RANDOMKNOTSIGORRIVIN,UNIVERSITYOFSTANDREWSANDTEMPLEUNIVERSITY

WHYRANDOMTOPOLOGY?

• Weknowwhat(forexample)theaveragelifeexpectancyofarandomhuman(orrandomAmerican,orrandomFrenchmanbornin1797)is.Wealsoknowthestandarddeviationofthisquantity,sowecansaythatifsomeonelivesuntiltheageof120,theyareveryunusual,whichcanbeinterpretedinanumberofways(ifweareaninsurancecompany,weprobablydon’thavetoworrytoomuchaboutsomeonelivingto120,ontheotherhand,ifweareadoctor,wemightwanttostudythe120yearold,toseewhatabouthim(or,morecommonly,her)isdifferent.

• So,it’sbotheasierthanunderstandingeverysingleperson,andinformative,inthatitgivesusanoverviewofthesituation.

• Plus,therandomnessletsusgeneratelargeexamples,anddefeatthe“lawofsmallnumbers”.

EASYISGOOD

WHATDOES”RANDOM”MEAN?GOODTHINGABOUTHUMANSISTHATTHEREAREONLYFINITELYMANYOFTHEM.SADLY,NOTSOFORMOSTMATHEMATICALSTRUCTURES.

WHATISARANDOMKNOT?

Whatisarandomknot?

Firstquestionis,whatisaknot?

WHATISAKNOT?

• Aknotisasimpleclosedcurvein𝑅"

• Asimpleclosedcurvein𝑅" isamapfrom𝑆$ to𝑅".

• Amapfrom𝑆$ to𝑅".Isatripleofmapfrom𝑆$ toR.

• Amapfrom𝑆$ toRisaperiodicfunction.

• WhichisgivenbyaFourierseries:𝑓 𝜃 = ∑ 𝑎+,+-. sin 2𝑘𝜋𝜃 +𝑏+ cos 2𝑘𝜋𝜃

• So,aknotisatripleofFourierseries,andarandomknotisatripleofrandom Fourierseries!

HISTORICALNOTE

• AsfarasIknow,thismodelwasintroducedbyBillThurstonintheearly1980s.

• Hewantedtoknowthedistributionofknottypes,andwasproposingtousethe(then)newJonespolynomialtostudyit.

• Hehadagraduatestudent(BruceRamsay)workingonit,butnothingcameofthis.

RANDOMTRIGONOMETRICSERIES

• Wewanttokeepthingssimple,soweassumethatthecoefficients(the𝑎+ andthe𝑏+)arerandomvariables.Forsimplicity,weassumetheyareindependent,Gaussian,butnot necessarilyidentical– wecanmakethevarianceafunctionofk.

• Wecanalsoassumethattheyare identical,butthereareonlyN frequencies(so,aFourierpolynomialofdegreeN).

HOWMANYDIFFERENTKNOTTYPES DOWEGET?

• Howdowetellthemapart?

• Bysimplifyingthequestion:welookattheknotprojectionontosomeplane(forourFourierknots,thefirsttwocoordinatesformaniceplane).

• Insteadofknottypes(whicharehardtotellapart),welookatthenumberofself-intersectionsoftheprojection– thisupper-boundsthecrossingnumberoftheknot.

• Andnow?

CROSSINGNUMBEROFAKNOT

• Thecrossingnumberistheminimal numberofself-intersectionsofanytopologicallyequivalentknotontoaplane(wecanassumethattheplaneisalwaysthexy plane,byrotatingtheknot).

• KnowntobeinNP(Hass,Lagarias,Pippenger) andco-NP(Lackenby),notknowntobepolynomialtime.

RANDOMPLANECURVES

• Whatdoesdecayofcoefficientstranslateinto?

PLANECURVE,QUADRATICDECAY

-1000 -500 500 1000

-500

500

1000

PLANECURVE,𝑘9"/; DECAY

-1000 -500 500 1000

-1000

-500

500

PLANECURVE,LINEARDECAY(SHOULDLOOKFAMILIAR)

-2000 -1000 1000

-1500

-1000

-500

500

1000

1500

PLANECURVE,𝑘9$/; DECAY(NOTREALLY)

-4000 -2000 2000 4000 6000

-6000

-4000

-2000

2000

4000

6000

MAINOBSERVATIONS:

• Thelimitingcurveiscontinuouslydifferentiable(almostsurely)ifthedecayis𝑘9(=>?@) forsomepositive

𝜖.

