Colored sl ( N ) link homology via matrix factorizations Hao Wu George Washington University
Overview The Reshetikhin-Turaev sl ( N ) polynomial of links colored by wedge powers of the defining representation has been categorified via several different approaches. I’ll talk about the categorification using matrix factorizations, which is a direct generalization of the Khovanov-Rozansky homology. I’ll also also review deformations and applications of this categorification.
Abstract MOY graphs An abstract MOY graph is an oriented graph with each edge colored by a non-negative integer such that, for every vertex v with valence at least 2, the sum of the colors of the edges entering v is equal to the sum of the colors of the edges leaving v .
Abstract MOY graphs An abstract MOY graph is an oriented graph with each edge colored by a non-negative integer such that, for every vertex v with valence at least 2, the sum of the colors of the edges entering v is equal to the sum of the colors of the edges leaving v . A vertex of valence 1 in an abstract MOY graph is called an end point.
Abstract MOY graphs An abstract MOY graph is an oriented graph with each edge colored by a non-negative integer such that, for every vertex v with valence at least 2, the sum of the colors of the edges entering v is equal to the sum of the colors of the edges leaving v . A vertex of valence 1 in an abstract MOY graph is called an end point. A vertex of valence greater than 1 is called an internal vertex.
Abstract MOY graphs An abstract MOY graph is an oriented graph with each edge colored by a non-negative integer such that, for every vertex v with valence at least 2, the sum of the colors of the edges entering v is equal to the sum of the colors of the edges leaving v . A vertex of valence 1 in an abstract MOY graph is called an end point. A vertex of valence greater than 1 is called an internal vertex. An abstract MOY graph Γ is said to be closed if it has no end points.
Abstract MOY graphs An abstract MOY graph is an oriented graph with each edge colored by a non-negative integer such that, for every vertex v with valence at least 2, the sum of the colors of the edges entering v is equal to the sum of the colors of the edges leaving v . A vertex of valence 1 in an abstract MOY graph is called an end point. A vertex of valence greater than 1 is called an internal vertex. An abstract MOY graph Γ is said to be closed if it has no end points. An abstract MOY graph is called trivalent is all of its internal vertices have valence 3.
Embedded MOY graphs An embedded MOY graph, or simply a MOY graph, Γ is an embedding of an abstract MOY graph into R 2 such that, through each vertex v of Γ, there is a straight line L v so that all the edges entering v enter through one side of L v and all edges leaving v leave through the other side of L v . · · · i 1 i 2 i k ■ ❑ ✒ L v v i 1 + i 2 + · · · + i k = j 1 + j 2 + · · · + j l ✒ ✕ · · · ■ j 1 j 2 j l Figure: An internal vertex of a MOY graph
Trivalent MOY graphs and their states ■ ✒ ✻ e e 1 e 2 or ✻ ✒■ e 1 e 2 e Let Γ be a closed trivalent MOY graph, and E (Γ) the set of edges of Γ. Denote by c : E (Γ) → N the color function of Γ. That is, for every edge e of Γ, c ( e ) ∈ N is the color of e .
Trivalent MOY graphs and their states ■ ✒ ✻ e e 1 e 2 or ✻ ✒■ e 1 e 2 e Let Γ be a closed trivalent MOY graph, and E (Γ) the set of edges of Γ. Denote by c : E (Γ) → N the color function of Γ. That is, for every edge e of Γ, c ( e ) ∈ N is the color of e . Define N = {− N + 1 , − N + 3 , · · · , N − 3 , N − 1 } and P ( N ) to be the set of subsets of N .
Trivalent MOY graphs and their states ■ ✒ ✻ e e 1 e 2 or ✻ ✒■ e 1 e 2 e Let Γ be a closed trivalent MOY graph, and E (Γ) the set of edges of Γ. Denote by c : E (Γ) → N the color function of Γ. That is, for every edge e of Γ, c ( e ) ∈ N is the color of e . Define N = {− N + 1 , − N + 3 , · · · , N − 3 , N − 1 } and P ( N ) to be the set of subsets of N . A state of Γ is a function σ : E (Γ) → P ( N ) such that (i) For every edge e of Γ, # σ ( e ) = c ( e ). (ii) For every vertex v of Γ, as depicted above, we have σ ( e ) = σ ( e 1 ) ∪ σ ( e 2 ). (In particular, this implies that σ ( e 1 ) ∩ σ ( e 2 ) = ∅ .)
Weight ■ ✒ ✻ e e 1 e 2 or ✻ ✒■ e 1 e 2 e For a state σ of Γ and a vertex v of Γ as depicted above, the weight of v with respect to σ is defined to be c ( e 1) c ( e 2) − π ( σ ( e 1 ) ,σ ( e 2 )) , wt ( v ; σ ) = q 2 where π : P ( N ) × P ( N ) → Z ≥ 0 is define by π ( A 1 , A 2 ) = # { ( a 1 , a 2 ) ∈ A 1 × A 2 | a 1 > a 2 } for A 1 , A 2 ∈ P ( N ) .
Rotation number Given a state σ of Γ, ◮ replace each edge e of Γ by c ( e ) parallel edges, assign to each of these new edges a different element of σ ( e ),
Rotation number Given a state σ of Γ, ◮ replace each edge e of Γ by c ( e ) parallel edges, assign to each of these new edges a different element of σ ( e ), ◮ at every vertex, connect each pair of new edges assigned the same element of N .
