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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
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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
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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
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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
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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
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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
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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
= + –
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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
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Chronological cobordismsCritical points cannot be permuted:
Critical points do not vanish:
Arrows cannot be reversed:
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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:
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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
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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) = (, ).
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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
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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
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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
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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) = +
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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!
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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?
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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
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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
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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
=
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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
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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:
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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)
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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
=
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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)
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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))
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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.
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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
= + –
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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