• Theexpectednumberofselfintersectionsisfiniteunderpreciselythesameconditions.

HOWDOYOUPROVETHELATTER?

FORUS,THEFUNCTIONIS

• (F(s)-F(t))/(s-t).

COROLLARY

• TheprobabilitythatarandomsmoothcurvehasmorethanNcrossingsdecaysatleastlike1/N.

• NOTthetruth:thetruthisthatthedecayisatleastexponentialinN– notclearhowtoproveyet(concentrationofmeasure?)

FACT

• Thereareexponentiallymany(inN)knotswithatmostN crossings,soifweorderknotsbycrossingnumber,wewillgeta(cumbersome)statement.

BUTWHATDOTHESEKNOTS“LOOKLIKE”

• Problem:hardtocomputeinterestinginvariants.

• Exceptone:theAlexanderpolynomial(definingwhichwilltakeustoofarafield,butcanbedefinedasagraphdeterminantwithavariablethrownin).Thiscanbecomputedinpolynomialtimeforpolygonalcurves(notfastpolynomialtime).

COEFFICIENTDISTRIBUTIONOFALEXANDERPOLYNOMIALSOFRANDOMKNOTS

• Forthis,wewantcrazyknotssooner,soweusethesecondmodel(trigonometricpolynomialsofdegreeN withidenticalindependentcoefficients.

COEFFICIENTDISTRIBUTION,REVEALED

REALLY?

• Apparentlyso…

WHATDOTHEZEROSLOOKLIKE?

• Noticethezero-freeregion

NOTHINGLIKERANDOMRECIPROCALPOLYNOMIALS

• Noticeconcentrationofzerosontheunitcircle

HOWMODEL-SPECIFICISTHIS?

• Let’stryanothermodel

• Thinkofaknotasa(closed)path.

• Now,takeasampleoftheuniformdistributionontheunitsphere,andconnectpoint1topoint2,point2,topoint3,…,pointN topoint1.(almostsurelynon-self-intersecting).

HISTORICALNOTE

• ThisisavariantofamodelduetoKenMillett– heproposedtakingasampleoftheuniformdistributionintheunitcube.

• TheexactmodelIamlookingatwasthesubjectofaquestionofJosephO’RourkeonMathOverflow(thequestionwas:whatwasthecrossingnumberofarandomsuchknotobtainedbytakingN pointsontheunitsphere.TheobviousguessisΩ(𝑁;),butitwasconjecturedbyBillThurston(again)thatthehyperbolicvolumegrowsas𝑁"/;,andhesuggestedlookingattheAlexanderpolynomialdegreeasanestimatorofthenumberofcrossings.

DEGREEOFALEXANDERPOLYNOMIALASAFUNCTIONOFN

• Myguessisoftheorderof𝑁"/;

• ButIcan’treallysayforsure.

WHATDOWEGETFORCOEFFICIENTSOFALEXANDERPOLYNOMIALS?

UNIVERSALITY?

WHATOTHERMODELSARETHERE?

• Takearandom4-regularplanargraph,anEulercycle,andmakeeveryvertexanover- orunder-crossing,randomly.

• Takearandomwalkon𝑍",andwhenyougetaclosedloop,seewhattheknottypeis.

• Petaldiagrams(giventhroughrandompermutations)

• Randombraids(randomwordsinthebraidgroup)

• Etc

WHICHMODELISPHYSICALLY/BIOLOGICALLYAPPROPRIATE?

• Studywhatweseeinnature,andthen,trytoseewhichmodelmatches.

WHATINVARIANTSTOSTUDY?

• Crossingnumber

• Hyperbolicvolume

• Conformalenergy

• Coefficients/zerosofotherknotpolynomials

COMPUTATIONALCAVEATS

• TheFouriermodelisnotsoeasytocomputewith.Why?

• WeneedapolygonalcurveforAlexanderpolynomialcomputation

• (wesimplifyitusingReidemeister moves,butstill)

• Thecomplexityisbad(𝑁F orso)inthenumberofsegments.

• So,weneedasophisticatedsubdivisionalgorithm(betterthanMathematica’s)

• Notsurewhatthebestmethodis.

• Secondmodelisveryeasy,bycontrast,butseemslessnatural.