14 Pages
English

A SHORT INTRODUCTION TO THE ALEXANDER POLYNOMIAL

-

Gain access to the library to view online
Learn more

Description

Niveau: Supérieur, Doctorat, Bac+8
A SHORT INTRODUCTION TO THE ALEXANDER POLYNOMIAL GWENAEL MASSUYEAU Abstract. These informal notes accompany a talk given in Grenoble for the work- shop “Representations de Uq(sl2) et invariants d'Alexander” (December 2008). We introduce the Alexander polynomial of links following Milnor and Turaev, who inter- preted this classical invariant as a kind of Reidemeister torsion. As shown by Turaev, this approach allows for an intrinsic construction of the Conway function of links, which is a refinement of the Alexander polynomial. Contents 1. Introduction 1 2. The Alexander function 2 2.1. The order of a module 2 2.2. The Alexander function of a finite connected CW-complex 3 2.3. The Alexander function of a link 5 3. The Milnor torsion 6 3.1. The torsion of a based acyclic chain complex 7 3.2. The Milnor torsion of a finite connected CW-complex 8 3.3. The Milnor torsion of a link 10 4. The Conway function 10 4.1. Turaev's sign-refinement of the Reidemeister torsion 10 4.2. Construction of the Conway function 11 References 14 1. Introduction James Alexander introduced his link invariant in the paper [1] published in 1928. Since then, the Alexander polynomial has been extensively studied by hundreds of authors, and several fundamental properties of the Alexander polynomial have been shown. See any of the classical references in knot theory, including [16] and [3].

  • links following

  • group

  • polynomial has

  • reidemeister torsion

  • alexander polynomial

  • approach via reidemeister torsions

  • unique factorization domain

  • ring z


Subjects

Informations

Published by
Reads 12
Language English
ASHORTINTRODUCTIONTOTHEALEXANDERPOLYNOMIALGWE´NAE¨LMASSUYEAUAbstract.TheseinformalnotesaccompanyatalkgiveninGrenobleforthework-shop“Repre´sentationsdeUq(sl2)etinvariantsd’Alexander”(December2008).WeintroducetheAlexanderpolynomialoflinksfollowingMilnorandTuraev,whointer-pretedthisclassicalinvariantasakindofReidemeistertorsion.AsshownbyTuraev,thisapproachallowsforanintrinsicconstructionoftheConwayfunctionoflinks,whichisarefinementoftheAlexanderpolynomial.Contents1.Introduction2.TheAlexanderfunction2.1.Theorderofamodule2.2.TheAlexanderfunctionofaniteconnectedCW-complex2.3.TheAlexanderfunctionofalink3.TheMilnortorsion3.1.Thetorsionofabasedacyclicchaincomplex3.2.TheMilnortorsionofaniteconnectedCW-complex3.3.TheMilnortorsionofalink4.TheConwayfunction4.1.Turaevssign-renementoftheReidemeistertorsion4.2.ConstructionoftheConwayfunctionReferences1223567801010111411.IntroductionJamesAlexanderintroducedhislinkinvariantinthepaper[1]publishedin1928.Sincethen,theAlexanderpolynomialhasbeenextensivelystudiedbyhundredsofauthors,andseveralfundamentalpropertiesoftheAlexanderpolynomialhavebeenshown.Seeanyoftheclassicalreferencesinknottheory,including[16]and[3].TheAlexanderpolynomialofalinkisdefineduptosomeindeterminacy.In1967,JohnConwayintroducedin[5]arefinementoftheAlexanderpolynomialandexplainedhowtocomputeitrecursivelyusingsomeadditiverelationsofalocalnature(whicharenowadayscalled“skeinrelations”).Conway’sapproachwasfixedbyLouisKauffmanintheone-variablecase[10]andbyRichardHartleyinthemulti-variablecase[9].But,thosetwoconstructionsoftheConwayfunctionareextrinsic.(KauffmanneedsaSeifertsurfaceforthelink,whileHartleyneedsadiagrampresentationofthelink.)In1962,JohnMilnornoticedin[11]averycloseconnectionbetweentheAlexanderpolynomialofalinkandacertainkindofReidemeistertorsion.ThisnewviewpointDate:November28,2008.1