Related papers: Some generalizations of Satoh's Tube map

Recent work on the loop representation of quantum gravity has revealed previously unsuspected connections between knot theory and quantum gravity, or more generally, 3-dimensional topology and 4-dimensional generally covariant physics. We…

Branch points of a real 2-surface S in a 4-manifold M generalize the branch points of complex curves in complex surfaces: for example, they can occur as singularities of minimal surfaces. We investigate such a branch point p when S is…

This article presents a survey of some recent results in the theory of spatial graphs. In particular, we highlight results related to intrinsic knotting and linking and results about symmetries of spatial graphs. In both cases we consider…

The present paper is an introduction to a combinatorial theory arising as a natural generalisation of classical and virtual knot theory. There is a way to encode links by a class of `realisable' graphs. When passing to generic graphs with…

We describe a Python module that we developed to calculate cobordism maps induced on Khovanov homology. As applications of our program, we compute these maps for all incompressible Seifert surfaces for prime knots up to 10 crossings, and…

Inspired by the paper on quantum knots and knot mosaics [23] and grid diagrams (or arc presentations), used extensively in the computations of Heegaard-Floer knot homology [2,3,7,24], we construct the more concise representation of knot…

We investigate Feynman diagrams which are calculable in terms of generalized one-loop functions, and explore how the presence or absence of transcendentals in their counterterms reflects the entanglement of link diagram constructed from…

We determine the relationship between the contact structure induced by a fibered knot, K, in the three-sphere and the contact structures induced by its various cables. Understanding this relationship allows us to classify fibered cable…

Two welded (respectively virtual) link diagrams are homotopic if one may be transformed into the other by a sequence of extended Reidemeister moves, classical Reidemeister moves, and self crossing changes. In this paper, we extend Milnor's…

A mechanical linkage is a mechanism made of rigid rods linked together by flexible joints, in which some vertices are fixed and others may move. The partial configuration space of a linkage is the set of all the possible positions of a…

Budney recently constructed an operad that encodes splicing of knots. He further showed that the space of (long) knots is generated over this operad by the space of torus knots and hyperbolic knots, thus generalizing the satellite…

In this paper we introduce a new invariant of virtual knots and links that is non-trivial for infinitely many virtuals, but is trivial on classical knots and links. The invariant is initially be expressed in terms of a relative of the…

We introduce the notion of a ribbon-clasp surface-link, which is a generalization of a ribbon surface-link. We generalize the notion of a normal form on embedded surface-links to the case of immersed surface-links and prove that any…

We report on new numerical computations of the set of self-contacts in tightly knotted tubes of uniform circular cross-section. Such contact sets have been obtained before for the trefoil and figure eight knots by simulated annealing -- we…

We reveal a hierarchical, multilayer organization of finite components -- i.e., tendrils and tubes -- around the giant connected components in directed networks and propose efficient algorithms allowing one to uncover the entire…

Every link in R^3 can be represented by a one-vertex ribbon graph. We prove a Markov type theorem on this subset of link diagrams.

In this paper, we consider local moves on classical and welded diagrams of string links, and the notion of welded extension of a classical move. Such extensions being non-unique in general, the idea is to find a topological criterion which…

This is a survey article about the connections between knot theory and four-dimensional topology. Every four-manifold can be represented in terms of a link, by a Kirby diagram. This point of view has led to progress in computing invariants…

We introduce an algebraic structure we call semiquandles whose axioms are derived from flat Reidemeister moves. Finite semiquandles have associated counting invariants and enhanced invariants defined for flat virtual knots and links. We…

The notion of a pseudoknot is defined as an equivalence class of knot diagrams that may be missing some crossing information. We provide here a topological invariant schema for pseudoknots and their relatives, 4-valent rigid vertex spatial…