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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. XIX Oporto Meeting on Geometry, Topology and Physics July 20, 2010. What is covered ?. Even vs odd link homologies sketch of the constructions

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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 XIX Oporto Meeting on Geometry, Topology and Physics July20, 2010

  2. Whatiscovered? Evenvsodd link homologies • sketch of theconstructions • chronologicalcobordisms Dottedcobordismswithchronologies • chronologiesseedots • neck-cuttingrelation and delooping lemma ChronologicalFrobeniusalgebras • dottedcobordismsareuniversal

  3. 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

  4. Mikhail Khovanov Khovanovcomplex 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

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

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

  7. EvenvsOdd 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

  8. 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 FactIfτ0τ1and dimW= 2,thereexistisotopies of M and Ithatinduce an isotopy of thesechronologies.

  9. Chronologicalcobordisms A change of a chronologyis a smoothhomotopyH. ChangesH and H’ areequivalentifH0 H’0 and H1  H’1. RemarkHtmight not be a chronology for somet (so calledcriticalmoments). Fact(Cerf, 1970) Everyhomotopyisequivalent to a homotopywithfinitely many criticalmoments of twotypes: type I: type II: Theorem (P, 2008) 2ChCobwithchanges of chronologiesis a 2-cate-gory. Thiscategoryisweaklymonoidalwith a strictsymmetry.

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

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

  12. 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) = (, ).

  13. 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) = (, ) evenodd XYZ 1 -1 YXZ 1 -1 ZYX 1 -1 1 / XY X / Y

  14. Chronologicalcobordisms A solutionin an R-additiveextension for changes: • type II: identity • general type I: • exceptionaltype I: 1 / XY or X / Y Theorem(P, 2010) Withtheabove: Aut(W) = {1} if#hdls(W) = 0 and #sphr(W)  1 Aut(W) = {1, XY} otherwise 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) = (, ) evenodd XYZ 1 -1 YXZ 1 -1 ZYX 1 -1

  15. Dottedchronologicalcobordisms MotivationCutting a neckdue to 4Tu: I may be 0! Z(X+Y) = + Adddotsformallyand assumetheusualS/D/Nrelations: I’mhomo-geneous! = 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

  16. Dottedchronologicalcobordisms MotivationCutting a neckdue to 4Tu: I may be 0! Z(X+Y) = + Adddotsformallyand assumetheusualS/D/Nrelations: I’mhomo-geneous! = 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!

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

  18. Dottedchronologicalcobordisms TheoremThereareisomorphisms Mor(, )  [X, Y, Z1, h, t]/(X2-1, Y2-1, (XY – 1)h, (XY – 1)t) =: R Mor(, )  v+R v-R=: A givenby bdeg(h) = (-1, -1) bdeg(t) = (-2, -2) bdeg(v+) = ( 1, 0) bdeg(v- ) = ( 0, -1) h t  = v+ v- = left module: right module:

  19. Dottedchronologicalcobordisms Algebra/coalgebrastructure: given by cobordisms = = XZ = = XZ   Operations areright-linear, but not left-linear! = Z2 =

  20. Universality of dottedcobordisms A chronologicalFrobenius system (R, A) inAisgiven by a monoidal2-functor F: 2ChCob A: R = F() A = F( ) • We furtherassume: • Risgraded, A = Rv+Rvisbigraded • bdeg(v+) = (1, 0) and bdeg(v) = (0, -1) A basechange: (R, A)  (R', A') whereA' := ARR' A twisting: (R, A)  (R, A')  ' (w) =  (yw) ' (w) = (y-1w) wherey  Aisinvertible and deg(y) = (1, 0). TheoremIf(R, A')is a twisting of (R, A)then C(D; A')  C(D; A) for any diagram D.

  21. Universality of dottedcobordisms Theorem (P, 2010) Any rank2chronologicalFrobenius system withgeneratorsindegrees(1, 0) and (0, -1)arrisesfrom(R, A)by a basechange anda twisting. CorollaryThereis no odd Lee theory: t = 1 X = Y CorollaryThereisonly one dotinoddtheoryover a field: X  Y  XY  1  h = t = 0

  22. EvenvsOdd Evenhomology (B-N, 2005) Oddhomology (P, 2010) Complexes for tanglesinChCob Dottedchronologicalcobordisms - only one dotover a 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

  23. 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 thedottedversion • The2-category nChCob of chronologicalcobordisms of dimensionncan be definedinthe same way. Each of themis a universalextension of nCobwith a strictsymmetryinthesense of A.Beliakova and E.Wagner

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