FOAM EVALUATION AND LINK HOMOLOGY THEORIES LECTURE NOTES FOR “PERSPECTIVES ON QUANTUM LINK HOMOLOGY THEORIES” LOUIS-HADRIEN ROBERT ABSTRACT. CONTENTS Introduction and disclaimer1 0.1. Conventions1 1. Lecture 1: glN -invariants via graph colorings1 1.1. Quantum binomials2 1.2. MOY graphs3 1.3. Link invariants6 2. Lecture 2: Coloring foams9 2.1. Foams and colorings9 2.2. Evaluation 11 2.3. Local relations 14 3. Lecture 3: Universal construction and glN -link homology 15 3.1. Universal construction 15 3.2. Rickard complexes and hyper-rectangles 17 3.3. Link homology 20 4. Lecture 4: Symmetric glN homology 21 4.1. Symmetric MOY calculus 21 4.2. Vinyl graphs and foams 23 4.3. Planar approach 27 References 28 INTRODUCTION AND DISCLAIMER 1. LECTURE 1: glN -INVARIANTS VIA GRAPH COLORINGS If H is a quantum group and L is a link colored with finite dimensonial H- modules, Reshetikhin–Turaev associate to this data a Laurent polynomial in q. In this first lecture, we aim to describe these link invariants in term of graph coloring in the case H Uq(gl ) and the H-module are (quantum) exterior powers of V, the Æ N standard representation of H. If N 2, the only relevant representation is V itself Æ and the the invariant is the Jones polynomial. For general N, if all components are colored by V, the invariant is called the glN polynomial and is denoted PN . It 1 2 LHR satisfies the following skein relation: 0 1 0 1 0 1 N N q¡ PN @ A q PN @ A PN @ A. ¡ Æ We will actually first focus on a framed version of these invariant: they satisfy a Reidemeister I up to normalization. The content of this lecture follows [MOY98] and its reinterpretation given in [Rob15]. 1.1. Quantum binomials. In all lectures, q is a formal parameter. The aim of this section is to give a quantum version of the following identity: {1,..., n} powerset cardinal à ! ¡ ¢ n #P a n . J K Æ a only subsets with a a choose n elements Namely: endow n with Ç quantum cardinal J K " # ¡ ¢ n #qP a n . J K Æ a quantum binomial In this formula and everywhere else, q is a formal variable. k k k k q q¡ X k i Y For k in Z, define [k] q 1 2 Z£q, q 1¤, if k 0, define [k]! [i] ¡ 1 ¡ ¡ Å ¡ Æ q q¡ Æ i 1 2 ¸ Æ i 1 2 £ 1¤ ¡ Æ £ Æ1¤ Z q, q¡ , with the usual convention that an empty product is equal to 1 Z q, q¡ . 2 Finally, if n,a Z, define 2 8 a " # Y [n 1 k] n <> Å ¡ if a 0, k 1 [k] ¸ a Æ Æ :>0 ortherwise. " # n Remark 1.1. (1) At this stage it is not clear that belongs to Z£q, q 1¤. a ¡ (2) For k { 1,0,1},[k] k; 2 ¡ Æ (3) For k Z,[ k] [k]; 2 ¡" Æ¡# " # n a n 1 (4) For n,a Z, ( 1)a ¡ ¡ ; 2 a Æ ¡ a " # n [n]! (5) For n,a Z 0, . 2 ¸ a Æ [a]![n a]! ¡ Lemma 1.2. The following identities hold: n m n m (1) [m n] q¡ [m] q [n] q [m] q¡ [n] Å Æ Å Æ Å FOAM EVALUATION AND LINK HOMOLOGIES 3 for any m, n in Z. " # " # " # " # " # n a n 1 a n n 1 a n 1 n a n 1 (2) q ¡ q ¡ ¡ q¡ ¡ q ¡ ¡ . a Æ a Å a 1 Æ a Å a 1 ¡ ¡ for any n,a in Z. Proof. Computations left to the reader. £ 1¤ Corollary 1.3. Quantum binomials are in Z q, q¡ and respect have a parity prop- erties: exponents appearing are either all even or all odd. Moreover, they are symmet- 1 ric under q q¡ . 