Journal of Knot Theory and Its Ramifications

We define invariants of surface-links, called generalized fundamental classes, and determine the values which the generalized fundamental classes of surface-links represented by diagrams with two or three triple points can take.

A generalization of the topological fundamental group is developed in order to construct a completion of Artin's braid group on infinitely many strands with respect to the following notion of convergence: b_n → id iff for each M > 0, eventually the first M strands of b_n are trivial.

The system of unoriented tangle equations and is completely solved for the tangles U and as a function of where K₁ and K₂ are 4-plats, and and rational tangles such that |f₁g₂ - g₁f₂| > 1.

The system of unoriented tangle equations and is completely solved for the tangles U and as a function of where K₁ and K₂ are 4-plats, and and rational tangles such that |f₁g₂ - g₁f₂| > 1. As an application, it is completely determined when one 4-plat can be obtained from another 4-plat via a signed crossing change.

Let V be the standard solid torus in S³. Let K_{p, 2} be the (p, 2)-torus knot in V such that K_{p, 2} meets a meridian disk D of V in two points with the winding number zero and the 2-string tangle TK_{p, 2} obtained by cutting along D is a rational tangle. We compute the Casson invariant of the cyclic covering space of S³ branched over a satellite knot whose companion is any 2-bridge knot D(b₁,…,b_2m) and pattern is (V, K_{p, 2}).

The Witten–Reshetikhin–Turaev invariant of classical link diagrams is generalized to virtual link diagrams. This invariant is unchanged by the framed Reidemeister moves and the Kirby calculus. As a result, it is also an invariant of the 3-manifolds represented by the classical link diagrams. This generalization is used to demonstrate that there are virtual knot diagrams with a non-trivial Witten–Reshetikhin–Turaev invariant and trivial 3-manifold fundamental group.

There are exactly four mutually non-isotopic unknotting tunnels τ_i, i = 1,2,3,4 for the pretzel knot P(-2,3,7). Moreover, there are at most 3 non-stabilized genus 3 Heegaard splittings.

We construct a linear representation of the group IA(F_n) of IA-automorphisms of a free group F_n, an extension of the Gassner representation of the pure braid group P_n. Although the problem of faithfulness of the Gassner representation is still open for n > 3, we prove that the restriction of our representation to the group of basis conjugating automorphisms Cb_n contains a non-trivial kernel even if n = 2. We construct also an extension of the Burau representation to the group of conjugating automorphisms C_n. This representation is not faithful for n ≥ 2.