Related papers: FlatKnotInfo: the first 1.24 million flat knots
In 2002, D. Hrencecin and L.H. Kauffman defined a filamentation invariant on oriented chord diagrams that may determine whether the corresponding flat virtual knot diagrams are non-trivial. A virtual knot diagram is non-classical if its…
Using computer calculations and working with representatives of pretzel tangles we established general adequacy criteria for different classes of knots and links. Based on adequate graphs obtained from all Kauffman states of an alternating…
We construct two complete invariants of oriented classical knots in space. The value of each invariant on any knot is a set, infinite for the first invariant and finite for the second. The finite set is computed algorithmically from any…
In this paper we discuss the applications of knotoids to modelling knots in open curves and produce new knotoid invariants. We show how invariants of knotoids generally give rise to well-behaved measures of how much an open curve is…
Families of alternating knots (links) and tangles are studied using as building block the conway defined as the twisting of two strands. The regular representation of knots assumes the projection has the minimal number of overpassings, and…
We introduce new skein invariants of links based on a procedure where we first apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using a…
We are giving tables of quasi-alternating knots with $8\le n \le 12$ crossings. As the obstructions for a knot to be quasialternating we used homology thickness with regards to Khovanov homology, odd homology, and Heegaard-Floer homology…
There are 46,972 prime knots with crossing number 14. Among them 19,536 are alternating and have arc index 16. Among the non-alternating knots, 17, 477, and 3,180 have arc index 10, 11, and 12, respectively. The remaining 23,762 have arc…
We present computational results about quasi-alternating knots and links and odd homology obtained by looking at link families in the Conway notation. More precisely, we list quasi-alternating links up to 12 crossings and the first examples…
We present a complete classification of spherical knotoids with up to six crossings and conjecture that our classification up to seven crossings is complete. Our work extends the tradition of knot tabulation to the setting of knotoids…
It is a major unsolved problem as to whether unknot recognition - that is, testing whether a given closed loop in R^3 can be untangled to form a plain circle - has a polynomial time algorithm. In practice, trivial knots (which can be…
We study factorizations of HOMFLY polynomials of certain knots and oriented links. We begin with a computer analysis of knots with at most 12 crossings, finding 17 non-trivial factorizations. Next, we give an irreducibility criterion for…
For any given number of crossings $c$, there exists a formula to determine the number of 2-bridge knots of $c$ crossings, and indeed it is a simple matter to actually construct presentations of these knots. However, the determination of…
We present an enhanced prime decomposition theorem for knots that gives the isotopy classes of composite knots that can be constructed from a given list of prime factors (allowing for the mirroring and orientation reversing for each…
Non-classical virtual knots may have non-isomorphic upper and lower quandles. We exploit this property to define the quandle difference invariant, which can detect non-classicality by comparing the numbers of homomorphisms into a finite…
The paper introduces Slope Conjecture which relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the…
We use Heegaard Floer homology to give obstructions to unknotting a knot with a single crossing change. These restrictions are particularly useful in the case where the knot in question is alternating. As an example, we use them to classify…
Using the techniques on annulus twists, we observe that $6_3$ has infinitely many non-characterizing slopes, which affirmatively answers a question by Baker and Motegi. Furthermore, we prove that the knots $6_2$, $6_3$, $7_6$, $7_7$, $8_1$,…
We introduce a new combinatorial method to encode knots and links with applications to knot invariants. Clasp diagrams defined in this paper are combinatorial blueprints for building knot diagrams out of full twists on two strings rather…
We present a class of knots associated with labelled generic immersions of intervals into the plane and compute their Gordian numbers and 4-dimensional invariants. At least 10% of the knots in Rolfsen's table belong to this class of knots.…