emmanuel wagner- khovanov-rozansky graph homology and composition product

8/3/2019 Emmanuel Wagner- Khovanov-Rozansky Graph Homology and Composition Product

1/11

arXiv:math/0702230v1

[math.G

T]8Feb2007

P R3

a = qn b = q q1 n q Pn(L) L Pn(D) D L

Pn (unknot) = [n]q =qn qn

q q1.

D m n 1

Pn+m(D) Pn(D1) Pm(D2)

D1 D2 D

R2

R2

) a1P(aP( ) = bP( )
http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1http://arxiv.org/abs/math/0702230v1


2/11

n 1 Pn() q

Pn+m() Pn(1) Pm(2) 1 2 m n 1 f {1, 2}

L()

v 1 2 v v

f L()

2

1

f,1

f,2

R2 m n 1,

Pn+m() =

fL()

q(,f) Pn(f,1)Pm(f,2),

(, f) = m,n(, f)

Pn( ) = qPn( ) qnPn( )

= q1Pn( ) qnPn( )

Pn


3/11

Pn R2 n 1

Z Cn()

Q

Cn() = iZ,jZ/2Z Ci,jn ().

d Z Z/2Z

Ci,jn ()d

Ci,j+1n ().

i Z j Z/2Z

KRi,jn () = (d : Ci,jn () C

i,j+1n ())/ (d : C

i,j1n () C

i,jn ())

Q i Z

KRin() = KRi,0n () KR

i,1n () KRn() = iZ,jZ/2ZKR

i,jn ().

Pn()

Pn() =iZ

QKRin() q

i.

{.} Z i, k Z

j Z/2Z KRi,jn (){k} = KRi+k,jn () k Z k

(Z/2Z) k

R2 m n 1 i Z j Z/2Z

KRi,jn+m()=

f L()

k, l Z, k + l + (, f) = ir, s Z/2Z, r + s = j

KRk,rn (f,1) Q KRl,sm (f,2){(, f)}

= Q (, f) = m,n(, f)

KRn+m() KRn(f,1) KRm(f,2) KRn()


4/11

Pn( ) =qn qn

q q1= [n]q

Pn( ) = [n 1]q Pn( )

Pn

= [2]q Pn( )

Pn( ) = Pn( ) + [n 2]q Pn( )

Pn

+ Pn

= Pn

+ Pn

Pn() Z[q, q1] n 1 R2

Pn()

R2

Pn

R2

Pn() Pn() Pn() n 1

R2

R2 r() +1 1

v||f Z v f


5/11

1

1

1

1

2 1

21

1

1

2

2

v||f = 0 v||f = 0 v||f = 0

1

2

2

1 2

2

1

1 2

2

2

2

v||f = 1 v||f = 1 v||f = 0

v||f

1 2 f

|f =

vv||f v m n 1

(, f) = m,n(, f) = , f + m r(f,1) n r(f,2) Z.

m, n 1

R2

Q() = Qn+m() =

fL()

q(,f)Pn(f,1)Pm(f,2) Z[q, q1].

Q

Q( ) = qmPn( ) + qnPm( )

= qm[n]q + qn[m]q = [n + m]q.

f0 L() E0

Q(f0,E0) =

fL(),f|E0=f0|E0

q(,f)Pn(f,1)Pm(f,2).


6/11

Q( ) =qn+m q(n+m)

q q1= [n + m]q

Q( ) = [n + m 1]q Q( )

Q

= [2]q Q( )

Q( ) = Q( ) + [n + m 2]q Q( )

Q

+ Q

= Q

+ Q

Q(1

1

1

)

Q( ) = Q(1

1

1

) + Q(2

1

1

) + Q(2

2

2

) + Q(1

2

2

).

Pm Q(2

1

1

)

Q(2

1

1

) = qn1[m]q Q( 1 ),

Pn Q(1

1

1

)

Q( 1

1

1 ) = qm

[n 1]q Q(1

).

Pm Q(2

2

2

)

Pn Q(1

2

2

)


7/11

qm[n 1]q Q( 1 ) + qn1[m]q Q( 1 )

+qn[m 1]q Q( 2 ) + qm+1[n]q Q( 2 )

=

qm[n 1]q + qn1[m]q

Q( 1 ) +

qn[m 1]q + q

m+1[n]q

Q( 2 )

= [n + m 1]q

Q( 1 ) + Q( 2 )

= [n + m 1]q Q( ).

