Catalan States of Lattice Crossing: Application of Plucking Polynomial

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1 Catalan States of Lattice Crossing: Application of Plucking Polnomial Mieczslaw K. Dabkowski Jozef H. Prztcki arxiv:7.0528v [math.gt] 4 Nov 207 Abstract For a Catalan state C of a lattice crossing L (m n) with no returns on one side we find its coefficient C (A) in the Relative Kauffman Bracket Skein Module epansion of L (m n). We show in particular that C (A) can be found using the plucking polnomial of a rooted tree with a dela function associated to C. Furthermore for C with returns on one side onl we prove that C (A) is a product of Gaussian polnomials and its coefficients form a unimodal sequence. Kewords: Catalan States Gaussian Polnomial Knot Kauffman Bracket Link Lattice Crossing Plucking Polnomial Rooted Tree Skein Module Unimodal Polnomial Mathematics Subject Classification 2000: Primar 57M99; Secondar 55N 20D Introduction Our work finds its roots in the theor of skein modules of -manifolds. Let us recall some standard notation and terminolog. F gn denotes an oriented surface of genus g with n boundar ( components I = [0 ] and M = F gn I. The KBSM of M has a structure of an algebra S ) 2 M (Skein Algebra ( of M ) with the multiplication defined b placing a link L above a link L 2. The algebra S ) 2 M appears naturall in a stud of quantizations of SL(2 C)-character varieties of fundamental groups of surfaces [ 5]. Therefore in several important cases (g n) {(0 2) (0 ) ( 0) ( ) ( 2)} these algebras have been well studied and understood (see for instance Theorem 2. Corollar.2 and Theorem 2.2 in [2]). Furthermore a ver elegant formula for the product in S 2 (F 0 I) was found b Charles Frohman and Razvan Gelca in [7]. Inspired b these results we started our quest for a formula of a similar tpe in S 2 (F 04 I) 2 (see [6]). In this paper we unravel a surprising connection between our skein algebra Gaussian polnomials and rooted trees. The Relative Kauffman Bracket Skein Module (RKBSM) was defined in []. As shown in [2] RKBSM of D 2 I with 2k fied points on ( D 2 I ) is a free module with a basis consisting of all crossingless connections C (called Catalan states) between boundar points. Let R 2 mn be an m n- rectangle and N = R 2 mn I. We fi 2 (m + n) points ( ) ( ) ( ) ( i 2 i 2 i = 2... n; j 2 j 2) j = 2... m on ( N ) so that their projection onto ( ) R 2 (mn) ( ) mn is as in Figure.. Let S 2 N denote the RKBSM of N with the 2 (m + n) points fied. Consider m n tangle L (m n) (m n-lattice crossing) obtained b placing n vertical parallel lines above m horizontal parallel lines as in Figure.2. Cat (m n) shall denote the set of all crossingless connections between 2 (m + n) points on ( N ). We can place an order on Cat (m n) so that it becomes a basis of S (mn) ( ) 2 N (see [2]) and now we have L (m n) = C (A) C C Cat(mn) For an oriented -manifold M Kauffman Bracket Skein Module (KBSM ) was defined in 987 b J. H. Prztcki ([]). 2 For M = F 04 I the presentation of the algebra S 2 ( M ) was found b D. Bullock and J.H. Prztcki (see Theorem. in [2]).

