Journal of Knot Theory and Its Ramifications

We recently discovered a relationship between the volume density spectrum and the determinant density spectrum for infinite sequences of hyperbolic knots. Here, we extend this study to new quantum density spectra associated to quantum invariants, such as Jones polynomials, Kashaev invariants and knot homology. We also propose related conjectures motivated by geometrically and diagrammatically maximal sequences of knots.

We give a foundational account on topological racks and quandles. Specifically, we define the notions of ideals, kernels, units, and inner automorphism group in the context of topological racks. Further, we investigate topological rack modules and principal rack bundles. Central extensions of topological racks are then introduced providing a first step toward a general continuous cohomology theory for topological racks and quandles.

The goal of this paper is twofold. First, we find a natural home for the double affine Hecke algebras (DAHA) in the physics of BPS states. Second, we introduce new invariants of torus knots and links called hyperpolynomials that address the “problem of negative coefficients” often encountered in DAHA-based approaches to homological invariants of torus knots and links. Furthermore, from the physics of BPS states and the spectra of singularities associated with Landau–Ginzburg potentials, we also describe a rich structure of differentials that act on homological knot invariants for exceptional groups and uniquely determine the latter for torus knots.

We will discuss a method for visual presentation of knotted surfaces in the four space, by examining a number and a position of its Morse’s critical points. Using this method, we will investigate surface-knot with one critical point of index $1$. Then we show infinitely many mutually distinct surface-knots that have an embedding with two critical points of index $1$. Next we define a long flat form of a banded link for any surface-knot and show diagrammatically a long flat form of $n$-twist-spun $(2,t)$-torus knots.

This paper shows how the Formal Knot Theory state model for the Alexander–Conway polynomial is related to Knot Floer Homology. In particular, we prove a parity result about the states in this model that clarifies certain relationships of the model with Knot Floer Homology.

Any finite-dimensional Hopf algebra $H$ is Frobenius and the stable category of $H$-modules is triangulated monoidal. To $H$-comodule algebras we assign triangulated module-categories over the stable category of $H$-modules. These module-categories are generalizations of homotopy and derived categories of modules over a differential graded algebra. We expect that, for suitable $H$, our construction could be a starting point in the program of categorifying quantum invariants of 3-manifolds.

Two different topics are shown to be related. Some group presentations generalizing certain symmetric presentations found by Coxeter, and the ideal compactification of the sets obtained by lifting knots in 3-manifolds to their universal covering spaces.

Marked vertex diagrams provide a combinatorial way to represent knotted surfaces in ${ℝ}^4$; including virtual crossings allows for a theory of virtual knotted surfaces and virtual cobordisms. Biquandle counting invariants are defined only for marked vertex diagrams representing knotted orientable surfaces; we extend these invariants to all virtual marked vertex diagrams by considering colorings by involutory biquandles, also known as bikei. We introduce a way of representing marked vertex diagrams with Gauss diagrams and use these to characterize orientability.

Quantum computing is a field of great interest, attracting, among others, the attention of many mathematicians. Although not all quantum mechanics is needed to successfully engage in research on quantum computing, the somewhat superficial approach usually applied by non-physicists is, in the opinion of the author of the lectures, not feasible. The following notes from lectures given at the mathematics department of George Washington University are meant to be a partial remedy to the situation, offering a very brief and slightly unorthodox introduction to one-particle quantum mechanics, and even shorter discussion of passage to multi-particle quantum mechanics, as needed for quantum computing.

This paper illustrates some classical non-cyclic milestones in mathematical knot theory.

Cobordisms are naturally bigraded and we show that this grading extends to Khovanov homology, making it a triply graded theory. Although the new grading does not make the homology a stronger invariant, it can be used to show that odd Khovanov homology is multiplicative with respect to disjoint unions and connected sums of links; same results hold for the generalized Khovanov homology defined by the author in his previous work. We also examine the module structure on both odd and even Khovanov homology, in particular computing the effect of sliding a basepoint through a crossing on the integral homology.

We construct an algebra of nontrivial homological operations on Khovanov homology with coefficients in ${\mathbb{Z}}_2$ generated by two Bockstein operations. We use the unified Khovanov homology theory developed by the first author to lift this algebra to integral Khovanov homology. We conjecture that these two algebras are infinite and present evidence in support of our conjectures. Finally, we list examples of knots that have the same even and odd Khovanov homology, but different actions of these homological operations. This confirms that the unified theory is a finer knot invariant than the even and odd Khovanov homology combined. The case of reduced Khovanov homology is also considered.

A conjecture of Riley about the relationship between real parabolic representations and signatures of two-bridge knots is verified for double twist knots.