homology
Jun Yoshida (joint work with Noboru Ito)
The University of Tokyo, Graduate School of Mathemtaical Sciences
December 19, 2019
Contents
1
Introduction Goal of the talk Background: Vassiliev invariants
Backgound: categorification Motivating Question Goal of the talk (continued) Main Theorem 1
Main Theorem 2
2
Review on Khovanov homology As a bigraded module On differential
3
Vassiliev skein relation on Khovanov homology
Mapping cone as subtraction Genus-1 map
Invariance under elementary moves
Cube of resolutions Example
4
Main Results
Khovanov homology on singular links
Example: “figure-eight”
FI-relation
Introduction
1
Introduction Goal of the talk
Background: Vassiliev invariants Backgound: categorification Motivating Question Goal of the talk (continued) Main Theorem 1
Main Theorem 2
2
Review on Khovanov homology
3
Vassiliev skein relation on Khovanov homology
4
Main Results
Goal of the talk
Goal
To investigate Vassiliev skein relation on Khovanov homology.
Recall
v: a polynomial link invariant.
Vassiliev skein relation:
v
!
= v
!
− v
! .
Extension of v to singular knots/links.
Main Question
What is the appropriate notion of “crossing change” on Khovanov homology?
What properties does extended Khovanov homology enjoy?
Background: Vassiliev invariants
Convention Singular knot/link
≡ immersed closed 1-manifold in S
3whose singular values are at worst finite and all transverse double points.
Ambient isotopy classes are mainly considered.
Definition
A polynomial invariant v is said to be of type n if v(K) = 0 for every knot K with at least n double points.
v is called a Vassiliev invariant if it is of type n for some n.
Quantum invariants yield Vassiliev invariants [Birman and Lin, 1993]
Jones
polynomial ∈ Quantum
Invariants Vassiliev
Invariants
Taylor expansion
Backgound: categorification
Khovanov homology Kh
i,j(L)
Link invariant values in bigraded abelian groups.
Theorem ([Khovanov, 2000])
For a link L, the graded Euler characteristic of Kh (L), i.e.
[Kh (L)]
q:= X
i,j
(−1)
iq
jdim Kh
i,j(L) ∈ Z [q, q
−1] ,
equals the unnormalized Jones polynomial of L.
Slogan: Khovanov homology categorifies Jones polynomial:
if V (L) is the Jones polynomial of a link L, then
[Kh(L)]
q= (q + q
−1)V (L) .
Motivating Question
Kh
V Vassiliev
Invariants
???
Taylor expansion decategorify categorify
???
???
Question
1
How does Khovanov homology relate to Vassiliev invariants?
2
Furthermore, does it produces categorifications of Vassiliev
invariants?
Goal of the talk (continued)
Target Question
What is the counterpart of Vassiliev skein relation on Khovanov homology?
Approach Consider long exact sequence instead of subtraction.
Recall that a long exact sequence of cohomologies
· · · → H
n(X ) → H
n(Y ) → H
n(Z) → H
n+1(X ) → . . . yields the identity on the Euler characteristics χ:
χ(X) − χ(Y ) + χ(Z) = 0 . Compare it with Vassiliev skein relation:
v
!
− v
! + v
!
= 0 .
Main Theorem 1
Theorem (Ito-Y. arXiv:1911.09308)
Khovanov homology (with coefficients in F
2) extends to a singular link invariant so that there is a long exact sequence
· · · → Kh
i,j; F
2!
Φb∗
−−→ Kh
i,j; F
2!
→ Kh
i,j; F
2!
→ Kh
i+1,j; F
2!
→ . . . .
Remark
The invariant seems to be NEW!
The long exact sequence can be seen as a categorified Vassiliev skein relation.
The map Φ b has an easy description.
Main Theorem 1
Corollary
Khovanov homology categorifes unnormalized Jones polynomial even on singular links.
Proof Observe that the identity
"
Kh ; F
2!#
q
−
"
Kh ; F
2!#
q
+
"
Kh ; F
2!#
q
= 0
arising from the long exact sequence is exactly the Vassiliev skein
relation for unnormalized Jones polynomial.
Main Theorem 2
Theorem (Ito-Y. arXiv:1911.09308)
Khovanov homology (with coefficients in F
2) satisfies FI-relation; i.e.
Kh ; F
2!
= 0 .
Equivalently
The genus-1 map Φ b is a quasi-isomorphism on such crossings:
Φ : b C
!
⊗ F
2−
∼→ C
!
⊗ F
2.
In particular, Φ b is non-trivial.
Review on Khovanov homology
1
Introduction
2
Review on Khovanov homology As a bigraded module On differential
3
Vassiliev skein relation on Khovanov homology
4
Main Results
As a bigraded module
D: a diagram of an (ordinary) link.
Definition
A state on D is a smoothng of D on all its crossings:
0-smoothing
←−−−−−−− c
1-smoothing−−−−−−−→ .
a state D
sis topologically of the form D
s∼ = (S
1)
qπ0(Ds).
Notation
For a state D
s, write |s| the number of 1-smoothings.
As a bigraded module
Definition
For a link diagram D, we set
C
i,j(D) :=
M
Ds:state
|s|=i
(V
⊗π0(Ds))
j−i,
where V := Z {1, x} is a graded abelian group with deg 1 = 1 and deg x = −1.
Basis
C
i,j(D) is a free abelian group generated by enhanced states.
Example
An enhanced state on trefoil 3
1: 1 x ∈ C
1,1
On differential
Observation
V underlies a (1+1)-TQFT (in the sense of [Atiyah, 1988]) associated to the Frobenius algebra Z [x]/(x
2).
