A 2-category of dotted cobordisms and a universal odd

link homology

III Knots in Poland, BędlewoJuly 27, 2010

Krzysztof PutyraColumbia University, New York

What is covered?

Even vs odd link homologies

Chronological cobordisms

Dotted cobordisms with chronologies

Chronological Frobenius algebras

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 resolutions

1

2

3110

101

011

100

010

001

000 111vertices

are smoothed diagrams

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

edges are cobordis

ms

Khovanov complex

Even homology (K, 1999)

Apply a graded functor

Odd homology (O R S, 2007)

Apply a graded pseudo-functor

ModCob2:KhF ModCob2:ORSF

Peter Ozsvath

Mikhail Khovano

v

Result: a cube of modules with commutative faces

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

see: arXiv:math/9908171 see: arXiv:0710.4300

Khovanov complex

0123 CCCC

direct sums create the complex

Theorem Homology groups of the complex C(D) are link invariants. The graded Euler characte-ristic of C(D) is the Jones polynomial JL(q). Peter

OzsvathMikhail

Khovanov

Even: signs given explicitely

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

Odd: signs given by homological properties

AA

AAAA

AA

2

223

2

000

100

010

001

110

101

011

111

Khovanov complex

1

2

3

Dror Bar-NatanTheorem (B-N, 2005) The complex is a link invariant under chain homotopies and some local relations.

edges are cobordisms with

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

Khovanov complex

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

= + –

Chronological cobordisms

A chronology: a separative Morse function τ.

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

An arrow: choice of a in/outcoming trajectory of a gradient flow of τ

Pic

k o

ne

Fact If τ0 τ1 and dimW = 2, there exist isotopies of M and I that induce an isotopy of these chronologies.

Chronological cobordisms

Theorem (P, 2008) 2ChCob with changes of chronologies is a 2-cate-gory. 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 (Cerf, 1970) Every homotopy is equivalent to a homotopy with finitely many critical moments of two types:

type I:

type II:

Chronological cobordismsCritical points cannot be permuted:

Critical points do not vanish:

Arrows cannot be reversed:

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

type II: identity

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

a b

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

type II: identity generic 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). Then

AB = X Y Z

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

Chronological cobordisms

where X 2 = Y 2 = 1

Note (X, Y, Z) → (-X, -Y, -Z) induces an isomorphism on complexes.

Some of the changes:

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

type II: identity generic 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

even oddXYZ 1 -1YXZ 1 -1ZYX 1 -1

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

type II: identity general type I:

exceptional type I: 1 / XY or X / Y

Theorem (P, 2010) With the above:• Aut(W) = {1} if #hdls(W) = 0 and #sphr(W) 1• Aut(W) = {1, XY} otherwise

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) = (, )

even oddXYZ 1 -1YXZ 1 -1ZYX 1 -1

Chronological cobordismscompare with Bar-Natan: arXiv:math/0410495

Theorem (P, 2008) The complex is invariant under Reidemeister moves up to chain homotopies and the following local relations:

where the critical points on the shown parts of cobordisms are consequtive, i.e. any other critical point appears earlier or later than the shown part.

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) = +

I’m homo-geneous!

I may be 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):

= 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!

Z(X+Y) = +

I’m homo-geneous!

I may be 0!

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(, ) R[h, t]/((XY – 1)h, (XY – 1)t) =: R

Mor(, ) v+R v-R =: A

given bybdeg(h) = (-1, -1)bdeg(t) = (-2, -2)bdeg(v+) = ( 1, 0)bdeg(v- ) = ( 0, -1)

h

v+ v-

t

= =

left module: right module:

A is a bimodule over R :

Dotted chronological cobordisms

Algebra/coalgebra structure: given by cobordisms

= XZ=

= XZ=

= Z2

=

Operations are right-linear, but not left-linear!

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

R = F()A = F( )

We further assume:• R is graded, A = Rv+ Rv is bigraded• bdeg(v+) = (1, 0) and bdeg(v) = (0, -1)

A base change: (R, A) (R', A') where A' := A R R'

A twisting: (R, A) (R, A') ' (w) = (yw)' (w) = (y-1w)

where y A is invertible and deg(y) = (1, 0).

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

for any diagram D.

Universality of dotted cobordisms

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

Theorem (P, 2010) Any rank 2 chronological Frobenius system with generators in degrees (1, 0) and (0, -1) arises from (R, A) by a base change and a twisting. Here, R = [X, Y, Z1]/(X2-1,Y2-1).Corollary Having a chronological Frobenius system F = (RF, AF), the homology HF(L) is a quotient of H(L).

Even vs Odd

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- only one dot over a 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 holds for the dotted version

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 with a strict symmetry in the sense of A.Beliakova and E.Wagner

A linear solution for chronological nested cobordisms exists and is given by 9 parameters (squares of 3 of them are equal 1)