d. w. sumners- some problems in applied knot theory, and some problems in geometric topology

8/3/2019 D. W. Sumners- Some Problems in Applied Knot Theory, and Some Problems in Geometric Topology

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TOPOLOGY PROCEEDINGS Volume 13 1988 163

SOME PROBLEMS IN APPLIED KNOT THEORY,AND SOME PROBLEMS IN GEOMETRIC TOPOLOGYD. W. Sumners

Modern knot theory was born ou t o f physics in the19th century . Gauss ' cons idera t ions on inductance inc i r c u l a r wires gave r i se to the "Gauss I n t eg r a l , " aformula fo r the l ink ing number o f two simple c losedcurves in 3-space [G]. William Thompson (Lord Kelvin) ,upon seeing experiments performed by P. G. T a i t involvingco l l id ing smoke r ings , conceived the "vortex theory ofatoms," in which atoms were modelled as conf igura t ions o fknot ted and l inked vor tex r ings in the ae the r [Th]. Int h i s contex t , a t ab le of th e elements was--you guessed it-a knot t ab le ! Ta i t s e t about cons t ruc t ing t h i s knot t a b l e ,and the r e s t i s h is to ry [Ta]!

Given th e ci rcumstances o f i t s b i r t h , it i s notsu rp r i s ing t h a t knot theory has , from t ime to t ime, beeno f use in sc i ence . One can th ink o f 3-dimensional knottheory as th e s tudy of f l ex ib le graphs in R 3 , withemphasis on graph entanglement (knot t ing and l i nk ing) . Amolecule can be represented by i t s molecular graph--atomsas ve r t i c e s , covalen t bonds as edges . A l a rge moleculecan be very f l ex i b l e . Such a f l ex ib le molecule does notusual ly mainta in a f ixed 3-dimensional conf igu ra t ion . Itcan assume a va r ie ty of conf igura t ions , driven from oneto th e o the r by thermal motion, so lven t e f f e c t s ,


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experimental manipulation, e tc . From an i n i t i a l configurat ion for a molecule (or se t of molecules), knot theory canhelp ident i fy a l l of the possible at ta inable configurat ionsof tha t molecular system. I t i s c lear tha t the notion oftopological equivalence of embeddings of graphs in R 3 isphysically unreal ist ic--one cannot s t re tch or shrinkmolecules a t wil l . Nevertheless, the topological def in i t ion of equivalence i s , on the one hand, broad enough togenerate a large body of mathematical knowledge, and, onthe other hand, precise enough to place useful and com-putable l imits on the physically possible motions andconfiguration changes of molecules. For molecules whichpossess complicated molecular graphs, knot theory canalso aid in the predict ion and detect ion of variousspa t i a l isomers [Si] . As evidence fo r the u t i l i t y of knottheory (and other mathematics) in chemistry and molecularbiology, see the excel lent survey ar t ic les [Wa,WC], andthe conference proceedings [ACG,KR,L].

Some of the problems posed below deal with configurat ions of random walks or self -avoiding (no s e l f -intersect ions) random walks on the in teger cubic l a t t i ce. lR 3 The s t a t i s t i c s of random walks on the l a t t ice areused to model configurations of l inear and c i rcula rmacromolecules. A macromolecule is a large moleculeformed by concatenating large numbers of monomers--such asthe synthet ic polymer polyethylene and the biopolymer DNA.Conversion of c i rcula r polymers from one topological s t a t e(say unknotted and unlinked) to another (say knotted and

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TOPOLOGY PROCEEDINGS VolUme 13 1988 165

l inked) can occur through the act ion of various agents,chemical or biologica l . Given cons tra ints (energetic,spa t i a l or temporal) , l inear polymers can exhibi t entanglement (knott ing and l inking) . Moreover, l inearpolymers can be converted to c i rcu lar polymers in variouscycl iza t ion react ions . I f one wants a random s a m p + . ~ __ o.L __the configuration space of a macromolecule in_ffi3 , one canmodel the spa t i a l configuration of a macromolecule as aself -avoiding random walk in R3, where the vert icesrepresent the posi t ions of carbon atoms, and adjacentvert ices are connected by s t ra ight l ine segments (a l l thesame length) , representing covalent bonds. A discre teversion of random walks in R3 is random walks on thein teger cubic l a t t i ce . One studies the s t a t i s t i c a lmechanics of large ensembles of these random walks inhopes of detect ing physical ly observable quanti t ies (suchas phase t ransi t ion) of the physical system being modelled.

