Journal of Knot Theory and Its Ramifications

In this sequel to [A colored 𝔰𝔩(N)-homology for links in S³, preprint (2009), arXiv:0907.0695v5], we construct an equivariant colored 𝔰𝔩(N)-homology for links, which generalizes both the above mentioned paper and the paper by [D. Krasner, Equivariant sl(n)-link homology, preprint (2008), arXiv:0804.3751v2]. The construction is a straightforward generalization of the paper, [A colored 𝔰𝔩(N)-homology for links in S³]. The proof of invariance is based on a simple observation which allows us to translate the proof in the above mentioned paper into the new setting.

As an application, we prove that deformations over ℂ of the colored 𝔰𝔩(N)-homology are link invariants. We also construct a spectral sequence connecting the colored 𝔰𝔩(N)-homology to its deformations over ℂ, which generalizes the spectral sequence given in the papers by [B. Gornik, Note on Khovanov link cohomology, preprint (2004), arXiv:math.QA/0402266 and E. Lee, An endomorphism of the Khovanov invariant, Adv. Math.197(2) (2005) 554–586].

We use the Heegaard Floer obstructions defined by Grigsby, Ruberman, and Strle to show that forty-five of the sixty-six knots through eleven crossings whose concordance orders were previously unknown have infinite concordance order.

In the previous paper, the representativity of a non-trivial knot K was defined as

In this paper, we will show that a (p, q, r)-pretzel knot has the representativity 3 if and only if (p, q, r) is either ±(-2, 3, 3) or ±(-2, 3, 5). We also show that an algebraic knot has the representativity less than or equal to 3.

Let M be a closed 3-manifold with a given Heegaard splitting. We show that after a single stabilization, some core of the stabilized splitting has arbitrarily high distance with respect to the splitting surface. This generalizes a result of Minsky, Moriah, and Schleimer for knots in S³. We also show that in the complex of curves, handlebody sets are either coarsely distinct or identical. We define the coarse mapping class group of a Heegaard splitting, and show that if (S, V, W) is a Heegaard splitting of genus ≥2, then the coarse mapping class group of (S, V, W) is isomorphic to the mapping class group of (S, V, W).

There is a classical correspondence between edge-signed plane graphs and link diagrams. Determining component number of links corresponding to plane graphs may be one of the first problems in studying links by using graphs. There has been several early studies in this aspect, for example, the component number of links formed from 2-dimensional square lattices (4⁴) has been determined. In this paper, we determine the component number of links corresponding to 2-dimensional triangular (3⁶) and honeycomb (6³) lattices with free or cyclic boundary condition.

We define virtual braid groups of type B and construct a morphism from such a group to the group of isomorphism classes of some invertible complexes of bimodules up to homotopy.

In this paper, we give a bound for the Δ-unknotting number of a Whitehead double in terms of the unknotting number and a certain integral invariant of its companion knot. As applications, we show that the Δ-unknotting number of m-twisted Whitehead doubles of certain knots does not remember its companion knot, and is equal to the twist number m. We also give possible Δ-unknotting number of m-twisted Whitehead doubles whose companions are knots with unknotting number 1, certain twist knots, amphicheiral knots, and positive knots.