2 2 n-2 n- n n vertical lines 2 n-2 n- n m- m -2 m-2 m- m m horizontal lines + marker - marker Figure. 2 n n-2 n- Figure. 2 n-2 n- n Figure.2 for some Laurent polnomials C (A) Z [ A ±]. In this paper we aim to determine C (A) for all Catalan states C with no returns on a fied side of R 2 mn. Let K (m n) be the set of all Kauffman states of L (m n) i.e. the set of all choices of positive and negative markers for mn crossings as in Figure.. For s in K (m n) denote b D s its corresponding diagram obtained b smoothing all crossings of L (m n) according to s. Furthermore let D s be the number of closed components of D s and C s be the Catalan state resulting from removal of said components. Define the function K : K (m n) Cat (m n) b K (s) = C s. For L (m n) the Kauffman state sum is given b: L (m n) = A p(s) n(s) ( A 2 A 2 ) Ds K (s) s K(mn) where p(s) and n(s) stand for the number of positive and negative markers determined b s respectivel. Hence for C Cat (m n) its coefficient is given b: C (A) = s K (C) A p(s) n(s) ( A 2 A 2 ) Ds if K (C) (C is a realizable Catalan state) otherwise C (A) = 0 (C is non-realizable). In [6] we gave necessar and sufficient conditions for K (C) and found a closed form formula for the number of realizable Catalan states. Furthermore for C with no arcs starting and ending on the same side of Rmn 2 a formula for C (A) was also obtained (see [6]). Let Cat F (m n) denote the subset of all realizable Catalan states of L (m n) that have no returns on the floor of R 2 mn. In this paper we consider a submodule of S (mn) ( ) 2 N generated b Cat F (m n) along with the projection L F (m n) of L (m n) onto the aforementioned submodule. In particular we show the following: ) For C Cat F (m n) its coefficient C (A) can be determined using the plucking polnomial Q of a rooted tree with a dela function 4 associated to C. 2) For C Cat F (m n) with returns allowed onl on the ceiling of R 2 mn the coefficient C (A) is a product of Gaussian polnomials and its coefficients form a unimodal sequence. The paper is organized as follows. In the second section we define a poset associated with Kauffman states s K (C) and use it to compute C (A). Net in section three for a Catalan state In fact it was derived b J. Hoste and J. H. Prztcki in 992 while writing [9] although it was not included in the final version of their paper. Also as we were told b M. Hajij a related formula was noted b S. Yamada [6] [8]. 4 Polnomial Q was defined in [] [4] and its properties were eplored in depth in [] [4] and [5]. 2

3 C Cat F (m n) we define a rooted tree T (C) with a dela function and the plucking polnomial Q (T (C)). We show that C (A) can be computed using Q (T (C)) the highest and the lowest coefficients of C (A) are equal to one and all its coefficients in between are positive. In section four we discuss several important properties of C (A). In particular using results of [] [4] and [5] for C with returns on its ceiling onl we obtain as corollaries that C (A) has unimodal coefficients and we give criterion for a Laurent polnomial to be C (A). Finall in the last section we give a closed formula for C (A) where C Cat (m ). 2 Poset of Kauffman States In this section for Catalan states with no returns on the floor we construct a poset of Kauffman states which we later use to compute C (A). Given a Catalan state C its boundar is C = X X Y Y where X = {... n } X = {... n} Y = {... } and Y = {... m} (see Figure 2.). Arcs e 0 joining and and e n joining n and as well as all arcs e i that join i and i+ (i = 2... n ) will be called the innermost upper cups. More generall for a Catalan state C we will refer to arcs with both ends in X or with one end in X and the other in either Y or Y as upper cups i- i i+ i+2 n- n e i l m i- i i+ i+2 n- n ( ) Figure 2.2 Figure 2. Let C be a Catalan state