The problems below are s ta ted in an informal s ty le ,and addresses of relevant people are included when known,in hopes tha t the in teres ted reader wil l contact them.


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PROBLEMS PROPOSED BY:J. L. BryantDepartment of MathematicsFlorida State UnIVersityTallahassee, FL 32306AndR. C. LacherDepartment of Computer ScienceFlorida State UniversityTallahassee, FL 32306

3Consider random walks on a cubic l a t t i c e in m t ha ts t a r t with 0 < Y < n, n > 1, and end when e i t he r y = 0 o ry n. An L-walk (R-walk) i s a walk t ha t s t a r t s withy 1 (y = n - 1 ) . (Think of an L-walk or R-walk as awalk t ha t s t a r t s on one o f the planes y = 0 o r y = n a n dtakes i t s f i r s t s tep in to the reg ion between the p lanes . )An L-loop (R-loop) i s an L-walk t h a t ends with y = 0(y = n) . Assume s tep pr obab i l i t i e s are a l l equa l to 1/6(pure i so t ropy) . Given an L-walk L and an R-walk R, def ine the of f s e t l ink ing number olk{L,R) as fo l lows: I feach of L a n d R is a loop, complete it to a c losed curveby j o in ing i t s endpoints wi th an a r b i t r a r y path in i t sbase plane , o f f s e t the l a t t i c e fo r R by the vec tor

1 1 1(-2' -2' -2) ' and def ine olk(L,R) to be the homologicall ink ing number of th e r e s u l t i ng (d i s jo in t ) closed curves .Otherwise , s e t olk{L,R) = O. We say L l i nks R ifolk{L,R) O.

Problem 1 . Given an L-walk L and a family n o f R-walkswith dens i ty of s t a r t s d, what i s the probab i l i ty Plink(n)t ha t L w i l l l ink a member o f 'R?


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Problem 2 . Compute l im Pl ink (n ) .n +

Problem 3 . Find th e expected value Dlink(n) of t he numbero f members o f t h a t L l i n k s .

Problem 4. Compute l im D l ink(n ) .n +

Problem 5 . Find th e expected sum Wl(n) of t he a bs o lu t eva lues o f th e o f f s e t l i nk i ng number o f L with th e memberso f ~

Problem 6 . Compute l im Wl(n) .n +

Problem 7. Find th e expected sum W (n) o f th e squares o f2th e o f f s e t l ink ing number o f L with th e members o f ~ (Comment: W (n ) should be e a s i e r to dea l with than Wl(n) . )2Problem 8. Compute l im w (n) .2n +

Given an L- loop t h a t s t a r t s a t (0 ,1 ,0 ) , def ine i t sreach to be its maximum y-va lue , i t s range to be i t smaximum x- o r z -va lue , and its breadth b = range / reach .By ana logy , def i ne t he breadth o f any loop .Problem 9 . Compute the expec ted value o f b as a func t iono f n and its asympto t i c s . (Comment: Simula t ion s t a t i s t i c s seem t o i nd ica t e t h a t b = 1 . 1 9 . See [BL].)

Represent a loop by an i so sce l e s t r i a n g l e p a r a l l e lto th e y axi s hav ing i t s base on th e base plane fo r


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the loop. I t s "breadth" b = a1t i tude/2base. Analogs ofDlink(n) and Plink(n) for these s implif ied loops are

P(n)= 1 - l in - En'_-11d1 r r ~ - l , [1-2b 2d(i+j+1/2-n)2 d , ] .1 J=n-1 ]Asymptotics for D(n) are given in [BL].Problem 10. Compute lim P(n) .n-+oo(Comment: We conjecture tha t n(P(n) O(log(n.)Problem 11. Show that lim P{n) lim Plink(n), and tha tn-+oo n-+oolim D(n) lim Dlink(n).n-+oo n-+oo

PROBLEMS PROPOSED BY:D. W. SumnersDepartment of MathematicsFlorida State UniversityTallahassee, FL 3-2306

