iii knots in poland, będlewo july 27, 2010 krzysztof putyra columbia university, new york
TRANSCRIPT
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)