Related papers: Petal Projections, Knot Colorings and Determinants

We introduce the warping matrix which is a new description of oriented knots from a viewpoint of warping degree.

For p=3 and for p=5 we prove that there are exactly p equivalence classes of p-coloured knots modulo (+/-1)--framed surgeries along unknots in the kernel of a p-colouring. These equivalence classes are represented by connect-sums of n…

The alternating knots, links and twists projected on the $S_2$ sphere were identified with the phase space of a Hamiltonian dynamic system of one degree of freedom. The saddles of the system correspond to the crossings, the edges correspond…

Explosive percolation in a network is a phase transition where a large portion of nodes becomes connected with an addition of a small number of edges. Although extensively studied in random network models and reconstructed real networks,…

In this paper, we introduce a new type of relation between knots called the descendant relation. One knot $H$ is a descendant of another knot $K$ if $H$ can be obtained from a minimal crossing diagram of $K$ by some number of crossing…

The jigsaw percolation process on graphs was introduced by Brummitt, Chatterjee, Dey, and Sivakoff as a model of collaborative solutions of puzzles in social networks. Percolation in this process may be viewed as the joint connectedness of…

We construct new knot polynomials. Let $V$ be the standard solid torus in 3-space and let $pr$ be its standard projection onto an annulus. Let $M$ be the space of all smooth oriented knots in $V$ such that the restriction of $pr$ is an…

The purpose of the paper is two-fold: to introduce a multivariable creative telescoping method, and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative $A$-polynomial of twist knots. Our multivariable…

This paper starts a systematic description of colored knot polynomials, beginning from the first non-(anti)symmetric representation R=[2,1]. The project involves several steps: (i) parametrization of big families of knots a la…

This paper extends the study of arc crossing change, a local operation on knot diagrams recently introduced by Cericola, from knot diagrams to link diagrams. We consider two types of arc crossing change on link diagrams and discuss when…

Generalizing unknotting number, $n$-adjacent knots have $n$ crossings such that changing any non-empty subset of them results in the unknot. In this paper, we determine the 2-adjacent knots through 12 crossings. Using Heegaard Floer…

We propose a statistical model defined on the three-dimensional diamond network where the splitting of randomly selected nodes leads to a spatially disordered network, with decreasing degree of connectivity. The terminal state, that is…

Percolation thresholds have recently been studied by means of a graph polynomial $P_B(p)$, henceforth referred to as the critical polynomial, that may be defined on any periodic lattice. The polynomial depends on a finite subgraph $B$,…

The signature function of a knot is a locally constant integer valued function with domain the unit circle. The jumps (i.e., the discontinuities) of the signature function can occur only at the roots of the Alexander polynomial on the unit…

A hypergraph consists of a set of vertices and a set of subsets of vertices, called hyperedges. In the metro map metaphor, each hyperedge is represented by a path (the metro line) and the union of all these paths is the support graph (metro…

We extend an approach of Beliakova for computing knot Floer homology and implement it in a publicly available computer program. We review the main programming and optimization methods used. Our program is then used to check that the Floer…

We study a class of combinatorial objects that we call "decorated trees". These consist of vertices, arrows and edges, where each edge is decorated by two integers (one near each of its endpoints), each arrow is decorated by an integer, and…

For any knot with genus one and unknotting number one, other than the figure-eight knot, we prove that there is exactly one way to unknot it by means of a crossing change. In the case of the figure-eight knot, we prove that there are…

We introduce natural language processing into the study of knot theory, as made natural by the braid word representation of knots. We study the UNKNOT problem of determining whether or not a given knot is the unknot. After describing an…

A knotted ribbon is one of physical aspect of a knot. A folded ribbon knot is a depiction of a knot obtained by folding a long and thin rectangular strip to become flat. The ribbonlength of a knot type can be defined as the minimum length…