Journal of Knot Theory and Its Ramifications

Using a Heegaard diagram for the pullback of a knot K ⊂ S³ in its double branched cover Σ₂(K), we give a combinatorial proof for the invariance of the associated knot Floer homology over ℤ.

We offer an alternative construction of Roberts' totally twisted Khovanov homology and prove that for knots, it agrees with δ-graded reduced characteristic-2 Khovanov homology.

Nelson and Kanenobu showed that forbidden moves unknot any virtual knot. Similarly a long virtual knot can be unknotted by a finite sequence of forbidden moves. Goussarov, Polyak and Viro introduced finite type invariants of virtual knots and long virtual knots and gave combinatorial representations of finite type invariants. We introduce Fⁿ-moves which generalize the forbidden moves. Assume that two long virtual knots K and K′ are related by a finite sequence of Fⁿ-moves. We show that the values of the finite type invariants of degree 2 of K and K′ are congruent modulo n and give a lower bound for the number of Fⁿ-moves needed to transform K to K′.

We show that there is a family of pseudo-Anosov braids independently parametrized by the braid index and the (canonical) length whose smallest conjugacy invariant sets grow exponentially in the braid index and linearly in the length.

We introduce the notion of quasi-triviality of quandles and define homology of quasi-trivial quandles. Quandle cocycle invariants are invariant under link-homotopy if they are associated with 2-cocycles of quasi-trivial quandles. We thus obtain a lot of numerical link-homotopy invariants.

Recent advances in understanding slicing properties of Bing doubles of knots have depended on properties of iterated covering links. We expand and refine this covering link calculus. Our main application here is a simplified proof of the following result of Cha and Kim: If the iterated Bing double of a knot K is slice, then K is algebraically slice. Further applications are included in joint work with Livingston studying 4-genera of Bing doubles. The techniques also appear in the work of Levine studying mixed Bing-Whitehead doubles.

Let L = K₁ ∪ K₂ be a 2-component link in S³ such that K₁ is a trivial knot. In this paper, we introduce for each n(∈ ℕ) a new bridge index b_K1=n([L]) of L called a constrained bridge index with respect to n-bridge K₁. Roughly speaking, b_K1=n([L]) is the minimum of the bridge numbers of the links ambient isotopic to L under the constraint that all of the bridge numbers of the components corresponding to K₁ are n. We give exact values b_K1=n([L]) (n = 1, 2, …) for particular constructed as follows: There exist unknotted solid tori V₁ ⊂ V₂ ⊂ ⋯ ⊂ V_m in S³, where the core of each V_j(j = 1, 2, …, m - 1) is a (1, α_j)-torus knot (α_j ∈ ℤ) in V_j+1, such that K₁ is the core of V₁, and is the core of the exterior of V_m.

We show that the crossing number of any link that is known to be quasi-alternating is less than or equal to its determinant. Based on this, we conjecture that the crossing number of any quasi-alternating link is less than or equal to its determinant. Thus if this conjecture is true, then it gives a new property of quasi-alternating links and easy obstruction to a link being quasi-alternating.