相关论文: Tabulation of knots up to five triple-crossings an…

In this paper we present a systematic method to generate prime knot and prime link minimal triple-point projections, and then classify all classical prime knots and prime links with triple-crossing number at most four. We also extend the…

In this article, we give a list of minimal grid diagrams of the 12 crossing prime alternating knots. This is a continuation of the work in https://doi.org/10.1142/S0218216520500765

A knot is a closed loop in space without self-intersection. Two knots are equivalent if there is a self homeomorphism of space bringing one onto the other. An arc presentation is an embedding of a knot in the union of finitely many half…

We will strengthen the known upper and lower bounds on the delta-crossing number of knots in therms of the triple-crossing number. The latter bound turns out to be strong enough to obtain (unknown values of) triple-crossing numbers for a…

A well-known algorithm for unknotting knots involves traversing a knot diagram and changing each crossing that is first encountered from below. The minimal number of crossings changed in this way across all diagrams for a knot is called the…

A partial order on the set of prime knots can be defined by the existence of an epimorphism between knot groups. We prove that all the prime knots with up to $6$ crossings are minimal. We also show that each fibered knot with the…

The warping degree of an oriented knot diagram is the minimal number of crossing changes which are required to obtain a monotone knot diagram from the diagram. The minimal warping degree of a knot is the minimal value of the warping degree…

It is known that the arc index of alternating knots is the minimal crossing number plus two and the arc index of prime nonalternating knots is less than or equal to the minimal crossing number. We study some cases when the arc index is…

We give a list of minimal grid diagrams of the 13 crossing prime nonalternating knots which have arc index 13. There are 9,988 prime knots with crossing number 13. Among them 4,878 are alternating and have arc index 15. Among the other…

Minimum braids are a complete invariant of knots and links. This paper defines minimum braids, describes how they can be generated, presents tables for knots up to ten crossings and oriented links up to nine crossings, and uses minimum…

This is the third paper in a series devoted to enumerating the prime alternating knots and links. This paper establishes a method for enumerating the prime alternating links. It is shown that one may choose any prime alternating link…

Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by V.~Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter…

We compose the table of knots in the thickened torus T x I having diagrams with at most 4 crossings. The knots are constructed by the three-step process. First we list regular graphs of degree 4 with at most 4 vertices, then for each graph…

Every knot can be embedded in the union of finitely many half planes with a common boundary line in such a way that the portion of the knot in each half plane is a properly embedded arc. The minimal number of such half planes is called the…

We show that if a classical knot diagram satisfies a certain combinatorial condition then it is minimal with respect to the number of classical crossings. This statement is proved by using the Kauffman bracket and the construction of atoms…

This is the first in a series of four papers wherein we enumerate all prime alternating knots and links. In this first paper, we introduce four operators on knots and show that, when used according to very simple rules on the prime…

A triple crossing is a crossing in a projection of a knot or link that has three strands of the knot passing straight through it. A triple crossing projection is a projection such that all of the crossings are triple crossings. We prove…

In this paper, we enumerate the number of oriented rational knots and the number of oriented rational links with any given crossing number and minimum genus. This allows us to obtain a precise formula for the average minimal genus 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.…

The unknotting number of a knot is the minimum number of crossings one must change to turn that knot into the unknot. The algebraic unknotting number is the minimum number of crossing changes needed to transform a knot into an Alexander…