Journal of Knot Theory and Its Ramifications

The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov–Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies as well to classical knots and classical knot cobordisms. We give an alternate formulation for the Manturov definition [34] of Khovanov homology [25], [26] for virtual knots and links with arbitrary coefficients. This approach uses cut loci on the knot diagram to induce a conjugation operator in the Frobenius algebra. We use this to show that a large class of virtual knots with unit Jones polynomial is non-classical, proving a conjecture in [20] and [10]. We then discuss the implications of the maps induced in the aforementioned theory to the universal Frobenius algebra [27] for virtual knots. Next we show how one can apply the Karoubi envelope approach of Bar-Natan and Morrison [3] on abstract link diagrams [17] with cross cuts to construct the canonical generators of the Khovanov–Lee homology [30]. Using these canonical generators we derive a generalization of the Rasmussen invariant [39] for virtual knot cobordisms and generalize Rasmussen’s result on the slice genus for positive knots to the case of positive virtual knots. It should also be noted that this generalization of the Rasmussen invariant provides an easy to compute obstruction to knot cobordisms in $S_g\times I\times I$ in the sense of Turaev [42].

Using the colored Kauffman skein relation, we study the highest and lowest $4n$ coefficients of the $n\mathrm{\text{th}}$ unreduced colored Jones polynomial of alternating links. This gives a natural extension of a result by Kauffman in regard with the Jones polynomial of alternating links and its highest and lowest coefficients. We also use our techniques to give a new and natural proof for the existence of the tail of the colored Jones polynomial for alternating links.

In this paper, we show that Montesinos links of the form $L(1/2,1/2,p/q;e)$, which we call $(2,2,r)$-Montesinos links, have finite involutory quandles. This generalizes an observation of Winker regarding the $(2,2,q)$-pretzel links. We also describe some properties of these quandles.

Quantum computing is closely related to $3$-dimensional topological quantum field theory through its approach in topological quantum computation. The following are notes based on lectures given by the author at the Mathematics Department of George Washington University. The idea is to give a brief introduction and survey of a few basic facts of quantum computing to an audience of non-experts interested in taking a quick look at some of the most important concepts of quantum computing. The author is a topologist with strong interest in the general ideas of quantum computing. The notes touch on how quantum topology enters the field but emphasizes the general ideas of quantum computing in the first place.

In [Simple ribbon fusions and genera for links, J. Math. Soc. Japan 68 (2016) 1033–1045], we introduced special types of fusions, so-called simple-ribbon fusions on links, and showed that they never decrease the genera of links. In this paper, we also gave a geometric condition for a simple-ribbon fusion to preserve a link type, and it turns out that a “complicated” simple-ribbon fusion may preserve a link type if the link is split. In this paper, we show that only “clearly trivial” simple-ribbon fusion preserves a link type if the link is non-split. This enables us to give a lower bound for the difference between genera of knots related by a simple-ribbon fusion.

Quantum computers by their nature are many particle quantum systems. Both the many-particle arrangement and being quantum are necessary for the existence of the entangled states, which are responsible for the parallelism of the quantum computers. Second quantization is a very important approximate method of describing such systems. This lecture will present the general idea of the second quantization, and discuss shortly some of the most important formulations of second quantization.

This paper studies equivariant Jones polynomials of periodic links. Namely, to every $n$-periodic link and any divisor $d$ of $n$, we associate a polynomial that is a graded Euler characteristic of $d$-graded equivariant Khovanov homology. The first main result shows that certain linear combinations of these polynomials, called the difference Jones polynomials, satisfy an appropriate version of the skein relation. This relation is used to generalize Przytycki’s periodicity criterion. We also provide an example showing that the new criterion is indeed stronger. The second main result gives a state-sum formula for the difference Jones polynomials. This formula is used to give an alternative proof of the periodicity criterion of Murasugi.

Fox coloring provides a combinatorial framework for studying dihedral representations of the knot group. The less well-known concept of Dehn coloring captures the same data. Recent work of Carter–Silver–Williams clarifies the relationship between the two focusing on how one transitions between Fox and Dehn colorings. In our work, we relate Dehn coloring to the dimer model for knots showing that Dehn coloring data is encoded by a certain weighted balanced overlaid Tait (BOT) graph. Using Kasteleyn theory, we provide graph theoretic methods for determining the structure of a knot’s coloring module. These constructions are closely related to Kauffman’s work on a state sum for the Alexander polynomial.

These five lectures — written for a summer school in 2006 — provide an introduction to Khovanov homology covering the basic definitions, important properties, a number of variants and some applications. At the end of each lecture the reader is referred to the relevant literature for further reading.

In 1993, Fenn, Rourke and Sanderson introduced rack spaces and rack homotopy invariants, and modifications to quandle spaces and quandle homotopy invariants were introduced by Nosaka in 2011. In this paper, we define the Cayley-type graph and the extended quandle space of a quandle in analogy to rack and quandle spaces. Moreover, we construct the shadow homotopy invariant of a classical link and prove that the shadow homotopy invariant is equal to the quandle homotopy invariant multiplied by the order of a quandle.