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A 2-category of dotted cobordisms and a universal odd link homology

A 2-category of dotted cobordisms and a universal odd link homology. Krzysztof Putyra Columbia University , New York. XXX Knots in Washington 21 st May 2010. What is covered ?. What are link homologies ? Cube of resolutions Even & odd link homologies via modules

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A 2-category of dotted cobordisms and a universal odd link homology

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  1. A 2-category of dottedcobordisms and a universalodd link homology Krzysztof Putyra Columbia University, New York XXX Knotsin Washington 21st May 2010

  2. Whatiscovered? Whatare link homologies? • Cube of resolutions • Even & odd link homologies • via modules • via chronologicalcobordisms Whydottedcobordisms? • chronology on dottedcobordisms • neck-cuttingrelation and delooping Whatis a chronologicalFrobenius algebra? • dottedcobordisms as a baby-model • universality of dottedcobordismswith NC

  3. Cube of resolutions A crossinghastworesolutions Example A 010-resolution of theleft-handedtrefoil Type0 (up) Type1 (down) 3 1 3 1 010 2 2 Louis Kauffman

  4. Cube of resolutions A change of a resolution is a cobordism Put a saddleovertheareabeingchanged:

  5. Cube of resolutions 110 100 1 3 2 verticesaresmootheddiagrams 000 111 101 010 011 001 edgesarecobordisms ObservationThisis a commutative diagram in a category of 1-mani-folds and cobordisms

  6. Mikhail Khovanov Khovanovcomplex, 1stapproach Evenhomology (K, 1999) Oddhomology (O R S, 2007) Apply a gradedpseudo-functor i.e. • Apply a gradedfunctor • i.e. Result: a cube of moduleswithbothcommutative and anticommutativefaces Result: a cube of moduleswithcommutativefaces Peter Ozsvath

  7. Mikhail Khovanov Khovanovcomplex, 1stapproach Even: signsgivenexplicitely Odd: signsgiven by homologicalproperties directsumscreatethecomplex {+2+3} {+3+3} {+0+3} {+1+3} TheoremHomologygroups of thecomplexCare link invariants. Peter Ozsvath

  8. Khovanovcomplex, 2ndapproach (even) Idea: • StayinCob as long as possible! • Build a complexin-Cob • Proveitisinvariant Applications: • Natural extensionovertangles • A categorification of the Jones polynomial for tangles • Planar algebra of complexes • Fastercomputations for nice links Dror Bar-Natan

  9. Khovanovcomplex, 2ndapproach (even) 110 100 1 3 2 000 111 101 010 011 001 edgesarecobordismswithsigns Objects: sequences of smootheddiagrams Morphisms: „matrices” of cobordisms Theorem (2005) Thecomplexis a link invariant under chainhomotopies and relations S/T/4Tu. Dror Bar-Natan

  10. Chronologicalcobordisms An arrow: choice of a in/outcomingtrajectory of a gradient flow of τ A chronology: aseparativeMorse function τ. An isotopy of chronologies:a smooth homotopyHs.th. Ht is a chronology Pick one ConjectureEveryisotopy of chronologiesisinduced by an isotopy of thecobordism and an isotopy of an interval. AlmostTheoremEveryisotopy of chronologiesisequivalent to one induced by an isotopy of thecobordism and an isotopy of an interval.

  11. Chronologicalcobordisms Criticalpointscannot be permuted: Critical pointsdo not vanish: Arrowscannot be reversed:

  12. Chronologicalcobordisms A changeof a chronologyis a smoothhomotopyH. ChangesH and H’ areequivalentifH0 H’0 and H1  H’1. RemarkHtmight not be a chronology for somet (so calledcriticalmoments). FactEveryhomotopyisequivalent to a homotopywithfinitely many criticalmoments of twotypes: type I: type II: Theorem2ChCobwithchanges of chronologiesis a 2-category. Thiscategoryisweaklymonoidalwith a strictsymmetry.

  13. Chronologicalcobordisms Remark Not everycobordismhas a trivialautomorphism group: RemarkThe problem does not existincase of embeddedornested cobordisms of genus zero.

  14. Chronologicalcobordisms A solutionin an R-additiveextension for changes: • type II: identity a b Any coefficientscan be replaced by 1’s due to scaling:

  15. Chronologicalcobordisms A solutionin an R-additiveextension for changes: • type II: identity • general type I: MM= MB= BM= BB= X X2 = 1 SS= SD= DS= DD= Y Y2 = 1 SM= MD= BS= DB= Z MS= DM= SB= BD= Z-1 CorollaryLetbdeg(W) = (B-M, D-S). Then AB= XYZ-  wherebdeg(A) = (, ) and bdeg(B) = (, ).

