Journal of Knot Theory and Its Ramifications

Rudimentary knots are invoked to generate a representation of the elementary particles, a model that endows the particles with visualizable structure. The model correlates with the basic tenets, taxonomy, and interactions of the Standard Model, but goes beyond it in a number of important ways, the most significant being that all particles (hadrons and leptons, fermions and bosons) and interactions share a common topology. Among other consequences of the modeling are the topological basis for isospin invariance and its connection to electric charge, the necessary identity of electron and proton charge magnitudes, and the existence of precisely three generations on the particle family tree. The salient feature of the model is that the elementary particles are viewed not as discrete, point-like objects in a vacuum but rather as sustainable, membrane-like distortions embodying curvature and torsion in and of an otherwise featureless continuum and that their manifest physical attributes correlate with the distortion. There are additional connections to the theories of fiber bundles, superstrings and instantons and, historically, to the work of Kelvin in the mid-nineteenth century and Cartan in the 1920s among others.

In 1990, John Berge described several families of knots in the three-dimensional sphere which have non-trivial Dehn surgeries yielding lens spaces. We study a subfamily of them from the view point of resolution of singularity of complex curves and surfaces, Kirby calculus in topology of four-dimensional manifolds and A'Campo's divide knot theory.

The classical knot groups are the fundamental groups of the complements of smooth or piecewise-linear (PL) locally-flat knots. For PL knots that are not locally-flat, there is a pair of interesting groups to study: the fundamental group of the knot complement and that of the complement of the "boundary knot" that occurs around the singular set, the set of points at which the embedding is not locally-flat. If a knot has only point singularities, this is equivalent to studying the groups of a PL locally-flat disk knot and its boundary sphere knot; in this case, we obtain a complete classification of all such group pairs in dimension ≥6. For more general knots, we also obtain complete classifications of these group pairs under certain restrictions on the singularities. Finally, we use spinning constructions to realize further examples of boundary knot groups.

Carter, Jelsovsky, Kamada, Langford and Saito introduced the quandle cocycle invariants of 2-knots, and calculated the cocycle invariant of a 2-twist-spun trefoil knot associated with a 3-cocycle of the dihedral quandle of order 3. Asami and Satoh calculated the cocycle invariants of twist-spun torus knots τ^rT(m,n) associated with 3-cocycles of some dihedral quandles. They used tangle diagrams of the torus knots. In this paper, we calculate the cocycle invariants of twist-spun 2-bridge knots τ^rS(α,β) by a similar method.

We discuss the strong invariant of virtual links proposed in [23]. This invariant is obtained as a generalization of the Jones–Kauffman polynomial (generalized Kauffman's bracket) by adding to the sum some equivalence classes of curves in two-dimensional surfaces. Thus, the invariant is valued in the infinite-dimensional free module over Z[q,q^-1].

We prove that this invariant can be decomposed into finite type Vassiliev invariant of virtual links (in Kauffman's sense); thus we present new infinite series of Vassiliev invariants.

It is also proved that this invariant is strictly stronger than the Jones–Kauffman polynomial for virtual knots proposed by Kauffman. Some examples when the invariant can recognize virtual knots that can not be recognized by other invariants are given.

We study, analytically and by numerical simulations, a mathematical model of closed unknotted polymer chains. The polymers are modeled by what we call "densely packed knots", which are knot diagrams entirely covering a long rectangular lattice, with randomly generated overpasses and underpasses at the nodes of the lattice. We are interested in the probability of the "knottedness" (the "knot complexity") of densily packed knots, which we measure via the degree of their Jones–Kauffman polynomial. In particular, we find the mean complexity of "daughter knots", obtained by cutting off a part of a trivial (i.e. unknotted) "parent" densily packed knot and closig up the "open tails" (the loose ends of the cut strands). We present arguments supporting the conjecture that the knot complexity n^* of a daughter knot of an unknotted parent one grows as where N is the total number of vertices on the lattice. This result gives a strong support for the conjectured "crumpled globule" structure of collapsed unknotted closed polymer chains, in which the polymer forms a system of densely packed folds, mutually separated in a broad range of scales.

We propose a conjecture concerning 3-manifolds (which implies the ℚ conjecture of Myers') and construct a countable sequence of examples which supports it.