Related papers: Amphicheiral links with special properties, II
It is well-known that a knot is Fox $n$-colorable for a prime $n$ if and only if the knot group admits a surjective homomorphism to the dihedral group of degree $n$. However, this is not the case for links with two or more components. In…
We say that a link $L_1$ is an s-major of a link $L_2$ if any diagram of $L_1$ can be transformed into a diagram of $L_2$ by changing some crossings and smoothing some crossings. This relation is a partial ordering on the set of all prime…
In a recent paper, Allen and Swenberg investigated which link polynomials are capable of detecting causality in (2+1)-dimensional globally hyperbolic spacetimes. They ultimately suggested it is likely that the Jones Polynomial accomplishes…
We study cosmetic crossings in knots of genus one and obtain obstructions to such crossings in terms of knot invariants determined by Seifert matrices. In particular, we prove that for genus one knots the Alexander polynomial and the…
We introduce twelve polynomial invariants for long virtual knots, called intersection polynomials, extending and refining the three intersection polynomials for virtual knots. They are defined via intersection numbers of cycles on a closed…
We present some elementary ideas to prove the following Sylvester-Gallai type theorems involving incidences between points and lines in the planes over the complex numbers and quaternions. (1) Let A and B be finite sets of at least two…
We provide an algorithm to determine whether a link L admits a crossing change that turns it into a split link, under some fairly mild hypotheses on L. The algorithm also provides a complete list of all such crossing changes. It can…
We relate the jumps of the signature function of a link to the roots of its first nonzero higher Alexander polynomial.
We generalise theorems of Cochran-Lickorish and Owens-Strle to the case of links with more than one component. This enables the use of linking forms on double branched covers, Heegaard Floer correction terms, and Donaldson's diagonalisation…
The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering…
We give a simple obstruction for a knot to be amphichiral, in terms of the homology of the 2-fold branched cover. We work with unoriented knots, and so obstruct both positive and negative amphichirality.
We introduce a class of links whose bracket polynomials admit an expansion over perfect matchings of a plane bipartite graph. This class includes 2-bridge links, pretzel links, and Montesinos links. Our first main result (Theorem A)…
We extend the table of Garoufalidis, Le and Zagier concerning conjectural Rogers-Ramanujan type identities for tails of colored Jones polynomials to all alternating knots up to 10 crossings. We then prove these new identities using q-series…
It follows from earlier work of Silver-Williams and the authors that twisted Alexander polynomials detect the unknot and the Hopf link. We now show that twisted Alexander polynomials also detect the trefoil and the figure-8 knot, that…
We show that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic crossings. As an application we prove the nugatory crossing conjecture for the…
In this work, we explain `Peiffer pairings' in the Moore (bi)complex of a bisimplicial group and give their applications for crossed modules and crossed squares.
We use matchings on Lyndon words to classify flat knots up to 8 crossings. Using flat knots invariants such as the based matrix, the $\phi$-invariant, the flat arrow polynomial, and the flat Jones-Krushkal polynomial, we distinguish all…
The Alexander polynomial (1928) is the first polynomial invariant of links devised to help distinguish links up to isotopy. Fox's conjecture (1962) -- stating that the absolute values of the coefficients of the Alexander polynomial for any…
We study $\mathrm{SL}_2(\mathbb{F})$-character varieties of knots over algebraically closed fields $\mathbb{F}$. We give a sufficient condition in terms of the double branched cover of a $2$-bridge knot (or, equivalently, of its Alexander…
We introduced concept of meander knots, 2-component meander links and multi-component meander links and derived different families of meander knots and links from open meanders with at most 16 crossings. We also defined semi-meander knots…