Journal of Knot Theory and Its Ramifications

S-equivalence of classical knots is investigated, as well as its relationship with mutation and the unknotting number. Furthermore, we identify the kernel of Bredon's double suspension map, and give a geometric relation between slice and algebraically slice knots. Finally, we show that every knot is S-equivalent to a prime knot.

The transforms of two oriented parallel strands to a k-half twist of two strands are called t_k-move and -move respectively depending on the orientations of the two strands. In this paper we give criterions to detect whether a knot K can be transformed to a knot K' by t_2k-moves and -moves respectively and if so, we give some results on how many moves are needed in these transformations respectively, by using some Vassiliev invariants. Moreover we give a relation between the Δ-move and the t_2k-move by considering the coefficient of z² in the Conway polynomial of a knot, which is a Vassiliev invariant of degree 2.

We exhibit a graph, G₁₂, that in every spatial embedding has a pair of non-splittable 2 component links sharing no vertices or edges. Surprisingly, G₁₂ does not contain two disjoint copies of graphs known to have non-splittable links in every embedding. We exhibit other graphs with this property that cannot be obtained from G₁₂ by a finite sequence of Δ-Y and/or Y-Δ exchanges. We prove that G₁₂ is minor minimal in the sense that every minor of it has a spatial embedding that does not contain a pair of non-splittable 2 component links sharing no vertices or edges.

An accidental surface in the exterior of a knot in the 3-sphere is a closed essential surface for which there is an annulus in the knot exterior X connecting a loop in the surface and a nontrivial loop in ∂X, the peripheral torus of the knot. The isotopy class of the loop in ∂X is called an accidental slope; each accidental surface has a unique accidental slope. It is known that accidental slopes are integral or 1/0, and there is a knot with two accidental slopes 0 and 1/0. We show that for any integer m≥0, there is a hyperbolic knot which has m+1 accidental surfaces with accidental slopes 0,1,…,m.

Two spatial embeddings of a graph are said to be edge-homotopic if they are transformed into each other by self-crossing changes and ambient isotopies. We show that two spatial embeddings of the complete graph on four vertices are edge-homotopic if and only if they have the same α-invariant.

The purpose of this note is to give a formula for the Vassiliev invariant of type three for 2-bridge knots, calculating without using a continued fraction expansion.

The paper computes the noncommutative A-ideal of the figure-eight knot, a noncommutative generalization of the A-polynomial. It is shown that if a knot has the same A-ideal as the figure-eight knot, then all colored Kauffman brackets are the same as those of the figure eight knot.

We show that for each positive integer n>1 a simple C_n-move on knots, introduced by Habiro, is equivalent to the (non-oriented) insertion in a knot via a tangle map the geometric pure braid n-commutator of the form p_n=[p_n-1,n,[p_n-2,n-1,…, [p_1,2,p_0,1]…]∈P_n+1, where P_n+1 is the subgroup of pure braids of the braid group B_n+1 on n+1 strands. We relate the insertions of the pure braid commutators in knots to the operations of inverting and mirroring the knots.

We show that there are infinitely many two component links in S³ whose complements have weakly reducible and irreducible non-minimal genus Heegaard splittings, yet the construction given in the theorem of Casson and Gordon does not produce an essential closed surface. The situation for manifolds with a single boundary component is still unresolved though we obtain partial results regarding manifolds with a non-minimal genus weakly reducible and irreducible Heegaard splitting.