There ex i s t natural ly occurring enzymes(topoisomerases and recornbinases) which, in order tomediate the v i ta l l i f e processes of repl icat ion , t ran-scr ip t ion and recombination, manipulate ce l lu la r DNA intopological ly in teres t ing and nont r iv ia l ways [we, 51].These enzyme act ions include promoting writhing (coi l ingup) of DNA molecules, passing one s t rand of DNA throughanother via an enzyme-bridged break in one of the s t rands ,and breaking a pai r of st rands and recombining to


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di f fe ren t ends. I f one regards DNA as very thin s t r ing ,these enzyme ac t iv i t i e s are the s tu f f of which recentcombinatorial knot theory is made! Moreover, re la t ive lynew experimental techniques (rec A enhanced elec t ronmicroscopy) [KS] make possible the unambiguous resolut ionof the DNA knots and l inks produced by react ing c i rcu larDNA with high concentrat ions of a puri f ied enzyme in vi t ro(in the l abora tory) . The experimental protocol is tomanufacture (by cloning techniques) a r t i f i c i a l c i rcu larDNA subs t ra te on which a par t icu la r enzyme wil l ac t . Asexperimental cont ro l var iables , one has the knot type(s)of the subst ra te , and the amount of wri thing (supercoiling)of the subs t ra te molecules. The product of an enzymereact ion i s an enzyme-specif ic family of DNA knots andl inks . The react ion products are f rac t ionated by gelelectrophoresis , in which the molecules migrate through ares i s t ive medium (the gel) under the forcing of an e lec-t r i c f ie ld (electrophoresis) . Molecules which are "al ike"group together and t r ave l toge ther in a band through thegel . Gel electrophoresis can be used to discr iminatebetween molecules on th e bas i s o f molecular weight .Given (as is the case here) tha t a l l molecules are thesame molecular weight, it then discr iminates betweenmolecules on the bas is of average 3-dimensional "shape".Following elec t rophores is , the molecules are f a t ~ e n e d witha prote in (rec A) coat ing, to enhance resolut ion ofcrossovers in an elec t ron micrograph of the molecule. Inth is manner, the knot (l ink) type of the various react ion


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products is an observable. This new observational powermakes possible the building of knot- theoret ic models[WC,WMC,ES] for enzyme act ion, in which one wishes toex t rac t information about enzyme mechanism from the DNAknots and l inks produced by an enzyme react ion.Problem 1: Build new models for enzyme act ion. Themodels now exis t ing involve signed crossover number [WC],polynomial invar iants [WMC], and 2-s t r ing ' tangles [ES].The si tuat ion i s basical ly th i s : as input to a black box(the enzyme), one has a family of DNA circ les (of knownknot type and degree of supercoi l ing) . The output of theblack box is another family of DNA knots and l inks . THEPROBLEM: What happened inside the box?Problem 2: Explain gel electrophoresis experimentalresul ts . Gel electrophoresis i s a race for molecules-they a ~ s t a r t together , and the to ta l distance t ravel ledby a molecule when the elec t r ic f ie ld is turned off isdetermined by i t s gel mobili ty. At the f inish of a gelrun, the molecules are grouped in bands, the slowest bandnearest the s ta r t ing posi t ion, the fas tes t band fa r thes taway. When relaxed (no supercoi ls) DNA circ les (a l l thesame molecular weight) run under cer ta in gel condit ions,the knotted DNA circ les t rave l according to the i r crossover number [DS]1 What i s it about crossover number (anar t i fac t of 2-dimensional knot projections) that determines how fas t a f lexible knot moves through a res is t ive


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medium? The theory o f gel mobi l i ty of molecules ( l inearor c i rcu la r ) i s r a t he r d i f f i c u l t to work out . See [LZJfo r some resu l t s on the gel mobi l i ty of unknotted c i r c u l a rmolecules under pulsed f i e ld e lec t rophores i s .

Problem 3: What are the proper t ies o f a random knot (off ixed l ength)? Chemists have long been i n t e res t ed in thesynthes i s o f molecules with exo t i c geometry; in pa r t i c u l a r ,the synthes i s o f knot ted and l inked molecules [WaJ. Onecan imagine such a synthes i s by means of a cyc1iza t ionreac t ion (random clos ing) of l i ne a r chain molecules [FW].Let N represen t the number of repea t ing un i t s in such al i nea r chain . A un i t may r ep resen t a monomer of th e subs t ance , o r the equiva len t s t a t i s t i c a l length o f the subs tance . For example, the equiva len t s t a t i s t i c a l lengthfo r polyethylene i s about 3 .5 monomers, and fo r duplexDNA, about 500 base pa i r s . A randomly closed chain o flength N i s a random piecewise l i nea r embedding of s l ,with a l l th e I - s implexes the same l ength . See [Rl,R2J fo ra discuss ion o f the topology o f the conf igura t ion space ofsuch PL ernbeddings. In orde r to make pred ic t ions aboutthe yie ld of such a cyc l iza t ion r eac t ion , one needsanswers to the fol lowing mathematical ques t ions [52J:

A. For random s imple closed curves of length N (as above) ,what i s the d i s t r i bu t i on of knot types , as a funct ion o f N?