The saddle operation assigns 1-smoothings to 0-smoothings:
: →
δ
c: C
i,j(D) → C
i+1,j(D) for each crossing c:
p 7→ p
(1)p
(2)p q
7→ pq
On differential
Definition
1
For a link diagram D, put d := X
±δ
c: C
i,j(D) → C
i+1,j(D) ,
where the sum is taken so that δ
capply on all the 0-smoothings.
2
n
+, n
−: the numbers of positive and negative crossings.
C
i,j(D) := C
i+n−,j+2n−−n+(D) .
C(D) = (C
∗,?(D), d) is called the Khovanov complex of D.
Theorem ([Khovanov, 2000])
For every abelian group M , the homology group
Kh
i,j(L; M ) := H
i(C
∗,j(D) ⊗ M )
Vassiliev skein relation on Khovanov homology
1
Introduction
2
Review on Khovanov homology
3
Vassiliev skein relation on Khovanov homology Mapping cone as subtraction
Genus-1 map
Invariance under elementary moves Cube of resolutions
Example
4
Main Results
Mapping cone as subtraction
Question
How can we realize the “subtraction”
Kh
!
“−” Kh
!
?
A Consider a map
f : C
!
→ C
!
and take the mapping cone Cone(f ).
Mapping cone as subtraction
Definition
Let f : X → Y be a chain map. Define a complex Cone(f ) by Cone(f )
i:= Y
i⊕ X
i+1with differential
d =
d
Yf 0 −d
X: Y
i⊕ X
i+1→ Y
i+1⊕ X
i+2.
Lemma
If f : X → Y is a chain map, then the sequence below is exact
· · · → H
n(X) −→
f∗H
n(Y ) −→
i∗H
n(Cone(f )) −→
p∗H
n+1(X) → . . . i : Y → Cone(f ) is the canonical inclusion;
p : Cone(f ) → X
∗+1is the canonical projection.
Genus-1 map
Proposition
The 2-dimensional cobordism
: →
gives rise to a chain map
Φ : b C
!
⊗ F
2→ C
!
⊗ F
2.
Remark
As a realization of crossing change, Φ b is not the standard one.
Indeed, the standard one is of degree 6= (0, 0).
Genus-1 map
Presentation Φ b : C
!
⊗ F
2→ C
!
⊗ F
2p 7→ 0
p q 7→ (pq)
(1)(pq)
(2)p q
7→ 0
p 7→ p
(1)p
(2)Invariance under elementary moves
Proposition
The chain map Φ b is invariant under elementary moves of double points.
Elementary moves [Bataineh-Elhamadi-Hajij-Youmans, 2018]
ΩIVa
←→
ΩIVe
←→
ΩV
←→
Invariance under elementary moves
Sketch of the proof of Proposition Show that the genus-1 map
Φ : C
!
→ C
!
is invariant instead of Φ b is.
Invariance is verified as the commutativity of appropriate diagrms: e.g.
C
i,j
C
i,j!
C
i,j
C
i,j−2
C
i,j−2!
C
i,j−2
Φ
RII RII
Φ
RII RII
.
23 / 31Cube of resolutions
Definition
Let D be a diagram of a singular link.
1
c
#(D): the set of double points in D.
2
A resolution of D is a diagram D
rwithout double points obtained by resolving double points of D:
b
− −-resolution←−−−−−−− b
+-resolution−−−−−−−→ b
+.
3
|r| the number of −-resolutions.
Cube of resolutions
Observation
Resolutions of D ←→
1:1subsets of c
#(D).
If D
r−and D
r+differ at only one double point, then Φ : b C(D
r−) ⊗ F
2→ C(D
r+) ⊗ F
2.
The maps Φ b at different double points commute with each other.
Lemma
For each singular link diagram D, we have a c
#(D)-cube of Khovanov complexes of resolutions of D and the genus-1 maps.
Definition
We denote by C
∗,?(D; F
2) the multiple mapping cone of the
c
#(D)-cube.
Example
Cube for b a :
C
a
−b
−
⊗ F
2 bΦa−−→ C
a
+b
−
⊗ F
2Φbb
↓ ↓
ΦbbC
a
−b
+
⊗ F
2 bΦa−−→ C
a
+b
+
⊗ F
2Main Results
1
Introduction
2
Review on Khovanov homology
3
Vassiliev skein relation on Khovanov homology
4
Main Results
Khovanov homology on singular links Example: “figure-eight”
FI-relation
Khovanov homology on singular links
Theorem (Ito-Y. arXiv:1911.09308) Let L be a singular link and D a diagram.
1
The homology Kh
i,j(L; F
2) := H
i(C
∗,j(D; F
2)) is independent of the choice of D.
2
If L has no double point, then Kh
i,j(L; F
2) agrees with the ordinary Khovanov homology (with coefficients in F
2).
3
For each double point of L, there is a long exact sequence
· · · → Kh
i,j; F
2!
Φb∗
−−→ Kh
i,j; F
2!
→ Kh
i,j; F
2!
→ Kh
i+1,j; F
2!
→ . . . .
Example: “figure-eight”
Kh
i,j
Φb
− → Kh
i,j
→ Kh
i,j
Proposition
The below is the table of the homology groups on the middle (resp. on the right):
i\j −5 −4 −3 −2 −1 0 1 2 3 4 5
−2 F
2F
2−1 F
2F
20 × F
2× F2
1 F
2F
22 F
2F
2FI-relation
Theorem (Ito-Y. arXiv:1911.09308)
Khovanov homology (with coefficients in F
2) satisfies FI-relation; i.e.
Kh ; F
2!
= 0 .
Proof Direct computation shows the diagram below commutes:
C ; F
2!
C ; F
2!
C ; F
2!
RI− RI+
bΦb
