A 2-category of dotted cobordisms and a universal odd
link homology
AMS Meeting, NewarkSpecial Session on Homology Theories for Knots and Skein Modules
22nd 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
What are 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
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-mani-folds and cobordisms
edges are cobordis
ms
Khovanov complex, 1st approach
Even 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
{+1+3} {+2+3} {+3+3}{+0+3}
Odd: signs given by homological properties
AA
AAAA
AA
3
233
3
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
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
= + –
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
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 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). Then
AB = 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
even oddXYZ 1 -1YXZ 1 -1ZYX 1 -1
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
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]/(X2, Y2, (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( )
Universality
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 deg(y) = (1, 0).
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