Journal of Knot Theory and Its Ramifications

We define A-polynomial n-tuple for a link of n-components and apply them to construct hyperbolic link manifolds with non-integral traces.

Any flat virtual link has a reduced diagram which satisfies a certain minimality, and reduced diagrams are related one another by a finite sequence of a certain Reidemeister move. The move preserves some numerical invariants of diagrams. So we can define numerical invariants for flat virtual links. One of them, the crossing number of a flat virtual knot K, coinsides with the self-intersection number of K as an essential geodesic loop on a hyperbolic closed surface. We also show an equation among these numerical invariants, basic properties by using the equation, and determine non-split flat virtual links with the crossing number up to three.

Let M be a compact, orientable, irreducible 3-manifold with an incompressible torus boundary T and γ a longitudinal slope on T, which bounds a surface F of genus 2. Suppose there exists a slope r that produces an essential 2-sphere S by Dehn filling. Let q be the minimal geometric intersection number between the essential 2-sphere and the core of the Dehn filling. Matignon and Sayari [5] proved that either q = 2 or the minimal geometric intersection number between γ and r is bounded by 3. Here, we construct an example of a hyperbolic 3-manifold realizing that bound.

How many Reidemeister moves do we need for unknotting a given diagram of the trivial knot? The absolute value of the writhe gives a lower bound of the number of Reidemeister I moves. That of a complexity of knot diagram "cowrithe" works for Reidemeister II, III moves.

In Appendix A, we give an example of an infinite sequence of diagrams D_n of the trivial knot with an O(n) number of crossings such that the author expects the number of Reidemeister moves needed for unknotting it to be O(n²). However, writhe and cowrithe do not prove this.

In 2002, Hrencecin and Kauffman defined a filamentation invariant on oriented chord diagrams that may determine whether the corresponding flat virtual knot diagrams are non-trivial. A virtual knot diagram is non-classical if its related flat virtual knot diagram is non-trivial. Hence filamentations can be used to detect non-classical virtual knots. We extend these filamentation techniques to virtual links with more than one component. We also give examples of virtual links that they can detect as non-classical.

The Jones polynomial is a well-defined invariant of virtual links. We observe the effect of a generalised mutation M of a link on the Jones polynomial. Using this, we describe a method for obtaining invariants of links which are also invariant under M.

The Jones polynomial of welded links is not well-defined in ℤ[q^1/4, q^-1/4]. Taking M = F_o allows us to pass to a quotient of ℤ[q^1/4, q^-1/4] in which the Jones polynomial is well-defined. We get the same result for M = F_u, so in fact, the Jones polynomial in this ring defines a fused isotopy invariant. We show it is non-trivial and compute it for links with one or two components.

In this article, we consider alternating knots on a closed surface in the 3-sphere, and show that these are not parallel to any closed surface disjoint from the prescribed one.

We relate the tree part of the Århus integral to Milnor's μ-invariants of string-links in homology balls thus generalizing results of Habegger and Masbaum.

In a paper of Menasco and Reid, it is conjectured that there exist no hyperbolic knots in S³ for which the complement contains a closed embedded totally geodesic surface. In this note, we show that one can get "as close as possible" to a counter-example. Specifically, we construct a sequence of hyperbolic knots {K_n} with complements containing closed embedded essential surfaces having principal curvatures converging to zero as n tends to infinity. We also construct a family of two-component links for which the complements contain closed embedded totally geodesic surfaces of arbitrarily large genera. In addition, we prove that a closed embedded surface with sufficiently small principal curvatures is not only quasi-Fuchsian (a result of Thurston's), but it is also either acylindrical or the boundary of a twisted I-bundle.