  16. Chronologicalcobordisms A solutionin an R-additiveextension for changes: • type II: identity • general type I: • exceptionaltype I: MM= MB= BM= BB= X X2 = 1 SS= SD= DS= DD= Y Y2 = 1 SM= MD= BS= DB= Z MS= DM= SB= BD= Z-1 AB= XYZ- bdeg(A) = (, ) bdeg(B) = (, ) 1 / XY X / Y

  17. Khovanovcomplex, 2ndapproach (odd) 110 100 3 1 2 edgesarechronologicalcobordismswithcoefficientsinR 000 111 101 010 011 001 FactThecomplexis independent of a choice of arrows and a signassignmentused to make itcommutative.

  18. Khovanovcomplex, 2ndapproach (odd) TheoremThecomplexC(D) isinvariant under chainhomotopies and thefollowingrelations: Dror Bar-Natan where X, Y and Z arecoefficients of chronologychangerelations.

  19. Khovanovcomplex, 2ndapproach Evenhomology (B-N, 2005) Oddhomology (P, 2008) Complexes for tanglesinChCob ? ?? ??? ???? • Complexes for tanglesinCob • Dottedcobordisms: • Neck-cuttingrelation: • Delooping and Gauss elimination: • Lee theory: = + – = {-1}  {+1} = 1 = 0

  20. Dottedchronologicalcobordisms MotivationCutting a neckdue to 4Tu: Z(X+Y) = + Adddotsformallyand assumetheusualS/D/Nrelations: = 1 (S) (D) bdeg(  ) = (-1, -1) = + – (N) A chronologytakescare of dots, coefficientsmay be derivedfrom (N): M M M= B= XZ S= D= YZ-1 = XY = = 0

  21. Dottedchronologicalcobordisms MotivationCutting a neckdue to 4Tu: Z(X+Y) = + Adddotsformallyand assumetheusualS/D/Nrelations: = 1 (S) (D) bdeg(  ) = (-1, -1) = + – (N) A chronologytakescare of dots, coefficientsmay be derivedfrom (N): M= B= XZ S= D= YZ-1 = XY = 0 RemarkT and 4Tucan be derivedfromS/D/N. Noticeallcoefficientsarehidden!

  22. Dottedchronologicalcobordisms Theorem (delooping) Thefollowingmorphismsaremutuallyinverse: {–1} {+1} ConjectureWe canuseit for Gauss elimination and a divide-conqueralgorithm. Problem How to keeptrack on signsduring Gauss elimination? –

  23. Dottedchronologicalcobordisms TheoremThereareisomorphisms Mor(, )  [X, Y, Z1, h, t]/((XY – 1)h, (XY – 1)t) =: R Mor(, )  v+R v-R=: A given by bdeg(h) = (-1, -1) bdeg(t) = (-2, -2) bdeg(v+) = ( 1, 0) bdeg(v- ) = ( 0, -1) h XZ t XZ  v+ v- CorollaryThereis no odd Lee theory: t = 1 X = Y CorollaryThereisonly one dotinoddtheoryover a field: X  Y  XY  1  h = t = 0

  24. ChronologicalFrobeniusalgebras A chronologicalFrobenius system (R, A) inAisgiven by a monoidal2-functor F: 2ChCob A: R = F() A = F( ) Baby model: dotted algebra R  = Mor(, ) A  = Mor(, ) Here, F(X) = Mor(, X).

  25. ChronologicalFrobeniusalgebras A chronologicalFrobenius system (R, A) = (F(), F( )) Baby model: dotted algebra (R , A ): F(X) = Mor(, X) • weaktensor productinChCob • productinR • bimodulestructure on A  =

  26. ChronologicalFrobeniusalgebras A chronologicalFrobenius system (R, A) = (F(), F( )) Baby model: dotted algebra (R , A ): F(X) = Mor(, X) • weaktensor productinChCob • productinR • bimodulestructure on A  leftproduct rightproduct

  27. ChronologicalFrobeniusalgebras A chronologicalFrobenius system (R, A) = (F(), F( )) Baby model: dotted algebra (R , A ): F(X) = Mor(, X) • weaktensor productinChCob • productinR • bimodulestructure on A left module: = right module: =

