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XXX Knots in Washington 21 st May 2010 Krzysztof Putyra Columbia University, New York

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What are link homologies? Cube of resolutions Even & odd link homologies via modules via chronological cobordisms Why dotted cobordisms? chronology on dotted cobordisms neck-cutting relation and delooping What is a chronological Frobenius algebra? dotted cobordisms as a baby-model universality of dotted cobordisms with NC

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A crossing has two resolutions Example A 010 -resolution of the left-handed trefoil Louis Kauffman Type 0 (up)Type 1 (down)

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A change of a resolution is a cobordism Put a saddle over the area being changed:

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vertices are smoothed diagrams Observation This is a commutative diagram in a category of 1-mani-folds and cobordisms edges are cobordisms

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Apply a graded functor i.e. Apply a graded pseudo-functor i.e. Peter Ozsvath Mikhail Khovanov Result: a cube of modules with commutative faces Result: a cube of modules with both commutative and anticommutative faces

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direct sums create the complex Theorem Homology groups of the complex C are link invariants. Peter Ozsvath Mikhail Khovanov Even: signs given explicitely Odd: signs given by homological properties {+1+3} {+2+3}{+3+3} {+0+3}

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Idea: Stay in Cob as long as possible! Build a complex in - Cob Prove it is invariant Applications: Natural extension over tangles A categorification of the Jones polynomial for tangles Planar algebra of complexes Faster computations for nice links Dror Bar-Natan

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Dror Bar-Natan Theorem (2005) The complex is a link invariant under chain homotopies and relations S/T/4Tu. edges are cobordisms with signs Objects: sequences of smoothed diagrams Morphisms: „matrices” of cobordisms

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A chronology: a separative Morse function τ. An isotopy of chronologies: a smooth homotopy H s.th. H t is a chronology Conjecture Every isotopy of chronologies is induced by an isotopy of the cobordism and an isotopy of an interval. An arrow: choice of a in/outcoming trajectory of a gradient flow of τ Pick one Almost Theorem Every isotopy of chronologies is equivalent to one induced by an isotopy of the cobordism and an isotopy of an interval.

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Critical points cannot be permuted: Critical points do not vanish: Arrows cannot be reversed:

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Theorem 2ChCob with changes of chronologies is a 2-category. This category is weakly monoidal with a strict symmetry. A change of a chronology is a smooth homotopy H. Changes H and H’ are equivalent if H 0 H’ 0 and H 1 H’ 1. Remark H t might not be a chronology for some t (so called critical moments). Fact Every homotopy is equivalent to a homotopy with finitely many critical moments of two types: type I: type II:

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Remark Not every cobordism has a trivial automorphism group: Remark The problem does not exist in case of embedded or nested cobordisms of genus zero.

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A solution in an R-additive extension for changes: type II: identity Any coefficients can be replaced by 1 ’s due to scaling: a a b b

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A solution in an R-additive extension for changes: type II: identity general type I: MM = MB = BM = BB = XX 2 = 1 SS = SD = DS = DD = YY 2 = 1 SM = MD = BS = DB = Z MS = DM = SB = BD = Z -1 Corollary Let bdeg(W) = (B-M, D-S). Then AB = X Y Z - where bdeg(A) = ( , ) and bdeg(B) = ( , ).

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A solution in an R-additive extension for changes: type II: identity general type I: exceptional type I: MM = MB = BM = BB = XX 2 = 1 SS = SD = DS = DD = YY 2 = 1 SM = MD = BS = DB = Z MS = DM = SB = BD = Z -1 AB = X Y Z - bdeg(A) = ( , ) bdeg(B) = ( , ) 1 / XY X / Y

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edges are chronological cobordisms with coefficients in R Fact The complex is independent of a choice of arrows and a sign assignment used to make it commutative

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Theorem The complex C(D) is invariant under chain homotopies and the following relations: where X, Y and Z are coefficients of chronology change relations. Dror Bar-Natan

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Complexes for tangles in Cob Dotted cobordisms: Neck-cutting relation: Delooping and Gauss elimination: Lee theory: Complexes for tangles in ChCob ? ?? ??? ???? = { -1 } { +1 } = 1 = 0 = + –

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Motivation Cutting a neck due to 4Tu: Add dots formally and assume the usual S/D/N relations: A chronology takes care of dots, coefficients may be derived from (N): MM = = 0 (S)(S) (N)(N) = + – = 1 (D)(D) bdeg( ) = (- 1, - 1 ) M = B = XZ S = D = YZ -1 = XY Z(X+Y) = +

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Motivation Cutting a neck due to 4Tu: Add dots formally and assume the usual S/D/N relations: A chronology takes care of dots, coefficients may be derived from (N): Z(X+Y) = + = 0 (S)(S) (N)(N) = + – = 1 (D)(D) bdeg( ) = (- 1, - 1 ) M = B = XZ S = D = YZ -1 = XY Remark T and 4Tu can be derived from S/D/N. Notice all coefficients are hidden!

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Theorem (delooping) The following morphisms are mutually inverse: {–1} {+1} – Conjecture We can use it for Gauss elimination and a divide-conquer algorithm. Problem How to keep track on signs during Gauss elimination?

