相关论文: Knots and words
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…
A classical knot is described by a one-stroke trajectory with entanglements of a string. The replica method appears as a powerful tool in statistical mechanics for a polymer or self-avoiding walk. We consider this replica N to 0 limit in…
For any smooth manifold $M$ of dimension $d\geq4$ we construct explicit classes in homotopy groups of spaces of embeddings of either an arc or a circle into $M$, in every degree that is a multiple of $d-3$, and show that they are detected…
In 2005 V. Turaev introduced the theory of topology of words and phrases. Turaev defined an equivalence relation on generalized words and phrases which is called homotopy. This is suggested by the Reidemeister moves in the knot theory. Then…
In a previous paper, we described the set of words that appear in the coding of smooth (resp. analytic) curves at arbitrary small scale. The aim of this paper is to compute the complexity of those languages.
Knots and links have been considered to be useful models for structural analysis of molecular chains such as DNA and proteins. One quantity that we are interested on molecular links is the minimum number of monomers necessary to realize…
The fundamental quandle is a powerful invariant of knots and links, but it is difficult to describe in detail. It is often useful to look at quotients of the quandle, especially finite quotients. One natural quotient introduced by Joyce is…
We discuss relations among various positivities of knots and links, such as strong quasipositivity and quasipositivity. We give several pieces of supporting evidence for conjectural statements concerning these positivities and the defect of…
A necklace is an equivalence class of words of length $n$ over an alphabet under the cyclic shift (rotation) operation. As a classical object, there have been many algorithmic results for key operations on necklaces, including counting,…
We analyze the connections between the mathematical theory of knots and quantum physics by addressing a number of algorithmic questions related to both knots and braid groups. Knots can be distinguished by means of `knot invariants', among…
An increasing sequence of integers is said to be universal for knots and links if every knot and link has a projection to the sphere such that the number of edges of each complementary face of the projection comes from the given sequence.…
Two categorifications are given for the arrow polynomial, an extension of the Kauffman bracket polynomial for virtual knots. The arrow polynomial extends the bracket polynomial to infinitely many variables, each variable corresponding to an…
This is an introductory article on high dimensional knots for the beginners. High dimensional knot theory is an exciting field. It is a field of knot theory, which is one of topology and is connected with many ones. In this article we use…
We define the concordance crosscap number of a knot as the minimum crosscap number among all the knots concordant to the knot. The four-dimensional crosscap number is the minimum first Betti number of non-orientable surfaces smoothly…
We classify Legendrian torus knots and figure eight knots in the tight contact structure on the 3-sphere up to Legendrian isotopy. As a corollary to this we also obtain the classification of transversal torus knots and figure eight knots up…
While the problem of computing the genus of a knot is now fairly well understood, no algorithm is known for its four-dimensional variants, both in the smooth and in the topological locally flat category. In this article, we investigate a…
We classify positive transversal torus knots in tight contact structures up to transversal isotopy.
Two string links are equivalent up to $2n$-moves and link-homotopy if and only if their all Milnor link-homotopy invariants are congruent modulo $n$. Moreover, the set of the equivalence classes forms a finite group generated by elements of…
Frequently, knots are enumerated by their crossing number. However, the number of knots with crossing number $c$ grows exponentially with $c$, and to date computer-assisted proofs can only classify diagrams up to around twenty crossings.…
This is a short survey of algebro-combinatorial link homology theories which have the Jones polynomial and other link polynomials as their Euler characteristics.