairns and Elton gave a condition for an abstract Gauss word to be realized as the Gauss word of a planar curve. In this paper, we extend their result to curves on surfaces of higher genus. In particular, we give a solution in the case of torus T².

We construct a quantum deformation of a family of the Yang-Baxter equation solutions naturally arising from a Lie algebra .