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Theorem There are isomorphisms Mor( , ) [X, Y, Z 1, h, t]/ ( (XY – 1 )h, (XY – 1 )t ) =: R Mor( , ) v + R v - R =: A given by Corollary There is no odd Lee theory: t = 1 X = Y Corollary There is only one dot in odd theory over a field: X Y XY 1 h = t = 0 bdeg(h) = (- 1, - 1 ) bdeg(t) = (- 2, - 2 ) bdeg(v + ) = ( 1, 0 ) bdeg(v - ) = ( 0, - 1 ) h XZ v+v+ v- v- t XZ

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Baby model: dotted algebra R = Mor( , ) A = Mor( , ) Here, F(X) = Mor( , X). A chronological Frobenius system (R, A) in A is given by a monoidal 2 -functor F: 2 ChCob A: R = F( ) A = F( )

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A chronological Frobenius system (R, A) = (F( ), F( )) Baby model: dotted algebra (R , A ): F(X) = Mor( , X) weak tensor product in ChCob product in R bimodule structure on A =

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A chronological Frobenius system (R, A) = (F( ), F( )) Baby model: dotted algebra (R , A ): F(X) = Mor( , X) weak tensor product in ChCob product in R bimodule structure on A left productright product

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A chronological Frobenius system (R, A) = (F( ), F( )) Baby model: dotted algebra (R , A ): F(X) = Mor( , X) weak tensor product in ChCob product in R bimodule structure on A = left module: right module:

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A chronological Frobenius system (R, A) = (F( ), F( )) Baby model: dotted algebra (R , A ): F(X) = Mor( , X) weak tensor product in ChCob changes of chronology torsion in R symmetry of A = XY = XZ -1 = YZ -1 no dots: XZ / YZ one dot: 1 / 1 two dots: XZ -1 / YZ -1 three dots: Z -2 / Z -2 ( 1 – XY)a = 0,bdeg(a) < 0 bdeg(a) = 2 n > 0 AB = X Y Z - bdeg(A) = ( , ) bdeg(B) = ( , ) cob: bdeg: ( 1, 1 )( 0, 0 )(- 1, - 1 ) (- 2, - 2 ) ( 1, 0 ) ( 0, - 1 )

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A chronological Frobenius system (R, A) = (F( ), F( )) Baby model: dotted algebra (R , A ): F(X) = Mor( , X) weak tensor product in ChCob changes of chronology algebra/coalgebra structure = XZ = XZ – +XZ h XZ v+v+ v- v- t XZ Recall: so that (v -, v - ) = v + t + v - h

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A chronological Frobenius system (R, A) = (F( ), F( )) Baby model: dotted algebra (R , A ): F(X) = Mor( , X) weak tensor product in ChCob changes of chronology algebra/coalgebra structure = XZ = = = Z 2 =

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A chronological Frobenius system (R, A) = (F( ), F( )) Baby model: dotted algebra (R , A ): F(X) = Mor( , X) weak tensor product in ChCob (right) product in R bimodule structure on A changes of chronology torsion in R: 0 = ( 1 –XY)t = ( 1 –XY)s 0 2 = … symmetry of A:tv + = Z 2 v + thv - = XZv - h… algebra/coalgebra structure right-linear, but not left We further assume: R is graded, A = R 1 Rα is bigraded bdeg( 1 ) = ( 1, 0 ) and bdeg(α) = ( 0, - 1 )

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A base change: (R, A) (R', A') where A' := A R R' Theorem If (R', A') is obtained from (R, A) by a base change then C(D; A') C(D; A) R' for any diagram D. Theorem (P, 2010) Any rank two chronological Frobenius system (R, A) is a base change of (R U, A U ), defined as follows: bdeg(c) = bdeg(e) = ( 1, 1 ) bdeg(h) = (- 1, - 1 ) bdeg( 1 A ) = ( 1, 0 ) bdeg(a) = bdeg(f) = ( 0, 0 )bdeg(t) = (- 2, - 2 ) bdeg( ) = ( 0, - 1 ) with ( 1 ) = –c ( 1 ) = (et–fh) 1 1 + f (YZ 1 + 1 ) + e ( ) = a ( ) = ft 1 1 + et( 1 + YZ -1 1 ) + (f + YZ -1 eh) A U = R[ ]/( 2 – h –t) R U = [X, Y, Z 1, h, t, a, c, e, f]/(ae–cf, 1 –af+YZ -1 (cet–aeh))

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A twisting: (R, A) (R', A') ' (w) = (yw) ' (w) = (y -1 w) where y A is invertible and Theorem If (R', A') is a twisting of (R, A) then C(D; A') C(D; A) for any diagram D. Theorem The dotted algebra (R , A ) is a twisting of (R U, A U ). Proof Twist (R U, A U ) with y = f + e, where v + = 1 and v – = . Corollary (P, 2010) The dotted algebra (R , A ) gives a universal odd link homology.

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Complexes for tangles in Cob Dotted cobordisms: Neck-cutting relation: Delooping and Gauss elimination: Lee theory: Complexes for tangles in ChCob Dotted chronological cobordisms - universal - only one dot over field, if X Y Neck-cutting with no coefficients Delooping – yes Gauss elimination – sign problem Lee theory exists only for X = Y = { -1 } { +1 } = 1 = 0 = + –

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Higher rank chronological Frobenius algebras may be given as multi-graded systems with the number of degrees equal to the rank For virtual links there still should be only two degrees, and a punctured Mobius band must have a bidegree (–½, –½) Embedded chronological cobordisms form a (strictly) braided monoidal 2-category; same for the dotted version unless (N) is imposed The 2-category nChCob of chronological cobordisms of dimension n can be defined in the same way. Each of them is a universal extension of nCob in the sense of A.Beliakova „Categorifying categorification” – Radmila’s categorification of [x] may be used to categorify Frobenius systems as well as this presentation

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