The Splittability and Triviality of 3-Bridge Links (original) (raw)

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.

Heegaard Splittings of Branched Coverings of S 3

Transactions of the American Mathematical Society, 1975

This paper concerns itself with the relationship between two seemingly different methods for representing a closed, orientable 3-manifold: on the one hand as a Heegaard splitting, and on the other hand as a branched covering of the 3-sphere. The ability to pass back and forth between these two representations will be applied in several different ways: 1. It will be established that there is an effective algorithm to decide whether a 3-manifold of Heegaard genus 2 is a 3-sphere. 2. We will show that the natural map from 6-plat representations of knots and links to genus 2 closed oriented 3-manifolds is injective and surjective. This relates the question of whether or not Heegaard splittings of closed, oriented 3manifolds are "unique" to the question of whether plat representations of knots and links are "unique". 3. We will give a counterexample to a conjecture (unpublished) of W. Haken, which would have implied that S could be identified (in the class of all simply-connected 3-manifolds) by the property that certain canonical presentations for 7TjS3 are always "nice". The final section of the paper studies a special class of genus 2 Heegaard splittings: the 2-fold covers of S which are branched over closed 3-braids. It is established that no counterexamples to the "genus 2 Poincare conjecture" occur in this class of 3-manifolds.

The many faces of cyclic branched coverings of 2-bridge knots and links

2001

We discuss 3-manifolds which are cyclic coverings of the 3-sphere, branched over 2-bridge knots and links. Different descriptions of these manifolds are presented: polyhedral, Heegaard diagram, Dehn surgery and coloured graph constructions. Using these descriptions, we give presentations for their fundamental groups, which are cyclic presentations in the case of 2-bridge knots. The homology groups are given for a wide class of cases. Moreover, we prove that each singly-cyclic branched covering of a 2-bridge link is the composition of a meridiancyclic branched covering of a determined link and a cyclic branched covering of a trivial knot.

On the classification of knots

Proceedings of the American Mathematical Society, 1974

Linking numbers between branch curves of irregular covering spaces of knots are used to extend the classification of knots through ten crossings and to show that the only amphicheirals in Reidemeister’s table are the seven identified by Tait in 1884. Diagrams of the 165 prime 10 10 -crossing knot types are appended. (Murasugi and the author have proven them prime; Conway claims proof that the tables are complete.) Including the trivial type, there are precisely 250 prime knots with ten or fewer crossings.

The incompatibility of crossing number and bridge number for knot diagrams

Discrete Mathematics

We define and compare several natural ways to compute the bridge number of a knot diagram. We study bridge numbers of crossing number minimizing diagrams, as well as the behavior of diagrammatic bridge numbers under the connected sum operation. For each notion of diagrammatic bridge number considered, we find crossing number minimizing knot diagrams which fail to minimize bridge number. Furthermore, we construct a family of minimal crossing diagrams for which the difference between diagrammatic bridge number and the actual bridge number of the knot grows to infinity.

Complementary Regions of Knot and Link Diagrams

Annals of Combinatorics, 2011

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. In this paper, it is proved that the following infinite sequences are all universal for knots and links: (3, 5, 7,. . .), (2, n, n + 1, n + 2,. . .) for all n ≥ 3 and (3, n, n + 1, n + 2,. . .) for all n ≥ 4. Moreover, the following finite sequences are also universal for knots and links: (3, 4, 5) and (2, 4, 5). It is also shown that every knot has a projection with exactly two odd-sided faces, which can be taken to be triangles, and every link of n components has a projection with at most n odd-sided faces if n is even and n + 1 odd-sided faces if n is odd.

Knots are determined by their complements

Bulletin of The American Mathematical Society, 1989

Two (smooth or PL) knots K, K' in S 3 are equivalent if there exists a homeomorphism h: S 3 -• S 3 such that h(K) = K'. This implies that their complements S 3 -K and S 3 -K' are homeomorphic. Here we announce the converse implication. THEOREM 1. If two knots have homeomorphic complements then they are equivalent. This answers a question apparently first raised by Tietze [T, p. 83]. It was previously known that there were at most two knots with a given complement [CGLS, Corollary 3]. Whitten [W] has shown that prime knots with isomorphic groups have homeomorphic complements. Hence we have COROLLARY 1.1. If two prime knots have isomorphic groups then they are equivalent.

Knots in Knots: a Study of Classical Knot Diagrams

Journal of Knot Theory and Its Ramifications, 2016

Looking at the structure of minimal prime knot presentations, one can notice that there are often, perhaps always, segments that present either the trefoil or the figure-eight knot. This note explores the question as to whether this is always the case, reporting on conversations with Jablan Slavik that began at a [Formula: see text] conference in Trieste, Italy and which never reached a conclusion. Evidence supporting this conjectured presence is reported and potential consequences are described.

A Partial Ordering of Knots and Links Through Diagrammatic Unknotting

Journal of Knot Theory and Its Ramifications, 2009

In this paper we define a partial order on the set of all knots and links using a special property derived from their minimal diagrams. A knot or link K is called a predecessor of a knot or link K if Cr(K ) < Cr(K) and a diagram of K can be obtained from a minimal diagram D of K by a single crossing change. In such a case we say that K < K. We investigate the sets of knots that can be obtained by single crossing changes over all minimal diagrams of a given knot. We show that these sets are specific for different knots and permit partial ordering of all the knots. Some interesting results are presented and many questions are posed.