Intersection Numbers for Twisted Homology

2004 ◽  
Vol 114 (2) ◽  
Author(s):  
Toyoshi Togi
Keyword(s):  
Intersection Numbers ◽  
Twisted Homology

THE JONES POLYNOMIAL AND THE INTERSECTION NUMBERS OF TWISTED CYCLES ASSOCIATED WITH A SELBERG TYPE INTEGRAL

2011 ◽  
Vol 20 (03) ◽  
pp. 469-496 ◽  
Author(s):  
KATSUHISA MIMACHI
Keyword(s):  
Special Functions ◽  
Jones Polynomial ◽  
Intersection Number ◽  
Homology Theory ◽  
Intersection Numbers ◽  
Type Integral ◽  
Jones Polynomials ◽  
Twisted Homology ◽  
Hypergeometric Type ◽  
Definition Of

We give a new definition of the Jones polynomial by means of the intersection number of loaded (or twisted) cycles associated with a Selberg type integral. Our definition is naturally formulated in the framework of the twisted homology theory, which is developd by Aomoto to study the special functions of hypergeometric type. The naturality of the definition leads to evaluate the Jones polynomials in several cases: well-known results in the case of two-bridge link, a formula for (3, s)-torus and that for the Prezel with 3 parameters. Our definition is motivated by the work of Bigelow.

Pseudo-Anosov Theory

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Braid Groups ◽  
Intersection Numbers ◽  
Branched Covers ◽  
Algebraic Integers ◽  
Dehn Twists ◽  
Mapping Classes ◽  
Dynamical Properties

This chapter focuses on the construction as well as the algebraic and dynamical properties of pseudo-Anosov homeomorphisms. It first presents five different constructions of pseudo-Anosov mapping classes: branched covers, constructions via Dehn twists, homological criterion, Kra's construction, and a construction for braid groups. It then proves a few fundamental facts concerning stretch factors of pseudo-Anosov homeomorphisms, focusing on the theorem that pseudo-Anosov stretch factors are algebraic integers. It also considers the spectrum of pseudo-Anosov stretch factors, along with the special properties of those measured foliations that are the stable (or unstable) foliations of some pseudo-Anosov homeomorphism. Finally, it describes the orbits of a pseudo-Anosov homeomorphism as well as lengths of curves and intersection numbers under iteration.

scholarly journals Profinite rigidity for twisted Alexander polynomials

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jun Ueki
Keyword(s):  
Rigidity Theorem ◽  
Homology Groups ◽  
Alexander Polynomials ◽  
Twisted Homology ◽  
Hyperbolic Volumes

AbstractWe formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a {{\mathbb{Z}}}-cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley’s parabolic representations of two-bridge knot groups and give a remark on hyperbolic volumes.

scholarly journals Twisted Homology of Quantum SL(2) - Part II

2009 ◽  
Vol 6 (1) ◽  
pp. 69-98 ◽  
Author(s):  
Tom Hadfield ◽  
Ulrich Krähmer
Keyword(s):  
Noncommutative Geometry ◽  
Hochschild Cohomology ◽  
Cyclic Homology ◽  
Coordinate Ring ◽  
Volume Form ◽  
Twisted Homology

AbstractWe complete the calculation of the twisted cyclic homology of the quantised coordinate ring = ℂq [SL(2)] of SL(2) that we began in [14]. In particular, a nontrivial cyclic 3-cocycle is constructed which also has a nontrivial class in Hochschild cohomology and thus should be viewed as a noncommutative geometry analogue of a volume form.

scholarly journals Intersection Numbers of Heegner Divisors on Shimura Curves

2008 ◽  
Vol 4 (4) ◽  
pp. 1165-1204
Author(s):  
Kevin Keating ◽  
David P. Roberts
Keyword(s):  
Shimura Curves ◽  
Intersection Numbers

scholarly journals Intersection numbers on the relative Hilbert schemes of points on surfaces

2017 ◽  
Vol 21 (3) ◽  
pp. 531-542 ◽  
Author(s):  
Amin Gholampour ◽  
Artan Sheshmani
Keyword(s):  
Hilbert Schemes ◽  
Intersection Numbers
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