Related papers: A simple algorithm to compute link polynomials def…
Recently, Bigelow defined a diagrammatic method for calculating the Alexander polynomial of a knot or link by resolving crossings in a planar algebra. I will present my multivariate version of Bigelow's calculation. The advantage to my…
A simple algorithm to compute all the zeros of a generic polynomial is proposed.
For any virtual link, a class of new links can be defined called stacks, in which copies of the virtual link are placed on top of one another. The resulting virtual link depends only on the virtual isotopy class of the original link, and…
We introduce birack brackets, skein invariants of birack-colored framed classical and virtual knots and links with values in a commutative unital ring. The multiset of birack bracket values over the homset from a framed link's fundamental…
In previous papers, the author realized the following principle for many knot theories: if a knot diagram is complicated enough then it reproduces itself, i.e., is a subdiagram of any other diagram equivalent to it. This principle is…
We present a series of algorithms for skein manipulation in a genus-2 handlebody, implementing a novel strand sorting method to reduce any skein to a skein in a 2-punctured disk. This reduction guarantees resolution as a linear combination…
Using the Fourier expansion of Markov traces for Ariki-Koike algebras over $\mathbb{Q}(q,u_{1},...,u_{e})$, we give a direct definition of the Alexander polynomials for mixed links. We observe that under the corresponding specialization of…
We show that there are only finitely many homogeneous links whose Conway polynomial has any given degree. Using this we give an example of an inhomogeneous, fibred knot. Secondly, we show how to compute the monodromy of a homogeneous link…
Trivial links are unique up to number of link components, but they can be hard to recognize from arbitrary diagrams. We define a new measure of the complexity of a link embedding, the crumple, and show how this may be used to measure…
Traditionally introduced in terms of advanced topological constructions, many link invariants may also be defined in much simpler terms given their values on a few initial links and a recursive formula on a skein triangle. Then the crucial…
We use Reidemeister torsion to study a twisted Alexander polynomial, as defined by Turaev, for links in the projective space. Using sign-refined torsion we derive a skein relation for a normalized form of this polynomial.
The Jones polynomial is a famous link invariant that can be defined diagrammatically via a skein relation. Khovanov homology is a richer link invariant that categorifies the Jones polynomial. Using spectral sequences, we obtain a skein-type…
Link prediction in complex networks has attracted increasing attention from both physical and computer science communities. The algorithms can be used to extract missing information, identify spurious interactions, evaluate network evolving…
We present a braid-theoretic approach to combinatorially computing knot Floer homology. To a knot or link K, which is braided about the standard disk open book decomposition for (S^3,\xi_std), we associate a corresponding multi-pointed nice…
A weak chord index $Ind'$ is constructed for self crossing points of virtual links. Then a new writhe polynomial $W$ of virtual links is defined by using $Ind'$. $W$ is a generalization of writhe polynomial defined in [6]. Based on $W$,…
A method for computing probabilistic propositions is presented. It assumes the availability of a single external routine for computing the probability of one instantiated variable, given a conjunction of other instantiated variables. In…
A Gauss diagram is a simple, combinatorial way to present a link. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting subdiagrams of certain combinatorial types. In this paper we…
A polynomial invariant of virtual links, arising from an invariant of links in thickened surfaces introduced by Jaeger, Kauffman, and Saleur, is defined and its properties are investigated. Examples are given that the invariant can detect…
In this paper, we construct invariants of braids, knots and links by studying dynamics of points in $\R^{2}$ and applying the Ptolemy relation $ac+bd=xy$.
We describe an algorithm that can effectively calculate the $s$-invariant of a link as defined by Beliakova and Wehrli. Our computations show that this cannot be done by merely calculating the $E_\infty$-page of the Bar-Natan--Lee--Turner…