of L (m n). We sa that C has no returns on the floor C Cat F (m n) if none of its (m + n) arcs have both ends in X. For such a state there are precisel arcs with none of its endpoints in X. Consider the set of Kauffman states K (C) of L (m n) which realize C Cat F (m n). Each Kauffman state s K (m n) can be identified with an m n matri (s ij ) where s ij = ±. Denote b K F (m n) the subset of K (m n) consisting of all states s with rows s i in the form: s i = ( ) }{{}}{{} b i n b i where 0 b i n (see Figure 2.2). In the net proposition we show that C (A) in L F (m n) can be computed using onl Kauffman states s K F (m n) with K (s) = C. In particular this significantl reduces the number of Kauffman states that one needs to consider 5. Proposition 2. The projection of L (m n) onto the submodule of S (mn) 2 is given b L F (m n) = A p(s) n(s) K (s) s K F (mn) 5 To compute L F (m n) we need to consider (n + ) m Kauffman states instead of 2 mn. ( M ) generated b Cat F (m n)

4 Proof. We compute L F (m n) starting from the top row of crossings to the bottom (row b row) using the Kauffman bracket skein relation and omitting states with returns on the floor (observe that we will never get trivial components). When doing so we should not take into the account Kauffman states with change of markers from to in a row (see Figure 2.) as the result in lower caps after the regular isotop of diagrams (see Figure 2.4). i i+ j j+ k k+ n- n i i+ j j+ k k+ n- n m m i i+ j j+ k k+ n- n ( ) i i+ j j+ k k+ n- n ( ) Figure 2. Figure 2.4 Therefore all Catalan states with no returns on the floor can be obtained using onl Kauffman states K F (m n). In particular we have L F (m n) = A p(s) n(s) K (s). s K F (mn) Hence the coefficient C (A) of the Catalan state C Cat F (m n) can be computed using the formula: C (A) = A p(s) n(s) s A(C) where A (C) consists of all Kauffman states representing C (i.e. A (C) = {s K F (m n) K (s) = C}). Let P (m n) = { n} m be the set of all sequences b = (b b 2... b m ) where 0 b j n and j { 2... m}. The sets K F (m n) and P (m n) are in bijection so denote b b (s) P (m n) the sequence corresponding to s K F (m n); analogousl b s (b) K F (m n) we denote the Kauffman state corresponding to b P (m n). Given b P (m n) we denote b C (b) the Catalan state obtained from b (i.e. C (b) = K (s (b))). Proposition 2.2 For ever C Cat F (m n) there is a sequence b P (m n) such that C (b) = C. Proof. Ever realizable Catalan state C (not necessaril in Cat F (m n)) has at least one innermost upper cup e b (see Figure 2.5). Otherwise as shown in Figure 2.6 C has no upper cups below the dotted line which also cuts C in n + 2 points. Hence it follows that C is non-realizable 6. The statement of Proposition 2.2 follows b induction on m. If m = there are n + Catalan states with no returns on the floor and a single innermost upper cup (see Figure 2.7) so Proposition 2.2 holds with b = (b ). Let m 2 and C Cat F (m n). Assume that the statement holds for all numbers smaller than m. As we noted before C has an innermost upper cup e b and we can deform the diagram of C to that in Figure 2.5. Consider the Catalan state C shown in the bottom of Figure 2.5. Since C is in Cat F (m n) then b the inductive hpothesis there is a sequence (b 2... b m ) that realizes it. We conclude that b = (b b 2... b m ) realizes C which finishes our proof. 6 As we have shown in [6] (see Lemma 2. and Theorem 2.5) a Catalan state C Cat (m n) is realizable if and onl if ever vertical line cuts C at most m times and ever horizontal line cuts C at most n times. 4