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B. What i s the probab i l i ty of kno t t i ng , as a func t ion ofN? One can show t ha t , fo r simple c losed curves of lengthN inscr ibed on the cub i ca l l a t t i c e in R3 , the knotprobab i l i ty goes to one exponent i a l ly rap id ly with N [SW].

PROBLEMS PROPOSED BY:R. F. WilliamsDepartment of MathematicsUniversity of TexasAustin, Texas 73713

~ EXPANSIVE VS. PSEUDO-ANOSOVThe r e fe rences here a re two p re p r in t s : [H] by

K. Hira ide , Department of Mathematics , Tokyo Metropol i tanUnivers i ty , Fukasawa 2-1-1 , Setagaya , Tokyo 158, Japan ,and [Le] by Jorge Lewowicz, I n s t i t u t o de Mathematica ,C a s i l l a de Correo 30, Montevideo, Uruguay. In [H] and[Le] , the authors independent ly prove t h a t the concepts"expans ive" and "pseudo-Anosov" coinc ide fo r su r f aces .

A. What i s th e s i t ua t i on fo r 3-manifolds?B. Find a good example o f a 3-manifold (such as

53) which does not suppor t an Anosov dif feomorphism.c. Prove some o f the beginning lemmas o f

Lewowicz-Hiraide fo r 3-manifolds .

I I . DYNAMICAL SYSTEMS

A. The two top ics of ze ta func t ions in dynamicalsystems and Alexander polynomials in knot theory a re

S 3 dOc a s e 1y re d see ] For ows onate: [M . f l , p e r 1 o 1Cor b i t s a re knots ; thus there should be a combination


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such as a 2 variable polynomial, combining knot theory(e .g . , the degree of the Alexander polynomial) anddynamical systems (the length of the orb i t ) . See [BW].

B. Branched surfaces can support Anosov endomorphisrns. However, a l l tha t are known are sh i f t equivalen t to l inear maps on the 2-torus, such as that inducedby the 2x2 matrix (i i)

Conjecture. Given an Anosov endomorphism g: K K,there is a l inear map f: T T, T the 2-torus, such tha tf is sh i f t equivalent to g. See the Northwestern thesisof Lan Wen, Department of Mathematics, Beijing University,P ~

Defini t ion. f : X X and g: Y Yare sh i f t equivalen t provided tha t there ex i s t maps r : X Y and s: Y Xand an in teger m such that r f = gr, sgand rs = gm.

Defini t ion. g: K K i s Anosov, provided there isa sub-bundle E of the tangent bundle TK, such tha t dgl e a v e ~ E i n v a r i a n t and con t r ac t s vec to r s , and such t h a tthe map induced on TK/E by dg expands vectors .

C. Hassler Whitney gives an example which i s dearto the hear t of a l l continuum theoris ts tha t know i t - both of us! I t i s a careful ly constructed arc A in theplane and a smooth function f: A Reals with grad f = 0(both part ia ls are 0), yet f has di f fe ren t values a t A's


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endpoints . Contac t Alec Norton, Boston Universi ty fo rh is prep r in t s and ideas on th i s sub jec t . (Don' t beaf ra id of smooth func t ions on manifolds . They havebeau t i f u l pathology and are cry ing out fo r continuumt heor i s t s t o look a t them. And they a re r ea l ly and t ru lyeasy to ge t the hang of . )

References[ACG] A. Amann, L. Cederbaum, W. Gans, eds . , Fractals ,

quasicrystals , chaos, knots and algebraic quantummechanics, NATO ASI Ser ies C: Mathematical andPhys ica l Sc iences , v. 235, Kluwer (1988).

[BL] J . L. Bryant , R. C. Lacher , Topological s t ructureso f semicrys ta l l ine polymers, these Proceedings , (1-16) .