7! Sketch of proof. If n is nonnegative, argue by induction on n, using that £ n ¤ £ n ¤ 1 n Æ 0 Æ which follows from the definition for all nonnegative n. If n is negative, use Re- mark 1.1(4) and the result for n 0. ¸ Definition 1.4. A weighted set is a set X together with a map wX : X Z. If X is a ! finite weighted set, the quantum cardinal of X is given by the following formula: X wX (x) £ 1¤ #q X q Z 0 q, q¡ . Æ x X 2 ¸ 2 Exercise 1.5. If X and Y a two finite weighted sets, X Y is weighted by declaring £ that wX Y ((x, y)) wX (x) wY (y). Prove that #q(X Y ) #q X#qY . £ Æ Å £ Æ Definition 1.6. Let (X, ) be a finite ordered set and Y P (X). Define the weight Á 2 of Y relatively to X by: w(Y ) w (X)(Y ): #{(x, y) (X \ Y ) Y such that x y} Æ P Æ 2 £ Á #{(x, y) (X \ Y ) Y such that y x}. ¡ 2 £ Á With the last definition, we see that if X is ordered, then for any nonnegative integer a, P a (X) is naturally weighted Proposition 1.7. For all a, n in Z 0, one has: ¸ " # ¡ ¢ n #qP a n . J K Æ a ¡ ¢ Idea of the proof. One shows that #qP a n satisfies relation (2) and conclude by J K induction. 1.2. MOY graphs. Definition 1.8. A MOY graph1 is a trivalent oriented plane2 graph ¡ (V(¡),E(¡)) Æ endowed with a thickness function `: E(¡) Z 0 such that the neighborhood of ! ¸ every vertex v is given by one of the two following local models: a b a b Å a b a b Å The first is called a split vertex, the second a merge vertex. The edge with thickness a b (resp. a, resp. b) is called the thick (resp. thin left, resp. thin right) edge at v. Å This edge is denoted et(v) (resp. el(v), resp. er(v)). Vertexless loops are allowed. 1MOY stands for Murakami–Ohtsuki–Yamada. These graphs are often called webs in the literature. 2Plane means embedded in R2 rather than embeddable in R2. 4 LHR Example 1.9. 3 2 1 1 4 1 2 1 2 3 Definition 1.10. Let ¡ be a MOY graph. A glN -coloring (or simply coloring) is a map ¡ ¢ c: E(¡) P N , such that: ! J K (1) For all edge e,#c(e) `(e). Æ (2) For every vertex v, c(et(v)) c(el(v)) c(er(v)). Æ t The set of glN -colorings of ¡ is denoted colglN (¡) or simply col(¡). For i j N , denote ¡i j(c) union of edges e of ¡ such that {i, j} c(e) 1. Keep Ç 2 J K \ Æ the orientation of edges containing j and reverse the ones containing i. ¡i j(c) is an oriented closed curve in R2. Example 1.11. {1,2,4} {2,4} {1} N 5 { Æ 1 , {4} 2 N {1,2,3,4,5} , 4 {5} {1,4} Æ , 5 J K } ¡2,4 is depicted {2} {1,4} w(¡2,4) 2 Æ {1,2,4} If γ is a finite disjoint union of oriented circles denote w(γ) the number of posi- tively oriented circles minus the number of negatively oriented circles in this collec- tion. For c a glN -coloring, define: X w(c) wglN (c) w(¡i j(c)). Æ Æ 1 i j N · Ç · By this mean, the set colglN (¡) is endowed with a weight function. Definition 1.12. Define the glN evaluation of a MOY graph ¡ by the following for- mula: X wgl (c) £ 1¤ ¡ gl #qcolgl (¡) q N Z 0 q, q¡ . h i N Æ N Æ 2 ¸ c colgl (()¡) 2 N Example 1.13. " # D E D E ¡ ¢ N a a #qP a N . glN Æ glN Æ J K Æ a FOAM EVALUATION AND LINK HOMOLOGIES 5 Lemma 1.14. (1) ¡ is invariant under ambient isotopy. h i 1 (2) For any MOY graph ¡, ¡ is symmetric under q q¡ . Dh iE 7! (3) For any MOY graph ¡, ¡ ¡ , where ¡ denotes a mirror image of ¡ Æ h i (4) For any MOY graphs ¡1 and ¡2, ¡1 ¡2 ¡1 ¡2 . h t i Æ h ih i Sketches of proof. Item (1) is obvious Item (2) follows from the fact that ¶: N N ! J kK JN K1 k 7! Å ¡ induces an involution on col(¡) such that w(¶(c)) w(c). Æ¡ Item (3): there is a natural bijection col(¡) c c col(¡) and w(c) w(c). 3 7! 2 Æ¡ Conclude with point (2). Item (4): there is a natural isomorphism of weighted sets between col(¡1 ¡2) and t col(¡1) col(¡2). The result follows from Exercise 1.5. £ Proposition 1.15. The following skein theoretic3 relations hold: * + " #* + a b a b (3) Å , Æ a a b a b Å Å a b c a b c * + * + (4) , Æ a b c a b c Å Å Å Å * + " #* + N a (5) b ¡ , Æ b a a 1 1 1 1 * a + * 1 + * + a 1 (6) a 1 a 1 [N a 1] ¡ , Å Å Æ Å ¡ ¡ 1 a a a a a a b a b * 1 + * 1 + * + (7) [b a] . Æ Å ¡ 1 1 a b a b a b Proof. We only prove some of them in some special cases, the general case and the rest of the relations are left to the reader. Relation (3) for a b 1.