Isotopic Knots Research Articles

Minimal generating sets of moves for diagrams of isotopic knots and spatial trivalent graphs

Polyak proved that all oriented versions of Reidemeister moves for knot and link diagrams can be generated by a set of just four oriented Reidemeister moves, and that no fewer than four oriented Reidemeister moves generate them all. We refer to a set containing four oriented Reidemeister moves that collectively generate all of the other oriented Reidemeister moves as a minimal generating set. Polyak also proved that a certain set containing two Reidemeister moves of type 1, one move of type 2 and one move of type 3 form a minimal generating set for all oriented Reidemeister moves. We expand upon Polyak’s work by providing an additional eleven minimal, 4-element, generating sets of oriented Reidemeister moves, and we prove that these 12 sets represent all possible minimal generating sets of oriented Reidemeister moves. We also consider the Reidemeister-type moves that relate oriented spatial trivalent graph diagrams with trivalent vertices that are sources and sinks and prove that a minimal generating set of oriented Reidemeister-type moves for spatial trivalent graph diagrams contains 10 moves.

A Reidemeister type theorem for petal diagrams of knots

We study petal diagrams of knots, which provide a method of describing knots in terms of permutations in a symmetric group S2n+1. We define two classes of moves on such permutations, called trivial petal additions and crossing exchanges, which do not change the isotopy class of the underlying knot. We prove that any two permutations which represent isotopic knots can be related by a sequence of these moves and their inverses.

Reidemeister moves and a polynomial of virtual knot diagrams

In 2006 C. Hayashi gave a lower bound for the number of Reidemeister moves in deformation of two equivalent knot diagrams by using writhe and cowrithe. It can be naturally extended for two virtually isotopic virtual knot diagrams. We introduce a polynomial qK(t) of a virtual knot diagram K and give lower bounds for the number of Reidemeister moves in deformation of two virtually isotopic knots by using qK(t). We give an example which shows that the polynomial qK(t) is useful to map out a sequence of Reidemeister moves to deform a virtual knot diagram to another virtually isotopic one.

On the normal form of knots

This paper is a report on the results of computer experiments with an algorithm that takes classical knots to what we call their “normal form” (and so can be used to identify the knot). The algorithm is implemented in a computer animation that shows the isotopy joining the given knot diagram to its normal form. We describe the algorithm, which is a kind of gradient descent along a functional that we define, present a table of normal forms of prime knots with 7 crossings or less, compare it to the knot table of normal forms of wire knots (obtained in [1] by mechanical experiments with real wire models) and (regretfully) present simple examples showing that normal forms obtained by our algorithm are not unique for a given knot type (sometimes isotopic knots can have different normal forms).

FIBER QUADRISECANTS IN KNOT ISOTOPIES

Fix a straight line L in Euclidean 3-space and consider the fibration of the complement of L by half-planes. A generic knot K in the complement of L has neither fiber quadrisecants nor fiber extreme secants such that K touches the corresponding half-plane at 2 points. Both types of secants occur in generic isotopies of knots. We give lower bounds for the number of these fiber secants in all isotopies connecting given isotopic knots. The bounds are expressed in terms of invariants calculable in linear time with respect to the number of crossings.

Normal holonomy and writhing number of polygonal knots

The normal holonomy of a polygonal knot is a geometrical invariant which is closely related to the writhing number. We show that normal holonomy fibers the space of knots over the circle and deduce that the writhing number fibers the space of knots over the real line. Consequently, two isotopic knots which have the same writhing number are isotopic through a family of knots having the same writhing number. In a similar vein, two isotopic knots having zero holonomy are isotopic through a family of such knots if and only if they have the same autoparallel linking number. More generally, the definition of normal holonomy makes sense for immersed polygonal knots. This time normal holonomy fibers the space of immersed knots over the circle, but now there are only two isotopy classes of immersed knots of zero holonomy.

INVARIANTS OF KNOT DIAGRAMS AND RELATIONS AMONG REIDEMEISTER MOVES

In this paper a classification of Reidemeister moves, which is the most refined, is introduced. In particular, this classification distinguishes some Ω3-moves that only differ in how the three strands that are involved in the move are ordered on the knot. To transform knot diagrams of isotopic knots into each other one must in general use Ω3-moves of at least two different classes. To show this, knot diagram invariants that jump only under Ω3-moves are introduced. Knot diagrams of isotopic knots can be connected by a sequence of Reidemeister moves of only six, out of the total of 24, classes. This result can be applied in knot theory to simplify proofs of invariance of diagrammatical knot invariants. In particular, a criterion for a function on Gauss diagrams to define a knot invariant is presented.

Knotted periodic orbits in suspensions of smale's horseshoe: Extended families and bifurcation sequences

We consider knotted periodic orbits in the “natural” suspension of Smale's horseshoe map as constructed by Holmes and Williams (1985) and Holmes (1986). We identify quartets and octets of periodic orbits which are isotopic knots and indicate how they are related via the sequences of bifurcations in which they are created in two-parameter families such as the Hénon map.

Non‐conjugate braids can define isotopic knots

Communications on Pure and Applied MathematicsVolume 22, Issue 2 p. 239-242 Article Non-conjugate braids can define isotopic knots Joan S. Birman, Joan S. BirmanSearch for more papers by this author Joan S. Birman, Joan S. BirmanSearch for more papers by this author First published: March 1969 https://doi.org/10.1002/cpa.3160220207Citations: 4AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume22, Issue2March 1969Pages 239-242 RelatedInformation