5 2 n- n en i i+ 2 n-2 n- n e 0 e i i+2 n C m- 2 n- n C i i+ n i+2 C m- 2 n-2 n- n Figure n-2 n- n n+2 n-2 2 n- n Figure 2.6 It is clear (from our proof of Proposition 2.2) that for m 2 there might be several sequences 7 b P (m n) with C (b) = C. Therefore for C Cat F (m n) we let b (C) = {b P (m n) C (b) = C}. We note that for C Cat F (m n) there is a bijection between the set A (C) of all Kauffman states representing C and the set b (C). Definition 2. Let b = (b b 2... b m ) P (m n) be a sequence with b i < b i+ < n for some i ( i m). Let P i an operation defined b P i (b) = (b... b i b i+ + b i + b i+2... b m ) and P i be its inverse. Sequences b b P (m n) are called P -equivalent if b and b differ b a finite number of P ± i operations. Proposition 2.4 If b b (C) and b P (m n) is P -equivalent to b then b b (C). Proof. It suffices to show that if b b (C) and b = P i (b) then b b (C). This is evident from Figure 2.8 since operations P i correspond (geometricall) to a change of order in which arcs of C (with no ends on the floor) are realized. Using P i operations we define a poset structure on P (m n) as follows: b covers b (i.e. b b ) iff there is i < m such that b = P i (b). Let be the transitive closure of. Clearl (P (m n) ) is a poset. Hence (b (C) ) is also a poset. Proposition 2.5 Let C Cat F (m n) and G (C) = (V E) be the directed graph with vertices V = b (C) and directed edges E = {(b b ) V V b b }. Let G (C) be the graph obtained from G (C) b ignoring the edge orientations. Then G (C) is a connected graph. 7 The Catalan state C in Cat F (m n) has eactel one representative b iff b = (b b 2... b m) with b = b 2 =... = b k = n b k+ =... = b m = n for some k m or b = b 2 =... = b k = b k+ =... = b m = 0 for for some k m. 5

6 2 n- n i+ i 2 n-2 n- n n e n e i e 0 2 n- n 2 n- n 2 n- n Figure 2.7 k k+ j j+2 k k+ k+2 k+2 j j+2 j- j+ n j- j+ n s(b... b i- ) s(b... b i- ) b i b i + b i i b i+ + Pi b i+ b i+ + b i+ i+ bi + b+ i b +2 i b+ i+ b+2 i+ i i+ s(b i+2 m )... b b i < b i+ s(b i+2 m )... b k k+ k+2 j- j j+ j+2 n k k+ k+2 j- j j+ j+2 n Figure 2.8 Proof. Connectedness of the graph G (C) follows b induction on m. For m = the statement is obvious since b (C) has onl one element. Assume that for all C Cat F (m n) the graph G (C ) is connected and let b = (b b 2... b m ) b = (b b 2... b m) V. If b = b then sequences a = (b 2... b m ) a = (b 2... b m) b (C ) where C Cat F (m n) is obtained from C b removing its top corresponding to b (= b ). B inductive assumption the graph G (C ) is connected so a and a are P -equivalent and consequentl b and b are also P -equivalent. Therefore there is a path in G (C) joining vertices b and b. WLOG we can assume that 0 b < b n (in fact b b 2 because e b and e b are the innermost cups). In the sequence a = (b 2... b m) that represents C Cat F (m n) there is b k corresponding to the innermost upper cup e b of C. B inductive assumption one changes a to a = (b 2 b... b m) b (C ) using P ± i operations where b 2 = b represents the innermost upper cup e b in C. Now P operation changes the sequence (b b b 2 b... b m) to (b b + b... b m). Using the same argument as in the first case (i.e. b = b ) we see that b and b are in the same connected component of G (C). It follows that the graph G(C) is connected. We observe that if b b then b = b + 2 hence the directed graph G (C) has no directed ccles i.e. G (C) is a Hasse diagram of the poset (b (C) ). Let b = m i= b i denote the weight of sequence b P (m n). Define b m b M b (C) to be a minimal and a maimal sequence representing C in the leicographic order on b (C) 8 respectivel. Proposition 2.6 Let C Cat F (m n) and b m b M b (C) be defined as the above. Then b m b M are unique elements having the smallest and the largest weight respectivel. Proof. Let b M = (b b 2... b m ) and b = (b b 2... b m) be another sequence representing C (if it eists). We show that b < b M b induction on m. For m = the statement holds since b (C) has eactl one element. Assume that the statement is valid for all numbers smaller than m. If b = b we can compare shorter sequences (b 2... b m ) and ( b 2... b m). Using induction assumption we conclude that the first sequence has the larger weight i.e. b < b M. 8 Recall b = (b b 2... b m) le b = ( b b 2... b m) iff there is k such that bi = b i for i < k and b k < b k. 6