  28. ChronologicalFrobeniusalgebras A chronologicalFrobenius system (R, A) = (F(), F( )) Baby model: dotted algebra (R , A ): F(X) = Mor(, X) • weaktensor productinChCob • changes of chronology • torsioninR • symmetry of A AB= XYZ-  bdeg(A) = (, ) bdeg(B) = (, ) cob: bdeg: (1, 1) (0, 0) (-1, -1) (-2, -2) (1, 0) (0, -1) no dots: XZ / YZ one dot: 1 / 1 twodots:XZ-1/ YZ-1 threedots: Z-2 / Z-2 = XY = YZ-1 = XY = XY = XZ-1 (1 – XY)a = 0, bdeg(a) < 0 bdeg(a) = 2n > 0

  29. ChronologicalFrobeniusalgebras A chronologicalFrobenius system (R, A) = (F(), F( )) Baby model: dotted algebra (R , A ): F(X) = Mor(, X) • weaktensor productinChCob • changes of chronology • algebra/coalgebrastructure = = XZ = = XZ   = Z2 =

  30. ChronologicalFrobeniusalgebras A chronologicalFrobenius system (R, A) = (F(), F( )) Baby model: dotted algebra (R , A ): F(X) = Mor(, X) • weaktensor productinChCob (right) • productinR • bimodulestructure on A • changes of chronology • torsioninR:0 = (1–XY)t = (1–XY)s02= … • symmetry of A: tv+ = Z2v+t hv- = XZv-h … • algebra/coalgebrastructure • right-linear, but not left • We furtherassume: • Risgraded, A = R1 Rαisbigraded • bdeg(1) = (1, 0) and bdeg(α) = (0, -1)

  31. ChronologicalFrobeniusalgebras A basechange: (R, A)  (R', A') whereA' := ARR' TheoremIf(R', A')isobtainedfrom(R, A) by a basechangethen C(D; A')  C(D; A)  R' for any diagram D. Theorem (P, 2010) Any ranktwochronologicalFrobeniussystem (R, A)is a basechange of (RU, AU),defined as follows: RU= [X, Y, Z1, h, t, a, c, e, f]/(ae–cf, 1–af+YZ-1 (cet–aeh)) AU= R[]/(2 – h –t) with  (1) = –c (1) = (et–fh) 11+ f (YZ 1 + 1) + e  () = a () = ft11+ et(1 + YZ-1 1) + (f + YZ-1eh)  bdeg(c) = bdeg(e) = (1, 1) bdeg(h) = (-1, -1) bdeg(1A) = (1, 0) bdeg(a) = bdeg(f) = (0, 0) bdeg(t) = (-2, -2) bdeg() = (0, -1)

  32. ChronologicalFrobeniusalgebras A twisting: (R, A)  (R', A')  ' (w) =  (yw) ' (w) = (y-1w) wherey  Aisinvertible and TheoremIf(R', A')is a twisting of (R, A)then C(D; A')  C(D; A) for any diagram D. TheoremThedotted algebra (R, A)is a twisting of (RU, AU). Proof Twist (RU, AU) withy = f + e, where v+=1 and v– = . Corollary (P, 2010) Thedotted algebra (R, A)gives a universalodd link homology.

  33. Khovanovcomplex, 2ndapproach Evenhomology (B-N, 2005) Oddhomology (P, 2010) Complexes for tanglesinChCob Dottedchronologicalcobordisms - universal -only one dotover field, if X  Y Neck-cuttingwith no coefficients Delooping – yes Gauss elimination – sign problem Lee theoryexistsonly for X = Y • Complexes for tanglesinCob • Dottedcobordisms: • Neck-cuttingrelation: • Delooping and Gauss elimination: • Lee theory: = + – = {-1}  {+1} = 1 = 0

  34. Furtherremarks • HigherrankchronologicalFrobeniusalgebrasmay be given as multi-graded systems withthenumber of degreesequal to therank • For virtuallinkstherestillshould be onlytwodegrees, and a puncturedMobius band musthave a bidegree (–½, –½) • Embeddedchronologicalcobordisms form a (strictly) braidedmonoidal 2-category; same for thedottedversionunless (N) isimposed • The 2-category nChCob of chronologicalcobordisms of dimensionncan be definedinthe same way. Each of themis a universalextension of nCobinthesense of A.Beliakova • „Categorifyingcategorification” – Radmila’scategorification of [x] may be used to categorifyFrobenius systems as well as thispresentation

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