On Two Categorifications of the Arrow Polynomial for Virtual Knots

363 Pages·2010·4.95 MB·English

Contributions in Mathematical
and Computational Sciences (cid:2) Volume 1
Editors
HansGeorgBock
WilliJäger
Forothertitlespublishedinthisseries,goto
www.springer.com/series/8861
[email protected]
Markus Banagl (cid:2) Denis Vogel
Editors
The Mathematics
of Knots
Theory and Application
[email protected]
Editors
MarkusBanagl DenisVogel
Dept.ofMathematics Dept.ofMathematics
HeidelbergUniversity HeidelbergUniversity
ImNeuenheimerFeld288 ImNeuenheimerFeld288
69120Heidelberg 69120Heidelberg
Germany Germany
[email protected] [email protected]
ISBN978-3-642-15636-6 e-ISBN978-3-642-15637-3
DOI10.1007/978-3-642-15637-3
SpringerHeidelbergDordrechtLondonNewYork
MathematicsSubjectClassification(2010): 57M25,57Q45,92C37,81T45
©Springer-VerlagBerlinHeidelberg2011
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Preface to the Series
ContributionsinMathematicaland Computational Sciences
Mathematicaltheoriesandmethodsandeffectivecomputationalalgorithmsarecru-
cialincopingwiththechallengesarisinginthesciencesandinmanyareasoftheir
application. New concepts and approaches are necessary in order to overcome the
complexitybarriersparticularlycreatedbynonlinearity,high-dimensionality,mul-
tiplescalesanduncertainty.Combiningadvancedmathematicalandcomputational
methods and computer technology is an essential key to achieving progress, often
eveninpurelytheoreticalresearch.
Thetermmathematicalsciencesreferstomathematicsanditsgenuinesub-fields,
aswellastoscientificdisciplinesthatarebasedonmathematicalconceptsandmeth-
ods, including sub-fields of the natural and life sciences, the engineering and so-
cial sciences and recently also of the humanities. It is a major aim of this series
to integrate the different sub-fields within mathematics and the computationalsci-
ences,andtobuildbridgestoallacademicdisciplines,toindustryandotherfields
ofsociety,wheremathematicalandcomputationalmethodsarenecessarytoolsfor
progress.Fundamentalandapplication-orientedresearchwillbecoveredinproper
balance.
The series will further offer contributions on areas at the frontier of research,
providing both detailed information on topical research, as well as surveys of the
state-of-the-artinamannernotusuallypossibleinstandardjournalpublications.Its
volumes are intended to cover themes involving more than just a single “spectral
line”oftherichspectrumofmathematicalandcomputationalresearch.
TheMathematicsCenterHeidelberg(MATCH)andtheInterdisciplinaryCenter
forScientificComputing(IWR)withitsHeidelbergGraduateSchoolofMathemat-
icalandComputationalMethodsfortheSciences(HGS)areinchargeofproviding
andpreparingthematerialforpublication.Asubstantialpartofthematerialwillbe
acquiredinworkshopsandsymposiaorganizedbytheseinstitutionsintopicalareas
of research. The resulting volumes should be more than just proceedings collect-
ingpaperssubmittedinadvance.Theexchangeofinformationandthediscussions
duringthemeetingsshouldalsohaveasubstantialinfluenceonthecontributions.
v
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vi PrefacetotheSeries
Starting this series is a venture posing challenges to all partners involved.
Auniquestyleattractingalargeraudiencebeyondthegroupofexpertsinthesubject
areasofspecificvolumeswillhavetobedeveloped.
The first volume covers the mathematics of knots in theory and application, a
field that appears excellently suited for the start of the series. Furthermore, due to
therolethatfamousmathematiciansinHeidelberglikeHerbertSeifert(1907–1996)
playedinthedevelopmentoftopologyingeneralandknottheoryinparticular,Hei-
delbergseemedafittingplacetohostthespecialactivitiesunderlyingthisvolume.