7 e b Suppose b is smaller than b. There eists b k < n (k > ) which represents the innermost upper cup in C. We consider the following cases: (i) b 2 =... = b k = n. Changing order of cups e b and e b in (b n... n b k... b m) where b k = b + k results in a sequence (b n... n b + k... b m) with the weight larger b. (ii) There is b i different than n for some < i < k. Let j be the smallest such inde (so b i = n for < i < j). Notice that the arc giving the innermost upper cup e b in C on the level j has inde b +j. There are two possibilities: If b j < b + j then the sequence (b j.. b k... b m) represents the Catalan state with two innermost cups e b j and e b+j where e b j is to the right of e b+j. B inductive assumption the sequence does not have the maimal weight as it is not maimal in the leicographical order (i.e. b j < b + j ). If b j > b + j then b j represents the innermost upper cup e b j j+ in C which is to the right of e b. This however contradicts the maimalit of b M in the leicographic order. Therefore b M is a unique element with the maimal weight in b (C). A proof for b m is similar. Eample 2.7 Figure 2.9 (diagram on the right) shows the Hasse diagram G (C) associated to the Catalan state C Cat F (4 4) (see the diagram on the left) and the maimal sequence b M = ( 4 4 ) that realizes C (see the middle) (44) (42) () (04) (02) Figure 2.9 From the definition of b (C) it directl follows that C (A) = ( m ) A 2bi n = b b(c) i= For the Catalan state in Figure 2.9 in particular we have: b b(c) C (A) = + 2A 4 + A 8 + A 2. A 2 b mn. Coefficient C (A) and Plucking Polnomial In this section we eplore the relationship between coefficients of Catalan states and the plucking polnomial of associated rooted trees. We would like to stress the fact that the definition of the plucking polnomial was stronglotivated b [6]. 9. Rooted Tree with Dela Function Associated to C Cat F (m n) Let C be a Catalan state in Cat (m n). We define in a standard wa a planar tree T (C) b taking the dual graph to C. That is the set of (m + n) arcs of C splits the rectangle R 2 mn into m + n + bounded regions R i. For each region R i we have a verte and two vertices are adjacent in T (C) iff their corresponding regions share boundar in common (see Figure.). 9 In fact the plucking polnomial for rooted trees was discovered just after the paper [6] was finished. 7

8 Figure. For a Catalan state C with no returns on the floor (C Cat F (m n)) we use a modified version of the tree T (C) described above. Let A = {a a 2... a n } be the set of arcs of C with an end on the floor of R 2 mn and {c c 2... c m } be the set of arcs of C A. Denote b T (C) = (V E) the dual graph to C A and observe that T (C) is an embedded planar tree with m + vertices v V (corresponding to regions of C A) and m edges e E (corresponding to arcs 0 of C A). There is an obvious choice for the root v 0 V of T (C) i.e. v 0 is the verte assigned to the regions containing (as a part of its boundar) the floor of R 2 mn (see right of Figure.2) T`(C) T(C) Figure.2 For a verte u V (T ) let d (u) be the number of vertices adjacent to u (degree of u). A verte v v 0 of degree one (d (v) = ) is called a leaf. Denote b L (T v 0 ) the set of all leaves of (T v 0 ). Let h : C A {0... m} be defined b setting h (c) to be 0 if c has both ends in X; and h (c) to be the maimal inde i of the end point i or i of the arc c C A otherwise. Define the dela function f : L (T (C) v 0 ) { 2... m} b putting f (v) = ma { h (c)} where c corresponds to the edge e incident to v. Let T (C) = (T (C) v 0 f) be the rooted tree with a dela function