Knot projections with a single multi-crossing, preprint at ArXiv:1208.5742 (original) (raw)

Multi-Crossing Numbers for Knots

2019

We study the projections of a knot K that have only n-crossings. The n-crossing number of K is the minimum number of n-crossings among all possible projections of K with only n-crossings. We obtain new results on the relation between n-crossing number and (2n− 1)-crossing number for every positive even integer n.

Triple-crossing projections, moves on knots and links and their minimal diagrams

Journal of Knot Theory and Its Ramifications, 2020

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 table of known knots and links with triple-crossing number equal to five. By introducing a new type of diagrammatic move, we reduce the number of generating moves on triple-crossing diagrams, and derive a minimal generating set of moves connecting triple-crossing diagrams of the same knot.

Bounds on Übercrossing and Petal Numbers for Knots

Journal of Knot Theory and Its Ramifications, 2015

An n-crossing is a point in the projection of a knot where n strands cross so that each strand bisects the crossing. An übercrossing projection has a single n-crossing and a petal projection has a single n-crossing such that there are no loops nested within others. The übercrossing number, ü(K), is the smallest n for which we can represent a knot K with a single n-crossing. The petal number is the number of loops in the minimal petal projection. In this paper, we relate the übercrossing number and petal number to well known invariants such as crossing number, bridge number, and unknotting number. We find that the bounds we have constructed are sharp for (r, r + 1)-torus knots. We also explore the behavior of übercrossing number under composition.

Quadruple crossing number of knots and links

Mathematical Proceedings of the Cambridge Philosophical Society, 2013

A quadruple crossing is a crossing in a projection of a knot or link that has four strands of the knot passing straight through it. A quadruple crossing projection is a projection such that all of the crossings are quadruple crossings. In a previous paper, it was proved that every knot and link has a quadruple crossing projection and hence, every knot has a minimal quadruple crossing number c4(K). In this paper, we investigate quadruple crossing number, and in particular, use the span of the bracket polynomial to determine quadruple crossing number for a variety of knots and links.

Triple Crossing Number of Knots and Links

Journal of Knot Theory and Its Ramifications, 2013

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 that every knot and link has a triple crossing projection and then investigate c3(K), which is the minimum number of triple crossings in a projection of K. We obtain upper and lower bounds on c3(K) in terms of the traditional crossing number and show that both are realized. We also relate triple crossing number to the span of the bracket polynomial and use this to determine c3(K) for a variety of knots and links. We then use c3(K) to obtain bounds on the volume of a hyperbolic knot or link. We also consider extensions to cn(K).

Minimum Braids: A Complete Invariant of Knots and Links

arXiv (Cornell University), 2004

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 braids to study graph trees, amphicheirality, unknotting numbers, and periodic tables.

A one-parameter approach to knot theory

2006

To an oriented knot we associate a trace graph in a thickened torus in such a way that knots are isotopic if and only if their trace graphs can be related by moves of finitely many standard types. For closed braids with a fixed number of strands, we recognize trace graphs up to equivalence excluding one type of moves in polynomial time with respect to the braid length.

A Self-Linking Invariant of Virtual Knots

2004

In this paper we introduce a new invariant of virtual knots and links that is non-trivial for infinitely many virtuals, but is trivial on classical knots and links. The invariant is initially be expressed in terms of a relative of the bracket polynomial and then extracted from this polynomial in terms of its exponents, particularly for the case of knots. This analog of the bracket polynomial will be denoted K (with curly brackets) and called the binary bracket polynomial. The key to the combinatorics of the invariant is an interpretation of the state sum in terms of 2-colorings of the associated diagrams. For virtual knots, the new invariant, J(K), is a restriction of the writhe to the odd crossings of the diagram (A crossing is odd if it links an odd number of crossings in the Gauss code of the knot. The set of odd crossings is empty for a classical knot.) For K a virtual knot, J(K) non-zero implies that K is non-trivial, non-classical and inequivalent to its planar mirror image. T...