Infinitely many knots whose Whitehead doubles have the trivial first coefficient Kauffman polynomial

We recall a skein relation of the first coefficient Kauffman polynomial for knots. By using the skein relation, we show that there exist infinitely many knots whose Whitehead doubles have the trivial first coefficient Kauffman polynomial.

A generalized skein relation for Khovanov homology and a categorification of the θ-invariant

The Jones polynomial is a famous link invariant that can be defined diagrammatically via a skein relation. Khovanov homology is a richer link invariant that categorifies the Jones polynomial. Using spectral sequences, we obtain a skein-type relation satisfied by the Khovanov homology. Thanks to this relation, we are able to generalize the Khovanov homology in order to obtain a categorification of the θ-invariant, which is itself a generalization of the Jones polynomial.

Crossing change on Khovanov homology and a categorified Vassiliev skein relation

Khovanov homology is a categorification of the Jones polynomial, so it may be seen as a kind of quantum invariant of knots and links. Although polynomial quantum invariants are deeply involved with Vassiliev (aka. finite type) invariants, the relation remains unclear in case of Khovanov homology. Aiming at it, in this paper, we discuss a categorified version of Vassiliev skein relation on Khovanov homology. More precisely, we will show that the “genus-one” operation gives rise to a crossing change on Khovanov complexes. Invariance under Reidemeister moves turns out, and it enables us to extend Khovanov homology to singular links. We then see that a long exact sequence of Khovanov homology groups categorifies Vassiliev skein relation for the Jones polynomials. In particular, the Jones polynomial is recovered even for singular links. We in addition discuss the FI relation on Khovanov homology.

New skein invariants of links

We study new skein invariants of links based on a procedure where we first apply a given skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using the given invariant. A skein invariant can be computed on each link solely by the use of skein relations and a set of initial conditions. The new procedure, remarkably, leads to generalizations of the known skein invariants. We make skein invariants of classical links, [Formula: see text], [Formula: see text] and [Formula: see text], based on the invariants of knots, [Formula: see text], [Formula: see text] and [Formula: see text], denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. We provide skein theoretic proofs of the well-definedness of these invariants. These invariants are also reformulated into summations of the generating invariants ([Formula: see text], [Formula: see text], [Formula: see text]) on sublinks of a given link [Formula: see text], obtained by partitioning [Formula: see text] into collections of sublinks. These summations exhibit the tight and surprising relationship between our generalized skein-theoretic procedure and the structure of sublinks of a given link.

Application of a skein relation to difference topology experiments

Difference topology is a technique used to study any protein that can stably bind to DNA. This technique is used to determine the conformation of DNA bound by protein. Motivated by difference topology experiments, we use the skein relation tangle model as a novel technique to study experiments using topoisomerase to study SMC proteins, a family of proteins that stably bind to DNA. The oriented skein relation involves an oriented knot, [Formula: see text], with a distinguished positive crossing; a knot [Formula: see text], obtained by changing the distinguished positive crossing of [Formula: see text] to a negative crossing; a knot, [Formula: see text], resulting from the non-orientation persevering resolution of the distinguished crossing; and a link [Formula: see text], the orientation preserving resolution of the distinguished crossing. We refer to [Formula: see text] as the skein quadruple. Topoisomerases are proteins that break one segment of DNA allowing a DNA segment to pass through before resealing the break. Recombinases are proteins that cut two segments of DNA and recombine them in some manner. They can act on direct repeat or inverted repeat sites, resulting in a link or knot, respectively. Thus, the skein quadruple is now viewed as [Formula: see text] circular DNA substrate, [Formula: see text] product of topoisomerase action, [Formula: see text] product of recombinase action on directed repeat sites, and [Formula: see text] product of recombinase action of inverted repeat sites.

A new two-variable generalization of the Jones polynomial

We present a new 2-variable generalization of the Jones polynomial that can be defined through the skein relation of the Jones polynomial. The well-definedness of this invariant is proved both algebraically and diagrammatically as well as via a closed combinatorial formula. This new invariant is able to distinguish more pairs of nonisotopic links than the original Jones polynomial, such as the Thistlethwaite link from the unlink with two components.

The Alexander polynomial of links in lens spaces

We show how the Alexander polynomial of links in lens spaces is related to the classical Alexander polynomial of a link in the 3-sphere, obtained by cutting out the exceptional lens space fiber. It follows from this relationship that a certain normalization of the Alexander polynomial satisfies a skein relation in lens spaces.