f associated to C Cat F (m n) (see Figure.). We note that there might be several different Catalan states C with the same T (C) (compare Figure. and Figure.4)..2 Plucking Polnomial of Rooted Trees with Dela Function We recall after [ 4] the definition of the plucking polnomial of a plane rooted tree with a dela function from leaves to positive integers. Let (T v 0 ) be a plane rooted tree (we assume that our trees are growing upwards see Figure.5). For v L(T v 0 ) consider the unique path from v to v 0 and let r(t v) be the number of vertices of T to the right of the path. 0 Each edge e E is dual to a unique arc c C A. 8

9 c c C a 2 a c 2 a c 4 a 4 v 0 C-A A = {a a 2 a a 4 } C-A ={c c 2 c c 4 } v 2 v v (v ) = v 0 f f ( T( C) v 0 f ) f (v 2 ) = (v ) = a 2 a 5 a a a C Figure. Figure.4 v v 0 Figure.5 Definition. Let (T v 0 f) be a plane rooted tree T with the root v 0 and dela function f. Denote b L (T ) the set of all leaves v of T with f (v) =. The plucking polnomial Q (T f) of (T v 0 f) is a polnomial in variable q defined as follows: If T has no edges we put Q (T f) = ; otherwise Q (T f) = q r(tv) Q (T v f v ) v L (T ) where f v (u) = ma { f (u) } if u is a leaf of T and f v (u) = if u is a new leaf of T v. Remark.2 Clearl Q (T f) = 0 if L (T ) = and we note that this is never the case when T is a tree with the dela function associated to a Catalan state. Eample. In Figure.6 computations of Q (T (C)) are shown for T (C) associated to the Catalan state C in Figure.. 2 Q( ) qq( ) = + q Q( ) = qq( ) + q Q( ) + qq( ) + q 4 Q( ) = qq( ) + q 2 Q( ) + qq( ) + q Q( ) + q 4 Q( ) = qq( ) + q 2 Q( ) + qq( ) + q Q( ) + q 4 Q( ) = ( q+ + 2q 2 q 2 + q 4 )Q( ) q = + q + 2q + q 4 Figure.6 9

10 Our net result establishes a relationship between the coefficient C (A) (C Cat F (m n)) and the plucking polnomial Q (T (C)) of a rooted tree T (C) associated to C. Theorem.4 Let C be a Catalan state with no returns on the floor mindeg q Q (T (C)) be the minimum degree of q in Q (T (C)) and Q A (T (C)) its evaluation at q = A 4. Then the coefficient C (A) of C in L F (m n) is given b C (A) = A 2 b M mn 4 mindeg qq(t (C )) Q A (T (C)). In particular if C has returns onl on its ceiling or the left side then C (A) = A 2 b M mn Q A (T (C)). Proof. The above statement holds since in the definition of Q (T (C)) we just follow computations of C (A) outlined in Proposition 2. (using the row b row approach). In particular we observe that deleting b from the sequence b = (b b 2... b m ) b (C) (i.e. assigning markers according to b in the first row of L(m n)) results in removing a leaf v L (T (C)) which corresponds to the innermost upper cup e b of C. Coefficients of the Laurant polnomial C (A) can then be found b comparing it with Q (T (C)). That is we observe that if a P i move is applied on b the new sequence P i (b) adds a new monomial to C(A) obtained bultipling the monomial A 2 b mn b a factor of A 4. Since each sequence b b (C) ields a unique order of removing vertices from T (C) each b contributes a monomial q n(b) to Q (T (C)). Therefore a P i move on b results in adding a monomial q n(b) multiplied b q to Q (T (C)). We describe this in detail for a P move on b = (b b 2 b.. b m ). B definition the move changes b to (b 2 + b + b... b m ) hence the monomial A 2 b mn corresponding to b changes to A 4+2 b mn. Now we observe that a P move induces a change of order of leaves v L (T (C)) v 2 L (T (C) v ) which results in decreasing r(t (C) v ) + r(t (C) v v 2 ) b (i.e. the corresponding monomial is multiplied b q ). Since the graph G (C) is connected we conclude that C (A) = A u(c) Q A (T (C)) for some u (C) that depends onl on C. Now to find u (C) it suffices to compare the maimal power of C (A) with the minimal power of the variable q in Q (T (C)) ( i.e. mindeg q Q (T (C))). The formula given in the