Q

Q

= Q

1

1

1

1

1

1

+ Q

22

2

+ Q

2

1

1

1

2

2

+Q

2

1

1

2

2

1

+ Q

1

2

2

2

1

1

+ Q

2

2

2

1

1

1

+Q

1

2 1

1 2

2

+ Q

1

1 2

2 1

2

+ Q

2

2 1

1 2

1

+Q

1

2 1

1 2

2

= Q

11

1

1

1

1

+ Q

2

2

2 + q1Q

2

2

1

1 + qQ

2

2

1

1 + qQ

1

2

2

1

+ q1Q

1

2

2

1

+ qQ

2 1

21

+q1Q

2 1

21

+ q1Q

1

1

2

2

+ qQ

1

1

2

2

.

Pn Pm

Q

= Q

1

11

1

1

1

+ Q

2

1

11

12

+Q 22

2

22

2

+ Q1

2

2

2

2

1

+ Q

2

11

1

1 2

+ Q

2

1

1 2

2

2

+Q

2

1

1

22

2

+ Q

1

2

2

11

1

+ Q

2

12

21

1

+ Q

22

1

1

12

.


8/11

Pn Pm

Q 1

1

+ q

m

[n 2]q Q 11

+ q

n2

[m]q Q 11

+Q

2

2

+ qn[m 2]q Q

22

+ q2m[n]q Q

22

+q1m[n 1]q Q

1 2

+ qn1[m 1]q Q

1 2

+q1n[m 1]q Q

2 1

+ qm1[n 1]q Q

2 1

+ Q

2

1

+ Q

1

2

= Q

+ [n + m 2]q Q

.

Q

2

2 2

1

2

21 2

1 = Q

1

2 2

2

2

2

1

1 2

Q

1

2

2

2

2

1

1

1

1 = Q

1

2

2

1

1

1 .

1 2

Q

2

2 2 2

2

22

2 2

+ Q

2

2 2 2

22

= Q

2

2 2 2

2

2 2

22

+ Q

2

2 2

22

2 ,

Q

1

2 2

2

2

2 2

1

1

+ Q1

2 2

22

1 = Q1

2 2

2

2

22

2

1 ,

Q

2 1

1

1

1

1

1

12 = Q

2

2 1

2

1

1

1

1 1

+ Q

2

1

1

1

1

2 .


9/11

Q Q Pn+m Q = Pn+m

KRin+m()=

f L()

k, l Z, k + l + (, f) = i

KRkn(f,1) Q KRlm(f,2){(, f)}.

n 1 i Z

KRi,jn () = 0 j = r() + 1 ,

r()

f L()

r() = r(f,1) + r(f,2).

L() f,2

S() = {f,1|f L()}.

S() f L() = f,1 S() (, ) = , f , f k, l Z

Q{k}l = iZ,jZ/2Z Q{k}li,j ,

Q{k}li,j =

Q

i = k j = l ,0


10/11

R2 n 2

KRn() = 1S(),2S(1),...,n1S(n2)Q{(1, . . . , n1)}r()

(1, . . . , n1) =

n2i=0

((i, i+1) + (n i) r(i+1) (n 1 i) r(i))

=n2i=0

((i, i+1) + 2 r(i+1)) (n 1) r()

0 =

KR1() =

Q 0

m = 1

n = 2 m = 1

KR3

= KR2

KR2( ){3}1

KR2( ){3}1 KR2( ){1}1

KR2( ){1}1

= Q{2} Q{2} QQQ{2} Q{4}

Q{4} Q{2} Q{2} QQQ{2}

KRn()

KRn() = 1S(),2S(1),...,nS(n1)

Q{(1, . . . , n)}r().

KRn()


11/11
http://arxiv.org/abs/math/0508510http://arxiv.org/abs/math/0505056http://arxiv.org/abs/math/0401268http://arxiv.org/abs/math/0402266http://arxiv.org/abs/math/0410495

emmanuel wagner- khovanov-rozansky graph homology and composition product

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