FRAMED KNOTS IN 3-MANIFOLDS AND AFFINE SELF-LINKING NUMBERS

2005 ◽  
Vol 14 (06) ◽  
pp. 791-818 ◽  
Author(s):  
VLADIMIR CHERNOV TCHERNOV
Keyword(s):  
The Self ◽  
Compact Manifolds ◽  
Not Given ◽  
Connected Sum ◽  
Linking Number ◽  
Fundamental Fact ◽  
Linking Numbers

The number |K| of non-isotopic framed knots that correspond to a given unframed knot K ⊂ S3 is infinite. This follows from the existence of the self-linking number slk of a zero homologous framed knot. We use the approach of Vassiliev–Goussarov invariants to construct "affine self-linking numbers" that are extensions of slk to the case of nonzero homologous framed knots in 3-manifolds. As a corollary we get that |K| = ∞ for all knots in an oriented (not necessarily compact) 3-manifold M that is not realizable as a connected sum (S1 × S2)# M′. This result for compact manifolds was first stated by Hoste and Przytycki. They referred to the works of McCullough for the idea of the proof, however to the best of our knowledge prior to this work the proof of this fundamental fact was not given in literature or in a preprint form. Our proof is based on different ideas. For M = (S1 × S2)# M′ we construct K in M such that |K| = 2 ≠ ∞.

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 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.

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.

scholarly journals The number of framings of a knot in a 3-manifold

2014 ◽  
Vol 23 (13) ◽  
pp. 1450072 ◽  
Author(s):  
Patricia Cahn ◽  
Vladimir Chernov ◽  
Rustam Sadykov
Keyword(s):  
Normal Bundle ◽  
Closed Curve ◽  
The Self ◽  
Connected Sum ◽  
Orientable Manifold ◽  
Intersection Points

In view of the self-linking invariant, the number |K| of framed knots in S3 with given underlying knot K is infinite. In fact, the second author previously defined affine self-linking invariants and used them to show that |K| is infinite for every knot in an orientable manifold unless the manifold contains a connected sum factor of S1 × S2; the knot K need not be zero-homologous and the manifold is not required to be compact. We show that when M is orientable, the number |K| is infinite unless K intersects a nonseparating sphere at exactly one point, in which case |K| = 2; the existence of a nonseparating sphere implies that M contains a connected sum factor of S1 × S2. For knots in nonorientable manifolds we show that if |K| is finite, then K is disorienting, or there is an orientation-preserving isotopy of the knot to itself which changes the orientation of its normal bundle, or it intersects some embedded S2 or ℝP2 at exactly one point, or it intersects some embedded S2 at exactly two points in such a way that a closed curve consisting of an arc in K between the intersection points and an arc in S2 is disorienting.

Self-intersection of foliation cycles on complex manifolds

2017 ◽  
Vol 28 (08) ◽  
pp. 1750054 ◽  
Author(s):  
Lucas Kaufmann
Keyword(s):  
Cohomology Class ◽  
Kähler Manifold ◽  
Projective Spaces ◽  
The Self ◽  
Compact Manifolds ◽  
Complex Manifolds ◽  
Kahler Manifold ◽  
Compact Kähler Manifold

Let [Formula: see text] be a compact Kähler manifold and let [Formula: see text] be a foliation cycle directed by a transversely Lipschitz lamination on [Formula: see text]. We prove that the self-intersection of the cohomology class of [Formula: see text] vanishes as long as [Formula: see text] does not contain currents of integration along compact manifolds. As a consequence, we prove that transversely Lipschitz laminations of low codimension in certain manifolds, e.g. projective spaces, do not carry any foliation cycles except those given by integration along compact leaves.

scholarly journals Quaternions and hybrid nematic disclinations

2013 ◽  
Vol 469 (2156) ◽  
pp. 20130204 ◽  
Author(s):  
Simon Čopar ◽  
Slobodan Žumer
Keyword(s):  
Liquid Crystals ◽  
Homotopy Theory ◽  
Classification Scheme ◽  
Colloidal Suspensions ◽  
The Self ◽  
Topological Classification ◽  
Winding Number ◽  
Linking Number ◽  
Disclination Line ◽  
Geometric Decomposition

Disclination lines in nematic liquid crystals can exist in different geometric conformations, characterized by their director profile. In certain confined colloidal suspensions and even more prominently in chiral nematics, the director profile may vary along the disclination line. We construct a robust geometric decomposition of director profile in closed disclination loops and use it to apply topological classification to linked loops with arbitrary variation of the profile, generalizing the self-linking number description of disclination loops with the winding number . The description bridges the gap between the known abstract classification scheme derived from homotopy theory and the observable local features of disclinations, allowing application of said theory to structures that occur in practice.

ON THE UNORIENTED SATO-LEVINE INVARIANT

1993 ◽  
Vol 02 (03) ◽  
pp. 335-358 ◽  
Author(s):  
MASAHICO SAITO
Keyword(s):  
Equivalence Classes ◽  
Linking Number ◽  
Conway Polynomial ◽  
Linking Numbers ◽  
Arf Invariant ◽  
Orientable Surfaces

The Sato-Levine invariant for 2-component links with linking number zero is generalized to links with even linking numbers using not necessarily orientable surfaces. It is well defined modulo 4 and completely classifies “unoriented pass equivalence classes” of 2-component proper links. It is shown that it can be expressed in terms of the coefficients of the Conway polynomial and some applications to the Arf invariant are given.

scholarly journals LINKING NUMBER IN A PROJECTIVE SPACE AS THE DEGREE OF A MAP

2007 ◽  
Vol 16 (04) ◽  
pp. 489-497 ◽  
Author(s):  
JULIA VIRO DROBOTUKHINA
Keyword(s):  
Projective Space ◽  
Configuration Space ◽  
The Self ◽  
Real Projective Space ◽  
3 Dimensional ◽  
Linking Number ◽  
Degree Of A Map ◽  
Number Of Cycles ◽  
Dimensional Projective Space

For any two disjoint oriented circles embedded into the 3-dimensional real projective space, we construct a 3-dimensional configuration space and its map to the projective space such that the linking number of the circles is the half of the degree of the map. Similar interpretations are given for the linking number of cycles in a projective space of arbitrary odd dimension and the self-linking number of a zero homologous knot in the 3-dimensional projective space.

Sign in / Sign up

Export Citation Format

Share Document