iii knots in poland, będlewo july 27, 2010 krzysztof putyra columbia university, new york
TRANSCRIPT
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A 2-category of dotted cobordisms and a universal odd
link homology
III Knots in Poland, BędlewoJuly 27, 2010
Krzysztof PutyraColumbia University, New York
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What is covered?
Even vs odd link homologies
Chronological cobordisms
Dotted cobordisms with chronologies
Chronological Frobenius algebras
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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)
1
2
3 1
2
3010
Cube of resolutions
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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-manifolds and cobordisms
edges are cobordis
ms
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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
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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
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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
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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
= + –
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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
Fact If τ0 τ1 and dimW = 2, there exist isotopies of M and I that induce an isotopy of these chronologies.
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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:
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Chronological cobordismsCritical points cannot be permuted:
Critical points do not vanish:
Arrows cannot be reversed:
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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 by scaling:
a b
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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) = (, ).
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Chronological cobordisms
where X 2 = Y 2 = 1
Note (X, Y, Z) → (-X, -Y, -Z) induces an isomorphism on complexes.
Some of the changes:
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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
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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
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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.
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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) = +
I’m homo-geneous!
I may be 0!
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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):
= 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!
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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(, ) 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 :
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Dotted chronological cobordisms
Algebra/coalgebra structure: given by cobordisms
= XZ=
= XZ=
= Z2
=
Operations are right-linear, but not left-linear!
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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.
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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).
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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
= + –
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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 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)