SpringerVerlagdeservesourspecialappreciationforitsmostefficientsupportin
structuringandinitiatingthisseries.
HeidelbergUniversity,Germany WilliJäger
HansGeorgBock
[email protected]
Preface
This volume is based on the themes of, and records advances achieved as a re-
sultof,theHeidelbergKnotTheorySemester,heldinwinter2008/09atHeidelberg
UniversityunderthesponsorshipoftheMathematicsCenterHeidelberg(MATCH),
organizedbyM.BanaglandD.Vogel.Intheprecedingsummersemesteranintro-
ductoryseminaronknotsaimedatprovidingnon-expertsandyoungmathematicians
withsomeofthefoundationalknowledgerequiredtoparticipateintheeventsofthe
winter semester. These comprised expository lecture series by several leading ex-
perts,representingratherdiverseaspectsofknottheoryanditsapplications,anda
concludingworkshopheldDecember15to19,2008.
Knotsseemtobeadeepstructure,whosepeculiarfeatureitistosurfaceunex-
pectedlyinmanydifferentandaprioriunrelatedareasofmathematicsandthenat-
uralsciences,suchasalgebraandnumbertheory,topologyandgeometry,analysis,
mathematical physics (in particular statistical mechanics), and molecular biology.
Itsrelevanceintopology,apartfromitsintrinsicinterest,ispartlyduetothefactthat
every closed, oriented 3-manifold can be obtained by surgery on a framed link in
the3-sphere.Moderntopologyhasalsoobtainedinformationonhigh-dimensional
knots,thatis,embeddingsofann-sphereinan(n+2)-spherewithnlargerthanone.
Inalgebra,representationsofquantumgroupsleadtoamultitudeofknotinvariants.
BasedonideasofB.Mazurinnumbertheory,onecanassigntotwoprimeidealsof
anumberfieldalinkingnumberinanalogywithclassicalknottheory.Thisnumber-
theoreticlinkingnumberplaysaroleinstudyingthestructureofGaloisgroupsof
certainextensionsofthenumberfield.Analysistouchesonknottheorybymeansof
operatoralgebrasandtheirconnectiontotheJonespolynomial.Asfarasgeometry
isconcerned,resultsbyFenchelonthecurvatureofaclosedspacecurvedateback
tothe1920s.Milnorshowedin1949thatthecurvaturemustexceed4π ifthecurve
isknotted.Onealsoconsiders“real”knotsasphysicalobjectsin3-spaceandstud-
iesvariousnaturalenergyfunctionalsonthem.Sumstakenoverallstatesofsuitable
modelsoriginatinginstatisticalmechanics,describinglargeensemblesofparticles,
can express knot invariants such as L. Kauffman’s bracket polynomial. The dis-
coveryoftheJonespolynomialentailedtieswithmathematicalphysicsbasedona
curiouscongruityoffiverelations,namelytheArtin-relationinbraidgroups,afun-
damentalrelationincertainoperatoralgebrasduetoHecke,thethirdReidemeister
vii
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viii Preface
move,theclassicalYang-Baxterequation,anditsquantumversion.Thisleadtothe
construction of topological quantum field theories by Witten and Atiyah. Cellular
DNAisalongmolecule,whichmaybeclosed(ase.g.thegenomeofcertainbacte-
ria)andknottedorlinkedwithotherDNAstrands.Enzymessuchastopoisomerase
orrecombinaseoperateonDNAchangingthetopologicalknotorlinktype.
TheobjectiveoftheHeidelbergKnotTheorySemesterwastodojusticetothis
diversitybybringingtogetherrepresentativesofmostoftheaboveresearchavenues,
accompaniedbythehopethatsuchameetingmightfosterinspirationandsynergy
acrossthevariousquestionsandapproaches.Certainly,afairlycomprehensivepor-
traitofthecurrentstate-of-the-artinknottheoryanditsapplicationsemergedasa
result.
