Journal of Knot Theory and Its Ramifications

In this paper, we consider the nullification numbers $n_i(L)$ of the 249 prime knots up to 10 crossings to an unlink with $i$ components. This extends an existing table of knot nullification numbers up to nine crossings in [C. Ernst, A. Montemayor and A. Stasiak, Nullification of small knots, Prog. Theor. Phys. Suppl. 191 (2011) 66–77].

We give explicit resolutions of all finite-dimensional simple $U_q({𝔰𝔩}_3)$-modules. We use these resolutions to categorify the colored ${𝔰𝔩}_3$-invariant of framed links via a complex of complexes of graded $\mathbb{Z}$-modules.

We illustrate that there are knots for which Heegaard knot Floer homology and Khovanov homology are identical but the Alexander module and torsion invariants differ. The examples are certain symmetric unions. We also give examples of similar flavor, concerning the Kauffman and Q-polynomials in place of the classical homological invariants. This shows there are nonmutant knots with the same knot Floer and Khovanov homology.

Kauffman defines an affine index polynomial invariant for virtual knots. The invariant is induced from a numerical invariant called an $n$-writhe. In this paper, we provide the difference of the values obtained from invariants between two virtual knots which can be transformed into each other by a single forbidden move. As a result, we make it possible for many virtual knots to determine the unknotting numbers by forbidden moves.

In this paper, we construct an algorithm to produce canonical grid diagrams of cable links and Whitehead doubles, which give sharper upper bounds of the arc index of them. Moreover, we determine the arc index of $2$-cable links and Whitehead doubles of all prime knots with up to eight crossings.

All knots have been shown to be isotopic to the unknot using a process known as virtualization. We extend and adapt this process to show that, up to fused isotopy, classical links are classified by their linking numbers. We provide an algebraic proof, utilizing Alexander’s Theorem and some simple results about the pure braid group.

Let $K$ be a link of Conway’s normal form $C(m)$, $m\ge 0$, or $C(m,n)$ with $mn>0$, and let $D$ be a trigonal diagram of $K.$ We show that it is possible to transform $D$ into an alternating trigonal diagram, so that all intermediate diagrams remain trigonal, and the number of crossings never increases.

We study the degree of polynomial representations of knots. We obtain the lexicographic degree for two-bridge torus knots and generalized twist knots. The proof uses the braid theoretical method developed by Orevkov to study real plane curves, combined with previous results from [Chebyshev diagrams for two-bridge knots, Geom. Dedicata 150 (2010) 405–425; E. Brugallé, P.-V. Koseleff, D. Pecker, Untangling trigonal diagrams, to appear in J. Knot Theory and its Ramifications]. We also give a sharp lower bound for the lexicographic degree of any knot, using real polynomial curves properties.