A COMBINATORIAL APPROACH TO LINKING NUMBERS IN RATIONAL HOMOLOGY SPHERES

2000 ◽  
Vol 09 (05) ◽  
pp. 703-711
Author(s):  
CARL J. STITZ
Keyword(s):  
Combinatorial Approach ◽  
Rational Homology ◽  
Characteristic Curves ◽  
Combinatorial Description ◽  
Linking Number ◽  
Reidemeister Moves ◽  
Linking Numbers ◽  
Heegaard Diagram ◽  
Rational Homology Spheres ◽  
Matrix Invariants

In this paper we find a method to compute the classical Seifert-Threlfall linking number for rational homology spheres without using 2-chains bounded by the curves in question. By using a Heegaard diagram for the manifold, we describe link isotopy combinatorially using the three traditional Reidemeister moves along with a fourth move which is essentially a Kirby move along the characteristic curves. This result is mathematical folklore which we set in print. We then use this combinatorial description of link isotopy to develop and prove the invariance of linking numbers. Once the linking numbers are in place, matrix invariants such as the Alexander polynomial can be computed.

Strongly-cyclic branched coverings and the Alexander polynomial of knots in rational homology spheres

2007 ◽  
Vol 142 (2) ◽  
pp. 259-268 ◽  
Author(s):  
YUYA KODA
Keyword(s):  
Fundamental Group ◽  
Homology Group ◽  
Alexander Polynomial ◽  
Homology Sphere ◽  
Cyclic Covering ◽  
Rational Homology ◽  
Branched Coverings ◽  
Branched Covering ◽  
Rational Homology Spheres ◽  
Rational Homology Sphere

AbstractLet K be a knot in a rational homology sphere M. In this paper we correlate the Alexander polynomial of K with a g-word cyclic presentation for the fundamental group of the strongly-cyclic covering of M branched over K. We also give a formula for the order of the first homology group of the strongly-cyclic branched covering.

scholarly journals ON ALEXANDER MODULES AND BLANCHFIELD FORMS OF NULL-HOMOLOGOUS KNOTS IN RATIONAL HOMOLOGY SPHERES

2012 ◽  
Vol 21 (05) ◽  
pp. 1250042 ◽  
Author(s):  
DELPHINE MOUSSARD
Keyword(s):  
Isomorphism Class ◽  
The Other ◽  
Rational Homology ◽  
Rational Homology Spheres

In this paper, we give a classification of Alexander modules of null-homologous knots in rational homology spheres. We characterize these modules [Formula: see text] equipped with their Blanchfield forms ϕ, and the modules [Formula: see text] such that there is a unique isomorphism class of [Formula: see text], and we prove that for the other modules [Formula: see text], there are infinitely many such classes. We realize all these [Formula: see text] by explicit knots in ℚ-spheres.

scholarly journals A LINKING NUMBER DEFINITION OF THE AFFINE INDEX POLYNOMIAL AND APPLICATIONS

2013 ◽  
Vol 22 (12) ◽  
pp. 1341004 ◽  
Author(s):  
LENA C. FOLWACZNY ◽  
LOUIS H. KAUFFMAN
Keyword(s):  
Virtual Knots ◽  
Linking Number ◽  
Linking Numbers ◽  
Vassiliev Invariant ◽  
Definition Of

This paper gives an alternate definition of the Affine Index Polynomial (called the Wriggle Polynomial) using virtual linking numbers and explores applications of this polynomial. In particular, it proves the Cosmetic Crossing Change Conjecture for odd virtual knots and pure virtual knots. It also demonstrates that the polynomial can detect mutations by positive rotation and proves it cannot detect mutations by positive reflection. Finally it exhibits a pair of mutant knots that can be distinguished by a type 2 vassiliev invariant coming from the polynomial.

