Related papers: Stabilization in the Braid Groups (with applicatio…
The Markov Theorem Without Stabilization (MTWS) (see math.GT/0310279) established the existence of a calculus of braid isotopies that can be used to move between closed braid representatives of a given oriented link type without having to…
We introduce a new braid-theoretic framework with which to understand the Legendrian and transversal classification of knots, namely a Legendrian Markov Theorem without Stabilization which induces an associated transversal Markov Theorem…
We review recent developments in the theory of Thompson group representations related to knot theory.
Any knot $K$ in genus-$1$ $1$-bridge position can be moved by isotopy to lie in a union of $n$ parallel tori tubed by $n-1$ tubes so that $K$ intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal…
Given a family of groups admitting a braided monoidal structure (satisfying mild assumptions) we construct a family of spaces on which the groups act and whose connectivity yields, via a classical argument of Quillen, homological stability…
This paper has been withdrawn by the authors, as it was combined with "Conformally invariant energies of knots I" (math/0409396) to be "Conformally invariant energies of knots" which has replaced the former.
Covering moves relate colored link diagrams appearing as the branch sets of simple branched coverings of $S^3$ by the same 3-manifold. We provide a complete set of covering moves on plat closures of braids in each fixed degree $d \geq 4$,…
Braid groups may be defined for every Coxeter diagram. Artin's braid group is of type A. Analogs of Temperley-Lieb, Hecke and Birman-Wenzl algebras exist for B-type. Our general hypothethis is that the braid group of B-type replaces Artin's…
Construction of (colored) knot polynomials for double-fat graphs is further generalized to the case when "fingers" and "propagators" are substituting R-matrices in arbitrary closed braids with m-strands. Original version of arXiv:1504.00371…
Recently Witten introduced a type IIB brane construction with certain boundary conditions to study knot invariants and Khovanov homology. The essential ingredients used in his work are the topologically twisted N = 4 Yang-Mills theory,…
We prove that the homology of the mapping class group of any 3-manifold stabilizes under connected sum and boundary connected sum with an arbitrary 3-manifold when both manifolds are compact and orientable. The stabilization also holds for…
We explore properties of braids such as their fractional Dehn twist coefficients, right-veeringness, and quasipositivity, in relation to the transverse invariant from Khovanov homology defined by Plamenevskaya for their closures, which are…
We show that two knots have matching Vassiliev invariants of order less than n if and only if they are equivalent modulo the nth group of the lower central series of some pure braid group, thus characterizing Vassiliev's knot invariants in…
Topological gauge theories in four dimensions which admit surface operators provide a natural framework for realizing homological knot invariants. Every such theory leads to an action of the braid group on branes on the corresponding moduli…
The recent proof by Bigelow and Krammer that the braid groups are linear opens the possibility of applications to the study of knots and links. It was proved by the first author and Menasco that any closed braid representative of the unknot…
Torus knots are an important family of knots about which much is understood; invariants of torus knots often exhibit nice formulas, making them convenient and fundamental building blocks for examples in knot theory. Spiral knots, defined…
This is a chapter that is to appear in the "Handbook of Knot Theory", edited by William W. Menasco and Morwen B. Thistlethwaite.
The goal of this book is to characterize algebraically the closed 4-manifolds that fibre nontrivially or admit geometries in the sense of Thurston, or which are obtained by surgery on 2-knots, and to provide a reference for the topology of…
We discuss a semi-implicit numerical scheme that allows for minimizing the bending energy of curves within certain isotopy classes. To this end we consider a weighted sum of the bending energy and the tangent-point functional. Based on…
This is a facsimile of the circa 1990 unpublished manuscript with the same title. All the original text, figures and tables are included; although text has been reset in \TeX, the original hand-drawn figures have been redrawn digitally, and…