A secure slip knot is very important in the field of arthroscopy. The new giant knot, developed by the first author, has the properties of being a one-way self-locking slip knot, which is secured without additional half hitches and can tolerate higher forces to be untied.

Let K′ be a knot that admits no cosmetic crossing changes and let C be a prime, non-cable knot. Then any knot that is a satellite of C with winding number zero and pattern K′ admits no cosmetic crossing changes. As a consequence we prove the nugatory crossing conjecture for Whitehead doubles of prime, non-cable knots.

We present a topological interpretation of knot and braid contact homology in degree zero, in terms of cords and skein relations. This interpretation allows us to extend the knot invariant to embedded graphs and higher-dimensional knots. We calculate the knot invariant for two-bridge knots and relate it to double branched covers for general knots.

Using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum sl(m) knot polynomial. Our knot homology naturally satisfies the categorified MOY relations and is conjecturally isomorphic to Khovanov-Rozansky homology. Our construction is motivated by the geometric Satake correspondence and is related to Manolescu’s by homological mirr...

In my first year report [16] I described how to write down explicitly the chain complex C∗(X̃) for the universal cover of the knot exterior X, given a knot diagram. In this report we describe work to fit this chain complex into an algebraic group which we construct which measures a “2nd order slice-ness obstruction,” in the sense that it obstructs the concordance class of a knot lying in the lev...

Introduction. The group π1(S − k) of a knot k contains an extraordinary amount of information. From combined results of W. Whitten [Wh] and M. Culler, C. McA. Gordon, J. Luecke and P.B. Shalen [CuGoLuSh] it is known that there are at most two distinct unoriented prime knots with isomorphic groups. Unfortunately, knot groups are generally difficult to use. Knot groups are usually described by pr...

The stick index of a knot is the least number of line segments required to build the knot in space. We define two analogous 2-dimensional invariants, the planar stick index and the spherical stick index. We find bounds on these quantities in terms of other knot invariants, and give planar stick and spherical stick constructions for torus knots and for compositions of trefoils.

Given a grid presentation of a knot (or link) K in the three-sphere, we describe a Heegaard diagram for the knot complement in which the Heegaard surface is a torus and all elementary domains are squares. Using this diagram, we obtain a purely combinatorial description of the knot Floer homology of K.

In mathematics there is a wide class of knot invariants that may be expressed in the form of multiple line integrals computed along the trajectory C describing the spatial conformation of the knot. In this work it is addressed the problem of evaluating invariants of this kind in the case in which the knot is discrete, i. e. its trajectory is constructed by joining together a set of segments of ...

Negami found an upper bound on the stick number s(K) of a nontrivial knot K in terms of the minimal crossing number c(K) of the knot, which is s(K) ≤ 2c(K). Furthermore, McCabe proved that s(K) ≤ c(K) + 3 for a 2-bridge knot or link, except in the cases of the unlink and the Hopf link. In this paper we construct any 2-bridge knot or link K of at least six crossings by using only c(K) + 2 straig...