Minimal Plat Representations of Prime Knots and Links are not Unique

1976 ◽  
Vol 28 (1) ◽  
pp. 161-167 ◽  
Author(s):  
José M. Montesinos
Keyword(s):  
Infinite Family ◽  
Equivalence Classes ◽  
Covering Space ◽  
Heegaard Splittings ◽  
Cyclic Covering ◽  
Knots And Links

Letdenote the 2-fold cyclic covering space branched over a linkLin S3. We wish to describe an infinite family of prime knots and links in which each memberLexhibits two minimal 6-plat representations, where the associated Heegaard splittings ofare minimal and inequivalent. Thus each knot or link of that family admits at least two equivalence classes of 6-plat representations which are minimal.

Piecewise-linear embeddings of knots and links with rotoinversion symmetry

2021 ◽  
Vol 77 (5) ◽  
Author(s):  
Michael O'Keeffe ◽  
Michael M. J. Treacy
Keyword(s):  
Piecewise Linear ◽  
Infinite Family ◽  
Knots And Links

This article describes the simplest members of an infinite family of knots and links that have achiral piecewise-linear embeddings in which linear segments (sticks) meet at corners. The structures described are all corner- and stick-2-transitive – the smallest possible for achiral knots.

scholarly journals Isogonal Weavings on the Sphere: Knots, Links, Polycatenanes

2020 ◽  
Author(s):  
Michael O'Keeffe ◽  
Michael Treacy
Keyword(s):  
Piecewise Linear ◽  
Infinite Family ◽  
Torus Knots ◽  
Vertex Transitive ◽  
Knots And Links ◽  
Complete Account

<p>We describe mathematical knots and links as piecewise linear – straight, non-intersecting sticks meeting at corners. Isogonal structures have all corners related by symmetry ("vertex" transitive). Corner- and stick-transitive structures are termed <i>regular</i>. We find no regular knots. Regular links are cubic or icosahedral and a complete account of these is given, including optimal (thickest-stick) embeddings. Stick 2-transitive isogonal structures are again cubic and icosahedral and also encompass the infinite family of torus knots and links. The major types of these structures are identified and reported with optimal embeddings. We note the relevance of this work to materials- and bio-chemistry.</p>

THE CASSON INVARIANT OF THE CYCLIC COVERING BRANCHED OVER SOME SATELLITE KNOT

2005 ◽  
Vol 14 (08) ◽  
pp. 1029-1044 ◽  
Author(s):  
YASUYOSHI TSUTSUMI
Keyword(s):  
Solid Torus ◽  
Covering Space ◽  
Winding Number ◽  
Cyclic Covering ◽  
Torus Knot ◽  
Satellite Knot

Let V be the standard solid torus in S3. Let Kp, 2 be the (p, 2)-torus knot in V such that Kp, 2 meets a meridian disk D of V in two points with the winding number zero and the 2-string tangle TKp, 2 obtained by cutting along D is a rational tangle. We compute the Casson invariant of the cyclic covering space of S3 branched over a satellite knot whose companion is any 2-bridge knot D(b1,…,b2m) and pattern is (V, Kp, 2).

Homology Invariants

1978 ◽  
Vol 30 (03) ◽  
pp. 655-670 ◽  
Author(s):  
Richard Hartley ◽  
Kunio Murasugi
Keyword(s):  
Homology Group ◽  
Covering Space ◽  
Simple Relationship ◽  
Cyclic Covering ◽  
Covering Spaces ◽  
Homology Groups ◽  
Branched Covering ◽  
The Relationship

There have been few published results concerning the relationship between the homology groups of branched and unbranched covering spaces of knots, despite the fact that these invariants are such powerful invariants for distinguishing knot types and have long been recognised as such [8]. It is well known that a simple relationship exists between these homology groups for cyclic covering spaces (see Example 3 in § 3), however for more complicated covering spaces, little has previously been known about the homology group, H1(M) of the branched covering space or about H1(U), U being the corresponding unbranched covering space, or about the relationship between these two groups.

scholarly journals Universal and Near-Universal Cycles of Set Partitions

10.37236/5051 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Zach Higgins ◽  
Elizabeth Kelley ◽  
Bertilla Sieben ◽  
Anant Godbole
Keyword(s):  
Analogous Result ◽  
Infinite Family ◽  
Equivalence Classes ◽  
Set Partitions ◽  
Universal Cycles

We study universal cycles of the set $\mathcal{P}(n,k)$ of $k$-partitions of the set $[n]:=\{1,2,\ldots,n\}$ and prove that the transition digraph associated with $\mathcal{P}(n,k)$ is Eulerian. But this does not imply that universal cycles (or ucycles) exist, since vertices represent equivalence classes of partitions. We use this result to prove, however, that ucycles of $\mathcal{P}(n,k)$ exist for all $n \geq 3$ when $k=2$. We reprove that they exist for odd $n$ when $k = n-1$ and that they do not exist for even $n$ when $k = n-1$. An infinite family of $(n,k)$ for which ucycles do not exist is shown to be those pairs for which ${{n-2}\brace{k-2}}$ is odd ($3 \leq k < n-1$). We also show that there exist universal cycles of partitions of $[n]$ into $k$ subsets of distinct sizes when $k$ is sufficiently smaller than $n$, and therefore that there exist universal packings of the partitions in $\mathcal{P}(n,k)$. An analogous result for coverings completes the investigation.  