scholarly journals LINEAR INDEPENDENCE IN THE RATIONAL HOMOLOGY COBORDISM GROUP

2019 ◽  
pp. 1-12 ◽  
Author(s):  
Marco Golla ◽  
Kyle Larson
Keyword(s):  
Infinite Order ◽  
Double Cover ◽  
Rational Homology ◽  
Cobordism Group ◽  
Connected Sums ◽  
Knot Concordance ◽  
Linearly Independent ◽  
Rational Homology Spheres ◽  
Homology Cobordism ◽  
Correction Terms

We give simple homological conditions for a rational homology 3-sphere $Y$ to have infinite order in the rational homology cobordism group $\unicode[STIX]{x1D6E9}_{\mathbb{Q}}^{3}$ , and for a collection of rational homology spheres to be linearly independent. These translate immediately to statements about knot concordance when $Y$ is the branched double cover of a knot, recovering some results of Livingston and Naik. The statements depend only on the homology groups of the 3-manifolds, but are proven through an analysis of correction terms and their behavior under connected sums.

scholarly journals Towards Noncommutative Linking Numbers via the Seiberg-Witten Map

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
H. García-Compeán ◽  
O. Obregón ◽  
R. Santos-Silva
Keyword(s):  
Three Dimensional ◽  
Simons Theory ◽  
Dimensional Manifold ◽  
First Order ◽  
Linking Number ◽  
Linking Numbers ◽  
Bohm Effect ◽  
Aharonov Bohm ◽  
Chern Simons ◽  
Noncommutative Parameter

Some geometric and topological implications of noncommutative Wilson loops are explored via the Seiberg-Witten map. In the abelian Chern-Simons theory on a three-dimensional manifold, it is shown that the effect of noncommutativity is the appearance of6nnew knots at thenth order of the Seiberg-Witten expansion. These knots are trivial homology cycles which are Poincaré dual to the higher-order Seiberg-Witten potentials. Moreover the linking number of a standard 1-cycle with the Poincaré dual of the gauge field is shown to be written as an expansion of the linking number of this 1-cycle with the Poincaré dual of the Seiberg-Witten gauge fields. In the process we explicitly compute the noncommutative “Jones-Witten” invariants up to first order in the noncommutative parameter. Finally in order to exhibit a physical example, we apply these ideas explicitly to the Aharonov-Bohm effect. It is explicitly displayed at first order in the noncommutative parameter; we also show the relation to the noncommutative Landau levels.

Representations of Artin’s braid groups and linking numbers of periodic orbits

1995 ◽  
Vol 04 (02) ◽  
pp. 197-212 ◽  
Author(s):  
John Guaschi
Keyword(s):  
Lower Bound ◽  
Periodic Orbit ◽  
Topological Entropy ◽  
Periodic Orbits ◽  
Braid Groups ◽  
Matrix Representations ◽  
Linking Number ◽  
Linking Numbers ◽  
Small Period

Let P be a periodic orbit of period n≥3 of an orientation-preserving homeomorphism f of the 2-disc. Let q be the least integer greater than or equal to n/2−1. Then f admits a periodic orbit Q of period less than or equal to q such that the linking number of P about Q is non-zero. This answers a question of Franks in the affirmative in the case that P has small period. We also derive a result regarding matrix representations of Artin’s braid groups. Finally a lower bound for the topological entropy of a braid in terms of the trace of its Burau matrix is found.

ON A LOCAL MOVE FOR VIRTUAL KNOTS AND LINKS

2009 ◽  
Vol 18 (11) ◽  
pp. 1577-1596 ◽  
Author(s):  
TOSHIYUKI OIKAWA
Keyword(s):  
Sufficient Conditions ◽  
Virtual Link ◽  
Link Invariant ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Linking Number ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Necessary And Sufficient ◽  
Knots And Links

We define a local move called a CF-move on virtual link diagrams, and show that any virtual knot can be deformed into a trivial knot by using generalized Reidemeister moves and CF-moves. Moreover, we define a new virtual link invariant n(L) for a virtual 2-component link L whose virtual linking number is an integer. Then we give necessary and sufficient conditions for two virtual 2-component links to be deformed into each other by using generalized Reidemeister moves and CF-moves in terms of a virtual linking number and n(L).

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