History Link polynomials Link homologies An introduction to link homology Marco Mackaay CAMGSD and Universidade do Algarve 2 September, 2013 1/39 Marco Mackaay An introduction to link homology
History Link polynomials Link homologies Outline History 1 2/39 Marco Mackaay An introduction to link homology
History Link polynomials Link homologies Outline History 1 Link polynomials 2 2/39 Marco Mackaay An introduction to link homology
History Link polynomials Link homologies Outline History 1 Link polynomials 2 Link homologies 3 Khovanov link homology sl ( 3 ) link homology sl ( N ) link homology 2/39 Marco Mackaay An introduction to link homology
History Link polynomials Link homologies Section outline History 1 Link polynomials 2 Link homologies 3 Khovanov link homology sl ( 3 ) link homology sl ( N ) link homology 3/39 Marco Mackaay An introduction to link homology
History Link polynomials Link homologies Link polynomials and homologies L − → P ( L ) polynomial ( Jones ’84 + HOMFLY-PT ’85)
History Link polynomials Link homologies Link polynomials and homologies L − → P ( L ) polynomial ( Jones ’84 + HOMFLY-PT ’85) L − → H ( L ) homology ( Khovanov ’99 + Khovanov-Rozansky ’04 ) 4/39 Marco Mackaay An introduction to link homology
History Link polynomials Link homologies Link polynomials and homologies L − → P ( L ) polynomial ( Jones ’84 + HOMFLY-PT ’85) L − → H ( L ) homology ( Khovanov ’99 + Khovanov-Rozansky ’04 ) χ q � � H ( L ) = P ( L ) 4/39 Marco Mackaay An introduction to link homology
History Link polynomials Link homologies Short history of foams sl ( 2 ) link homology 1 Khovanov (1999), Bar-Natan (2005), Clark-Morrison-Walker (2007) and Caprau (2007), Blanchet (2010) 5/39 Marco Mackaay An introduction to link homology
History Link polynomials Link homologies Short history of foams sl ( 2 ) link homology 1 Khovanov (1999), Bar-Natan (2005), Clark-Morrison-Walker (2007) and Caprau (2007), Blanchet (2010) 5/39 Marco Mackaay An introduction to link homology
History Link polynomials Link homologies Short history of foams sl ( 2 ) link homology 1 Khovanov (1999), Bar-Natan (2005), Clark-Morrison-Walker (2007) and Caprau (2007), Blanchet (2010) sl ( 3 ) link homologies using sl ( 3 ) -foams 2 Khovanov (2004), M.-Vaz (2007), Morrisson-Nieh (2008), Lauda-Queffelec-Rose (2012) 5/39 Marco Mackaay An introduction to link homology
History Link polynomials Link homologies Short history of foams sl ( 2 ) link homology 1 Khovanov (1999), Bar-Natan (2005), Clark-Morrison-Walker (2007) and Caprau (2007), Blanchet (2010) sl ( 3 ) link homologies using sl ( 3 ) -foams 2 Khovanov (2004), M.-Vaz (2007), Morrisson-Nieh (2008), Lauda-Queffelec-Rose (2012) sl ( N ) ( N > 3 ) link homologies using foams 3 Khovanov-Rozansky (2004), M.-Stoˇ si´ c-Vaz (2007). 5/39 Marco Mackaay An introduction to link homology
History Link polynomials Link homologies Section outline History 1 Link polynomials 2 Link homologies 3 Khovanov link homology sl ( 3 ) link homology sl ( N ) link homology 6/39 Marco Mackaay An introduction to link homology
History Link polynomials Link homologies Reidemeister moves R1 R2 R3 7/39 Marco Mackaay An introduction to link homology
� � History Link polynomials Link homologies The sl ( N ) -link polynomial Resolutions in MOY-webs (Murakami-Ohtsuki-Yamada) � � ����������� � � 1 � 0 � � � � � � � � ����������� � � � � � � � 0 � 1 � � � 8/39 Marco Mackaay An introduction to link homology
History Link polynomials Link homologies � N � � Γ ⊔ Γ ′ � = � Γ �� Γ ′ � ��� = [ N ] , � � = , 2 = [ 2 ] = [ N − 1 ] = +[ N − 2 ] + = + 9/39 Marco Mackaay An introduction to link homology
� � � � History Link polynomials Link homologies The Jones polynomial of the trefoil ( N = 2 ) q ( q + q − 1 ) q 2 ( q + q − 1 ) 2 2 1 100 110 � 3 + + � � � � � ( q + q − 1 ) 2 q ( q + q − 1 ) q 2 ( q + q − 1 ) 2 q 3 ( q + q − 1 ) 3 000 010 101 111 � � � + � + � � q ( q + q − 1 ) q 2 ( q + q − 1 ) 2 001 011 ( q + q − 1 ) 2 3 q ( q + q − 1 ) 3 q 2 ( q + q − 1 ) 2 q 3 ( q + q − 1 ) 3 − + − · ( − 1 ) n − qn + − 2 n − ( q + q − 1 ) − 1 = q − 2 + 1 + q 2 − q 6 − − − − − − − − − − − − − − − → J ( T )= q 2 + q 6 − q 8 . (with ( n + , n − ) = ( 3 , 0 ) ) 10/39 Marco Mackaay An introduction to link homology