Four lecture series were given: DeWitt Sumners gave 5 lectures on scientific
applications of knot theory, discussing DNA topology, a tangle model for DNA
site-specific recombination, random knotting, topoisomerase, spiral waves and vi-
ral DNA packing. Kent Orr’s 3 lectures explained knot concordance and surgery
techniques,whileLouisKauffman’s2lecturesintroducedvirtualknotsanddetailed
parallelstoelementaryparticles.ThetopicofMasanoriMorishita’s6lectureswere
theaforementionedanalogiesbetweenknottheoryandnumbertheory.
The21speakersofthefinalworkshop“TheMathematicsofKnots”reportedon
avarietyofinterestingcurrentdevelopments.Manyoftheseaccountsaremirrored
inthepapersofthepresentvolume.Amongthelow-dimensionaltopicswerevirtual
knotsandassociatedinvariantssuchasarrowandJonespolynomials,theHOMFLY
polynomial,questionsaboutDehnfilling,Legendrianknots,Khovanovhomology,
surface knots, slice knots, fibered knots and property R, colorings by metabelian
groups, singular knots, Gram determinants of planar curves, as well as geometric
structures such as surfaces associated with knots, and the fibering of 3-manifolds
when the product of the manifold with a circle is known to be symplectic. High-
dimensionaltopicsconcernedtheCohnnoncommutativelocalizationofringsandits
applicationtoknotsviaalgebraicK-andL-theory,aswellashigh-dimensionalnon-
locallyflatembeddingsandtheroleofknottheoryvis-à-vistransformationgroups.
Scientifictalksdiscussedrandomknotting,viralDNApacking,andthetopologyof
DNA-proteininteractions.
We wish to extend our sincere thanks to the contributors of this volume and to
allparticipantsoftheHeidelbergKnotTheorySemester,especiallytothelecturers
givingmini-courses,fortheenergyandtimetheyhavedevotedtothiseventandthe
preparation of the present collection. Paul Seyfert receives the editors’ thanks for
technicalhelpintypesettingthisvolume.Furthermore,wearegratefultoDorothea
Heukäuferforherefficienthandlingofnumerouslogisticalissues.Finally,wewould
like to express our gratitude to Willi Jäger and MATCH, whose financial support
madetheHeidelbergKnotTheorySemesterpossible.
HeidelbergUniversity,Germany MarkusBanagl
DenisVogel
[email protected]
Contents
1 Knots,SingularEmbeddings,andMonodromy . . . . . . . . . . . . 1
MarkusBanagl,SylvainE.Cappell,andJuliusL.Shaneson
2 LowerBoundsonVirtualCrossingNumberandMinimalSurface
Genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
KumudBhandari,H.A.Dye,andLouisH.Kauffman
3 ASurveyofTwistedAlexanderPolynomials . . . . . . . . . . . . . . 45
StefanFriedlandStefanoVidussi
4 OnTwoCategorificationsoftheArrowPolynomialforVirtual
Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
HeatherAnnDye,LouisHirschKauffman,andVassilyOlegovich
Manturov
5 AnAdelicExtensionoftheJonesPolynomial . . . . . . . . . . . . . 125
JesúsJuyumayaandSofiaLambropoulou
6 LegendrianGridNumberOneKnotsandAugmentations
ofTheirDifferentialAlgebras . . . . . . . . . . . . . . . . . . . . . . 143
JoanE.Licata
7 EmbeddingsofFour-valentFramedGraphsinto2-surfaces . . . . . 169
VassilyOlegovichManturov
8 GeometricTopologyandFieldTheoryon3-Manifolds . . . . . . . . 199
KishoreMarathe
9 FromGoeritzMatricestoQuasi-alternatingLinks . . . . . . . . . . 257
JózefH.Przytycki
ix
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x Contents
10 AnOverviewofProperty2R. . . . . . . . . . . . . . . . . . . . . . . 317
MartinScharlemann
11 DNA,KnotsandTangles . . . . . . . . . . . . . . . . . . . . . . . . . 327
DeWittSumners
WorkshopTalks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
[email protected]