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

XXX Knots in Washington21st May 2010

Krzysztof PutyraColumbia University, New York

What is covered?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

A crossing has two resolutions

Example A 010-resolution of the left-handed trefoil

Louis Kauffman

Type 0 (up) Type 1 (down)

1

2

3 1

2

3010

Cube of resolutions

Cube of resolutionsA change of a resolution is a cobordism

Put a saddle over the area being changed:

Cube of resolutions1

2

3110

101

011

100

010

001

000 111vertices

are smoothed diagrams

Observation This is a commutative diagram in a category of 1-mani-folds and cobordisms

edges are cobordis

ms

Khovanov complex, 1st approachEven homology (K, 1999)Apply a graded functor

i.e.

Odd homology (O R S, 2007)Apply a graded pseudo-

functor

i.e.

ModCob2:KhF ModCob2:ORSF

fFORS

XFORS

YFORS

ZFORS gfFORS

gFORS

fFKh

XFKh

YFKh

ZFKh gfFKh

gFKh

Peter Ozsvath

Mikhail Khovano

v

Result: a cube of modules with commutative faces

Result: a cube of modules with both commutative and anticommutative faces

Khovanov complex, 1st approach

0123 CCCC

direct sums create the complex

Theorem Homology groups of the complex C are link invariants.

Peter Ozsvath

Mikhail Khovano

v

Even: signs given explicitely

Odd: signs given by homological properties

AA

AAAA

AA

3

233

3

{+1+3} {+2+3} {+3+3}{+0+3}

Khovanov complex, 2nd approach (even)

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

000

100

010

001

110

101

011

111

Khovanov complex, 2nd approach (even)

1

2

3

Dror Bar-NatanTheorem (2005) The complex is a link invariant under chain homotopies and relations S/T/4Tu.

edges are cobordisms with

signs Objects: sequences of smoothed diagramsMorphisms: „matrices” of cobordisms

Chronological cobordismsA chronology: a separative Morse function τ.

An isotopy of chronologies: a smooth homotopy H s.th. Ht 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.

Chronological cobordismsCritical points cannot be permuted:

Critical points do not vanish:

Arrows cannot be reversed:

Chronological cobordisms

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 H0 H’0 and H1 H’1.

Remark Ht 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:

Chronological cobordismsRemark Not every cobordism has a trivial automorphism group:

Remark The problem does not exist in case of embedded or nestedcobordisms of genus zero.

Chronological cobordismsA solution in an R-additive extension for changes:

type II: identity

Any coefficients can be replaced by 1’s due to scaling:

a b

Chronological cobordismsA solution in an R-additive extension for changes:

type II: identity general type I:MM = MB = BM = BB = X X2 = 1

SS = SD = DS = DD = Y Y2 = 1SM = MD = BS = DB = ZMS = DM = SB = BD = Z-1

Corollary Let bdeg(W) = (B-M, D-S). ThenAB = X Y Z-

where bdeg(A) = (, ) and bdeg(B) = (, ).

Chronological cobordismsA solution in an R-additive extension for changes:

type II: identity general type I:

exceptional type I:

MM = MB = BM = BB = X X2 = 1SS = SD = DS = DD = Y Y2 = 1SM = MD = BS = DB = ZMS = DM = SB = BD = Z-1

AB = X Y Z- bdeg(A) = (, ) bdeg(B) = (, )

1 / XY

X / Y

Khovanov complex, 2nd approach (odd)

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.

1

2

3

000

100

010

001

110

101

011

111

Khovanov complex, 2nd approach (odd)

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

Khovanov complex, 2nd approach

Even homology (B-N, 2005)Complexes for tangles in CobDotted cobordisms:

Neck-cutting relation:

Delooping and Gauss elimination:

Lee theory:

Odd homology (P, 2008)Complexes for tangles in

ChCob

?

??

???

????

= {-1} {+1}

= 1 = 0

= + –

Dotted chronological cobordismsMotivation 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):

M M=

= 0(S)

(N) = + –

= 1(D) bdeg( ) = (-1, -1)

M = B = XZS = D = YZ-1

= XY

Z(X+Y) = +

Dotted chronological cobordismsMotivation 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)

(N) = + –

= 1(D) bdeg( ) = (-1, -1)

M = B = XZ S = D = YZ-1

= XYRemark T and 4Tu can be derived from S/D/N.Notice all coefficients are hidden!

Dotted chronological cobordismsTheorem (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?

Dotted chronological cobordismsTheorem There are isomorphisms

Mor(, ) [X, Y, Z1, 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-

t XZ

Chronological Frobenius algebras

Baby model: dotted algebraR = Mor(, ) A = Mor(, )

Here, F(X) = Mor(, X).

A chronological Frobenius system (R, A) in A is given by a monoidal 2-functor F: 2ChCob A:

R = F()A = F( )

Chronological Frobenius algebrasA 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

=

Chronological Frobenius algebrasA 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 product right product

Chronological Frobenius algebrasA 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:

Chronological Frobenius algebrasA 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

= XY

= XY

= XZ-1

= YZ-1no dots: XZ / YZone dot: 1 / 1two dots: XZ-1/ YZ-

1

three dots: Z-2 / Z-2(1 – XY)a = 0, bdeg(a) < 0bdeg(a) = 2n > 0

AB = X Y Z-

bdeg(A) = (, )bdeg(B) = (, )

cob:bdeg: (1, 1) (0, 0) (-1, -1) (-2, -2) (1, 0) (0, -1)

Chronological Frobenius algebrasA 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=

= Z2

=

Chronological Frobenius algebrasA 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)s0

2 = …• symmetry of A: tv+ = Z2v+t hv- = XZv-h …

algebra/coalgebra structure• right-linear, but not left

We further assume:• R is graded, A = R1 Rα is bigraded• bdeg(1) = (1, 0) and bdeg(α) = (0, -1)

Chronological Frobenius algebrasA base change: (R, A) (R', A') where A' := A R R'

Theorem If (R', A') is obtained from (R, A) by a base change thenC(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 (RU, AU), defined as follows:

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)

with (1) = –c (1) = (et–fh) 11+ f (YZ 1 + 1) + e () = a () = ft 11+ et(1 + YZ-1 1) + (f + YZ-1eh)

AU = R[]/(2 – h –t)RU = [X, Y, Z1, h, t, a, c, e, f]/(ae–cf, 1–af+YZ-1 (cet–aeh))

Chronological Frobenius algebrasA twisting: (R, A) (R', A')

' (w) = (yw)' (w) = (y-1w)

where y A is invertible and

Theorem If (R', A') is a twisting of (R, A) thenC(D; A') C(D; A)

for any diagram D.

Theorem The dotted algebra (R, A) is a twisting of (RU, AU).Proof Twist (RU, AU) with y = f + e, where v+=1 and v– = .

Corollary (P, 2010) The dotted algebra (R, A) gives a universal odd link homology.

Khovanov complex, 2nd approach

Even homology (B-N, 2005)Complexes for tangles in Cob

Dotted cobordisms:

Neck-cutting relation:

Delooping and Gauss elimination:

Lee theory:

Odd homology (P, 2010)Complexes for tangles in ChCob

Dotted chronological cobordisms- universal- only one dot over field, if X Y

Neck-cutting with no coefficients

Delooping – yesGauss elimination – sign problem

Lee theory exists only for X = Y= {-1} {+1}

= 1 = 0

= + –

Further remarks 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