History Khovanov link homology Link polynomials sl ( 3 ) link homology Link homologies sl ( N ) link homology Section outline History 1 Link polynomials 2 Link homologies 3 Khovanov link homology sl ( 3 ) link homology sl ( N ) link homology 11/39 Marco Mackaay An introduction to link homology
History Khovanov link homology Link polynomials sl ( 3 ) link homology Link homologies sl ( N ) link homology Khovanov link homology The main ideas Let A = Q [ X ] / ( X 2 ) , such that deg ( 1 ) = 0 and deg ( X ) = 2 . So qdim ( A {− 1 } ) = q + q − 1 . Associate to each resolution with s circles the vector space A ⊗ s { t } . Take direct sum over each column. Resolutions in consecutive columns can be obtained from one another by merging two circles into one or splitting one circle into two. Define a coproduct on A by 1 �→ 1 ⊗ X + X ⊗ 1 X �→ X ⊗ X . Use the product and coproduct in A to define differentials. 12/39 Marco Mackaay An introduction to link homology
History Khovanov link homology Link polynomials sl ( 3 ) link homology Link homologies sl ( N ) link homology Khovanov homology of the trefoil 001 011 : 0*1 1− 2− 3− *01 *11 00* 01* 000 010 101 111 0*0 1*1 *10 11* 10* *00 100 110 1*0 13/39 Marco Mackaay An introduction to link homology
History Khovanov link homology Link polynomials sl ( 3 ) link homology Link homologies sl ( N ) link homology Lemma A is a Frobenius algebra with trace ε ( 1 ) = 0 and ε ( X ) = 1 . 14/39 Marco Mackaay An introduction to link homology
History Khovanov link homology Link polynomials sl ( 3 ) link homology Link homologies sl ( N ) link homology Lemma A is a Frobenius algebra with trace ε ( 1 ) = 0 and ε ( X ) = 1 . Lemma A defines a 2-dimensional TQFT, i.e. a monoidal functor Oriented surfaces → Vect . 14/39 Marco Mackaay An introduction to link homology
History Khovanov link homology Link polynomials sl ( 3 ) link homology Link homologies sl ( N ) link homology Lemma A is a Frobenius algebra with trace ε ( 1 ) = 0 and ε ( X ) = 1 . Lemma A defines a 2-dimensional TQFT, i.e. a monoidal functor Oriented surfaces → Vect . Corollary We have d 2 = 0 . 14/39 Marco Mackaay An introduction to link homology
History Khovanov link homology Link polynomials sl ( 3 ) link homology Link homologies sl ( N ) link homology Theorem (Khovanov) Khovanov’s complex is invariant up to homotopy equivalence under the Reidemeister moves. Thus the homology is a link invariant. 15/39 Marco Mackaay An introduction to link homology
History Khovanov link homology Link polynomials sl ( 3 ) link homology Link homologies sl ( N ) link homology Theorem (Khovanov) Khovanov’s complex is invariant up to homotopy equivalence under the Reidemeister moves. Thus the homology is a link invariant. Question (Bar-Natan) What is the kernel of Khovanov’s TQFT? 15/39 Marco Mackaay An introduction to link homology
History Khovanov link homology Link polynomials sl ( 3 ) link homology Link homologies sl ( N ) link homology Bar-Natan’s cobordisms Theorem (Bar-Natan) The kernel of Khovanov’s TQFT is generated by: ( 2 D ) : = 0 ( S ) : = 0 , = 1 � � � � � � ( NC ) : = + �� �� �� �� �� �� 16/39 Marco Mackaay An introduction to link homology
� � � � � � � History Khovanov link homology Link polynomials sl ( 3 ) link homology Link homologies sl ( N ) link homology The first Reidemeister move � � d = : h = 1 i f 0 = ∑ g 0 = ( − 1 ) i 0 − 1 − i i = 0 � � 0 : 0 17/39 Marco Mackaay An introduction to link homology
History Khovanov link homology Link polynomials sl ( 3 ) link homology Link homologies sl ( N ) link homology Recovering Khovanov homology from Bar-Natan’s cobordisms Let Cob /ℓ be the category of cobordisms modulo the Bar-Natan relations. A cobordism f has degree q ( f ) = − χ ( f )+ 2 d . 18/39 Marco Mackaay An introduction to link homology
History Khovanov link homology Link polynomials sl ( 3 ) link homology Link homologies sl ( N ) link homology Recovering Khovanov homology from Bar-Natan’s cobordisms Let Cob /ℓ be the category of cobordisms modulo the Bar-Natan relations. A cobordism f has degree q ( f ) = − χ ( f )+ 2 d . Γ complete resolution: F ( Γ ) = Hom Cob /ℓ ( / 0 , Γ ) , F ( Γ ) graded 18/39 Marco Mackaay An introduction to link homology
Recommend
More recommend