Metacyclic Invariants of Knots and Links

1970 ◽  
Vol 22 (2) ◽  
pp. 193-201 ◽  
Author(s):  
R. H. Fox
Keyword(s):  
Symmetric Group ◽  
Permutation Group ◽  
Equivalence Class ◽  
Base Point ◽  
Covering Space ◽  
Transitive Permutation Group ◽  
Oriented Link ◽  
Inner Automorphism ◽  
Knots And Links

To each representation ρ on a transitive permutation group P of the group G = π(S – k) of an (ordered and oriented) link k = k1 ∪ k2 ∪ … ∪ kμ in the oriented 3-sphere S there is associated an oriented open 3-manifold M = Mρ(k), the covering space of S – k that belongs to ρ. The points 01, 02, … that lie over the base point o may be indexed in such a way that the elements g of G into which the paths from oi to oj project are represented by the permutations gρ of the form , and this property characterizes M. Of course M does not depend on the actual indices assigned to the points o1, o2, … but only on the equivalence class of ρ, where two representations ρ of G onto P and ρ′ of G onto P′ are equivalent when there is an inner automorphism θ of some symmetric group in which both P and P′ are contained which is such that ρ′ = θρ.

scholarly journals Isogonal weavings on the sphere: knots, links, polycatenanes

2020 ◽  
Vol 76 (5) ◽  
pp. 611-621
Author(s):  
Michael O'Keeffe ◽  
Michael M. J. Treacy
Keyword(s):  
Piecewise Linear ◽  
Infinite Family ◽  
Materials Chemistry ◽  
Torus Knots ◽  
Vertex Transitive ◽  
Knots And Links ◽  
Complete Account

Mathematical knots and links are described as piecewise linear – straight, non-intersecting sticks meeting at corners. Isogonal structures have all corners related by symmetry (`vertex'-transitive). Corner- and stick-transitive structures are termed regular. No regular knots are found. Regular links are cubic or icosahedral and a complete account of these (36 in number) is given, including optimal (thickest-stick) embeddings. Stick 2-transitive isogonal structures are again cubic and icosahedral and also encompass the infinite family of torus knots and links. The major types of these structures are identified and reported with optimal embeddings. The relevance of this work to materials chemistry and biochemistry is noted.

An infinite family of non-Haken hyperbolic 3-manifolds with vanishing Whitehead groups

1986 ◽  
Vol 99 (2) ◽  
pp. 239-246 ◽  
Author(s):  
Andrew J. Nicas
Keyword(s):  
Fundamental Group ◽  
Infinite Family ◽  
Universal Covering ◽  
Covering Space ◽  
Homotopy Equivalence ◽  
Universal Covering Space ◽  
Aspherical Manifold ◽  
Aspherical Manifolds ◽  
Whitehead Groups

A manifold M is said to be aspherical if its universal covering space is contractible. Farrell and Hsiang have conjectured [3]:Conjecture A. (Topological rigidity of aspherical manifolds.) Any homotopy equivalence f: N → M between closed aspherical manifolds is homotopic to a homeomorphism,and its analogue in algebraic K-theory:Conjecture B. The Whitehead groups Whj(π1M)(j ≥ 0) of the fundamental group of a closed aspherical manifold M vanish.

ON 4-MOVE EQUIVALENCE CLASSES OF KNOTS AND LINKS OF TWO COMPONENTS

2011 ◽  
Vol 20 (01) ◽  
pp. 47-90 ◽  
Author(s):  
M. K. DABKOWSKI ◽  
S. JABLAN ◽  
N. A. KHAN ◽  
R. K. SAHI
Keyword(s):  
Mirror Image ◽  
Equivalence Classes ◽  
The Family ◽  
Alternating Links ◽  
Alternating Link ◽  
Knots And Links ◽  
Trivial Link ◽  
Hopf Link

We study equivalence classes of knots and links of 2 components modulo 4-move. We show that all knots up to 12 crossings and knots in the family 6* reduce by 4-moves to the trivial knot. We also prove that links of 2 components with 11 crossings, and links 6* a1.a2.a3.a4.a5.a6 such that ai is a 2-algebraic tangle with no trivial components reduce to either the trivial link or to the Hopf link. For alternating links of 2-components with 12 we show that L reduces by 4-moves to either trivial link or to the Hopf link whenever L is different than 9*.2 : .2 : .2 (or its mirror image). We suggest the alternating link 9*.2 : .2 : .2 with 12 crossings as a potential example to answer the Problem 1.1(iii) in negative.

scholarly journals A FACTORIZATION OF THE CONWAY POLYNOMIAL AND COVERING LINKAGE INVARIANTS

2007 ◽  
Vol 16 (05) ◽  
pp. 631-640 ◽  
Author(s):  
TATSUYA TSUKAMOTO ◽  
AKIRA YASUHARA
Keyword(s):  
Taylor Expansion ◽  
The Other ◽  
Covering Space ◽  
Cyclic Covering ◽  
Quotient Field ◽  
Conway Polynomial ◽  
Linking Numbers ◽  
Two Factors

Levine showed that the Conway polynomial of a link is a product of two factors: one is the Conway polynomial of a knot which is obtained from the link by banding together the components; and the other is determined by the [Formula: see text]-invariants of a string link with the link as its closure. We give another description of the latter factor: the determinant of a matrix whose entries are linking pairings in the infinite cyclic covering space of the knot complement, which take values in the quotient field of ℤ[t, t-1]. In addition, we give a relation between the Taylor expansion of a linking pairing around t = 1 and derivation on links which is invented by Cochran. In fact, the coefficients of the powers of t - 1 will be the linking numbers of certain derived links in S3. Therefore, the first non-vanishing coefficient of the Conway polynomial is determined by the linking numbers in S3. This generalizes a result of Hoste.

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