Rotation number Given a state σ of Γ, ◮ replace each edge e of Γ by c ( e ) parallel edges, assign to each of these new edges a different element of σ ( e ), ◮ at every vertex, connect each pair of new edges assigned the same element of N . This changes Γ into a collection { C 1 , . . . , C k } of embedded oriented circles, each of which is assigned an element σ ( C i ) of N .
Rotation number Given a state σ of Γ, ◮ replace each edge e of Γ by c ( e ) parallel edges, assign to each of these new edges a different element of σ ( e ), ◮ at every vertex, connect each pair of new edges assigned the same element of N . This changes Γ into a collection { C 1 , . . . , C k } of embedded oriented circles, each of which is assigned an element σ ( C i ) of N . The rotation number rot ( σ ) of σ is then defined to be k � rot ( σ ) = σ ( C i ) rot ( C i ) . i =1
The sl ( N ) MOY graph polynomial The sl ( N ) MOY polynomial of Γ is defined to be � � wt ( v ; σ )) q rot ( σ ) , � Γ � N := ( σ v where σ runs through all states of Γ and v runs through all vertices of Γ.
MOY relations (1–4) � N � 1. �� m � N = , where � m is a circle colored by m . m i j k � i j k ■ ■✒ ■ ✒ ✒ � � � ✒ ■ 2. = . j + k i + j ✻ ✻ N N i + j + k i + j + k ✻ ✻ m + n m + n � � � � ✻ ✻ � m + n � · 3. = . m n n N N ✻ m + n ✻ ✻ m m � � � � n � N − m � 4. = · . ✻ ❄ m + n n ✻ N N m
MOY relations (5–7) ■ � ✻ ■ ✠ 1 1 m m ✛ ✠ m +1 � � � � � 5. ✲✻ = + [ N − m − 1] · . m ❄ 1 1 m ❄ m − 1 N N N ✒ ✒ ❘ m +1 ❘ ❄ 1 m 1 m ✻ ✻ ✻ ✻ ❪ ✣ ✲ m l m m l l ✻ ✻ n ✻ � � � ✛ � � � 6. � m − 1 � � m − 1 � l + n m − n = · + · . n n − 1 ✻ ✻ m + l l − 1 N N N ✛ l + n − 1 ✣❪ ✻ ✻ 1 m + l − 1 1 m + l − 1 1 m + l − 1 ✻ ✻ ✻ ✻ m n + l m n + l ✲ ✛ j n + k − m � ✻ ✻ � � ✻ ✻ � 7. � � = � m l · . n + k m + l − k m − j n + l + j j =max { m − n , 0 } k − j ✛ ✲ N N ✻ ✻ ✻ ✻ k n + j − m n m + l n m + l
MOY relations (5–7) ■ � ✻ ■ ✠ 1 1 m m ✛ ✠ m +1 � � � � � 5. ✲✻ = + [ N − m − 1] · . m ❄ 1 1 m ❄ m − 1 N N N ✒ ✒ ❘ m +1 ❘ ❄ 1 m 1 m ✻ ✻ ✻ ✻ ❪ ✣ ✲ m l m m l l ✻ ✻ n ✻ � � � ✛ � � � 6. � m − 1 � � m − 1 � l + n m − n = · + · . n n − 1 ✻ ✻ m + l l − 1 N N N ✛ l + n − 1 ✣❪ ✻ ✻ 1 m + l − 1 1 m + l − 1 1 m + l − 1 ✻ ✻ ✻ ✻ m n + l m n + l ✲ ✛ j n + k − m � ✻ ✻ � � ✻ ✻ � 7. � � = � m l · . n + k m + l − k m − j n + l + j j =max { m − n , 0 } k − j ✛ ✲ N N ✻ ✻ ✻ ✻ k n + j − m n m + l n m + l The above MOY relations uniquely determine the sl ( N ) MOY graph polynomial.
Unnormalized colored Reshetikhin-Turaev sl ( N ) polynomial For a link diagram D colored by non-negative integers, define � D � N by applying the following at every crossing of D . ✻ ✻ m n ■ ✒ ✲ � n � m � � m n + k − m � ( − 1) m − k q k − m ✻ ✻ = , n + k m − k ✛ k =max { 0 , m − n } N N ✻ ✻ k n m ✻ ✻ m n ■ ✒ ✲ � n � m � � m n + k − m � ( − 1) k − m q m − k ✻ ✻ = . n + k m − k ✛ N k =max { 0 , m − n } N ✻ ✻ k n m
Normalized colored Reshetikhin-Turaev sl ( N ) polynomial For each crossing c of D , define the shifting factor s( c ) of c by ■ ✒ � m n ( − 1) − m q m ( N +1 − m ) if m = n , s = 1 if m � = n , ■ ✒ � m n ( − 1) m q − m ( N +1 − m ) if m = n , s = 1 if m � = n . The normalized Reshetikhin-Turaev sl ( N )-polynomial RT D ( q ) of D is � RT D ( q ) = � D � N · s( c ) , c where c runs through all crossings of D .
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