[BW] J . Birman, R. F. Wil l iams, in Contemporary Mathe-matics, v. 20, Am. Math. Soc. (1983).[OS] F. B. Dean, A. Stas iak , T. Kol ler , N. R. Cozza re l l i ,

Duplex DNA knots produced by escherichia col itopoisomerase I , J . BioI . Chern. 260 (1985), 4795-4983.

[ES] C. Erns t , D. W. Sumners, A calculus for ra t iona l tangles: applicat ions to DNA recombination, pre -pr i n t , Flor ida Sta te Univers i ty .

[FW] H. L. Fr i sch , E. Wasserman, Organic and b io log i ca l chemistry, J . Am. Chern. Soc. 83 (1961), 3789-3795.[G] K. F. Gauss, Geometria Si tus , werke konigl ichen

gese l lschaf t der wissenschaf ten zu got t ingen, (1877) ,v. 8, 271-286 (Reprinted 1973 by alms in Hi1desheim) .

[H] K. Hira ide , Ezpansive homeomorphisms o f sur faces , prepr in t , Tokyo Metropol i tan Univers i ty .

[KR] R. B. King, D. Rouvray, eds . , Graph Theory andTopology in Chemistry, Studies in Phys ica l andTheore t ica l Chemistry 51, Elsev ie r (1987).


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[KS] M. A. Krasnow, A. Stasiak, S. J . Spengler, F. Dean,T. Koller , N. R. Cozzarel l i , Determination o f theabsolute handedness o f knots and catenanes o f DNA,Nature 304, (1983), 559-560.

[LJ R. C. Lacher, ed. , MATH/CHEM/COMP 1987, Studies inPhysical and Theoret ical Chemistry 54, Elsevier(1988)

[LeJ J . Lewowicz, Expansive homeomorphisms o f surfaces,prepr in t , Ins t i tu to de Mathematica, Montivideo.

[LZJ S. D. Levine, B. H. Zirnrn, Separat ions o f open-circular DNA using pulsed- f ie ld electrophoresis ,Proc. N.AS. USA 84, (1987), 40 54- 4057.

[MJ J . Milnor, In f in i te cycl ic coverings, in Conferenceon the Topology of Manifolds, J . G. Hocking, ed. ,Prindle, Weber & Schmidt, (1968), 115-133.

[RIJ R. Randell, A molecular configuration space, inMATH/CHEM/COMP 1987, R. C. Lacher, ed . , Elsevier ,(1987), 125-140.

[R2J R. Randell, Conformation spaces o f molecular rings,in MATH/CHEM/COMP 1987, R. C. Lacher, ed . , Elsevier ,(1987), 141-156.

[SiJ J . Simon, Topological chiral i ty o f certain mole-cu les , Topology 25, 229-234.

[SlJ D. W. Sumners, The role o f knot theory in DNAresearch, in Geometry and Topology, Manifolds,Variet ies and Knots, C. McCrory, T. Schi f r in , eds . ,Marcel Dekker, (1987), 297-318.

[S2J D. W. Sumners, Knot t heory , s t a t i s t i c s and DNA, Kern.Ind. 35, (1986), 657-661

[SW] D. W. Sumners, S. G. Whittington, Knots in s e l f avoiding walks, J . Phys. A Math. Gen. 21, (1988),1689-1694.

[TaJ P. G. Tai t , On knots I , I I , I I I , Scient i f ic PapersVol. 1, Cambridge Universi ty Press, (1898), 273-347.

[ThJ W. Thompson (Lord Kelvin) , On vortex atoms, Philosophical Magazine 34, #227, (July 1867), 15-24.


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[WaJ D. M. Wa1ba, Topological stereochemistry,Tetrahedron 41 (1985), 3161-3212.

[Wc] S. A. Wasserman, N. R. C6zzare11i , Biochemicaltopology: applicat ions to DNA recombination andrepl icat ion, Science 232 (1986) , 951-960.

[WMC] J . H. White, K. C. M i l le t t , N. R. Cozzare1l iDescription o f the topological entanglement o f DNAcatenanes and 'knots by a pOlJerfu l method invo lvingstrand passage and recombination, J . Mol. Bio l . 197(1987), 585-603.

Florida State Univers i tyTal lahassee , Florida 32306

d. w. sumners- some problems in applied knot theory, and some problems in geometric topology

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