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Dotted Cobordisms and Odd Link Homology Overview

Dive into the realm of dotted cobordisms and understand the universal odd link homology concept. Explore the grid of resolutions, Khovanov complex, and chronological cobordisms highlighted by experts like Krzysztof Putyra and Peter Ozsvath. Discover the unique properties of these cobordisms and their implications in link invariants.

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Dotted Cobordisms and Odd Link Homology Overview

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  1. A 2-category of dottedcobordisms and a universalodd link homology Krzysztof Putyra Columbia University, New York IIIKnotsin Poland, Będlewo July 27, 2010

  2. Whatiscovered? Evenvsodd link homologies Chronologicalcobordisms Dottedcobordismswithchronologies ChronologicalFrobeniusalgebras

  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 110 100 1 3 2 verticesaresmootheddiagrams 000 111 101 010 011 001 edgesarecobordisms ObservationThisis a commutative diagram in a category of 1-manifolds and cobordisms

  5. Mikhail Khovanov Khovanovcomplex Evenhomology (K, 1999) Oddhomology (O R S, 2007) Apply a gradedpseudo-functor see: arXiv:math/9908171 • Apply a gradedfunctor see: arXiv:0710.4300 Result: a cube of moduleswithbothcommutative and anticommutativefaces Result: a cube of moduleswithcommutativefaces Peter Ozsvath

  6. Mikhail Khovanov Khovanovcomplex Odd: signsgiven by homologicalproperties Even: signsgivenexplicitely directsumscreatethecomplex {+2+3} {+3+3} {+0+3} {+1+3} TheoremHomologygroups of thecomplexC(D) are link invariants. Thegraded Euler characte-ristic of C(D) isthe Jones polynomialJL(q). Peter Ozsvath

  7. 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 somelocalrelations. Dror Bar-Natan

  8. Khovanovcomplex 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

  9. 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.

  10. 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.

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

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

  13. Chronologicalcobordisms A solutionin an R-additiveextension for changes: • type II: identity • generictype 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) = (, ).

  14. Chronologicalcobordisms Some of thechanges: whereX 2 = Y 2= 1 Note (X, Y, Z) → (-X, -Y, -Z) inducesan isomorphismon complexes.

  15. Chronologicalcobordisms A solutionin an R-additiveextension for changes: • type II: identity • generictype 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

  16. 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

  17. Chronologicalcobordisms compare with Bar-Natan: arXiv:math/0410495 Theorem(P, 2008) Thecomplexisinvariant under Reidemeistermovesup to chainhomotopiesand thefollowinglocalrelations: wherethecriticalpoints on theshown parts of cobordismsareconsequtive, i.e. anyothercritical point appearsearlierorlaterthantheshown part.

  18. 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

  19. 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!

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

  21. Dottedchronologicalcobordisms TheoremThereareisomorphisms Mor(, )  R[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 t  = v+ v- Ais a bimoduleoverR : = left module: right module:

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

  23. 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.

  24. Universality of dottedcobordisms Theorem (P, 2010) Any rank2chronologicalFrobenius system withgeneratorsindegrees(1, 0) and (0, -1)arisesfrom(R, A) by a basechange and a twisting. Here, R = [X, Y, Z1]/(X2-1,Y2-1). CorollaryHaving a chronologicalFrobenius system F = (RF, AF), thehomologyHF(L) is a quotient of H(L). CorollaryThereis no odd Lee theory: t = 1 X = Y CorollaryThereisonly one dotinoddtheoryover a field: X  Y  XY  1  h = t = 0

  25. 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

  26. 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) braidedmonoidal2-category; same holds 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 • A linearsolution for chronologicalnestedcobordismsexists and isgiven by 9parameters (squares of 3 of themareequal1)

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