theorem follows and in particular if C has onl returns on its top or the left side then mindeg q Q (T (C)) = 0 so C (A) = A 2 b M mn Q A (T (C)). Corollar.5 Let C be a Catalan state with no returns on the floor and C (A) = A mindeg A C(A) N i=0 a i A 4i be its coefficient in L F (m n) where N = 4 (madeg AC(A) mindeg A C(A)). Then a 0 = a N = and a i > 0 for all i = 2... N. 4 Coefficients of Catalan States with Returns on One Side The recursion for Q (T (C)) given in Definition. can be used to prove man important results about C (A). In particular we can find a closed form formula for coefficients C (A) of Catalan states C with returns on the ceiling onl. Furthermore for such states the polnomial Q (T (C)) depends onl on the tree Q (T (C)) rather than its particular planar embedding (see Theorem 4.). To simplif our notations let T stand for the rooted tree (T v 0 ). We recall also the notation of the q-analogue of an integer m and q-analogue of a multinomial coefficient. The q-analogue of an integer m is defined b [m] q = + q + q q m and let [m] q! = [] q [2] q... [m] q. Therefore the q-analogue of the multinomial coefficient ( a +a 2+...a k a a 2...a k is given b )q ( ) a + a a k a a 2... a kq q = [a + a a k ] q! [a ] q! [a 2 ] q!... [a k ] q!. 0

11 In particular the binomial coefficient (Gauss polnomial) is defined b ( ) ( ) a + a 2 a + a 2 = a a a 2 q Basic properties of the plucking polnomial Q(T ) are summarized in the following theorem [ 4]. Theorem 4. (i) Let T = T T 2 be the wedge product of rooted trees T and T 2 having the common root v 0. Then ( ) E (T ) Q (T ) = Q (T ) Q (T 2 ). E (T ) E (T 2 ) q. q T T 2 v 0 Figure 4. (ii) (Wedge Product Formula) If T = k T i is the wedge product of rooted trees T i i = 2... k which have the common root v 0 then i= ( ) E (T ) Q (T ) = E (T ) E (T 2 )... E (T k ) k Q (T i ). q i= (iii) (Product Formula) Let v V (T ) and denote b T v the rooted subtree of T with the root v. Assume that T v k(v) = Ti v where T i v are rooted trees with the common root v and let i= be the weight of v. Then Q (T ) = ( E (T v ) ) W (v) = ) E E (T v) E (T 2 (T v)... k(v) v v V (T ) W (v). In particular Q (T ) does not depend on a plane embedding. Remark 4.2 Since C (A) = A 2 bm mn Q A (T (C)) the result of Theorem 4. (ii) gives us a closed form formula for computing C (A) i.e. C (A) = A 2 bm mn ( W A (v)) v V (T ) where W A (v) is the Laurant polnomial in A obtained from W (v) b substituting q = A 4. Here we list some further important from our perspective properties of Q (T ) and its coefficients (see [] [4] and [5] for proofs). Naturall these impl the corresponding properties of C (A). Corollar. Let C be a Catalan state with the associated rooted tree T (C) as in Figure 4.2 (Figure 4. shows an eample of such a C) then ( ) C (A) = A 2 b a + a M mn a k a a 2... a k q=a 4 q

12 a a 2 ak- a k m points a 0 + k a i i= = m a 0 n points Figure 4.2 Figure 4. Theorem 4.4 Let C Cat F (m n) be a Catalan state with returns on its top onl and T = T (C) be the corresponding rooted tree. If N Q (T ) = a i q i then (i) a 0 = a N = and a i > 0 i = N. (ii) a i = a N i i = N i.e. Q (T ) is a palindromic polnomial (also called smmetric polnomial). (iii) The sequence {a i } N i=0 is unimodal i.e. there is k such that a 0 a... a k a k a k+... a N. (iv) Q (T ) is a product of q-binomial coefficients (Gaussian polnomials). (v) Q (T ) is a product of cclotomic polnomials. (vi) The degree of the polnomial Q (T ) is given b deg (Q (T )) = i=0 v V (T ) deg (W (v)) where W (v) is defined as in Theorem 4. (iii) and deg (W (v)) = i<j k(v) E (T v i ) E ( T v j ). We can characterize polnomials which can be epressed as Q (T ) for some rooted tree T. Theorem 4.5 [ 5] Let P (q) be a polnomial. Then P (q) = Q (T ) for some rooted tree T if the following conditions are satisfied: (i) P (q) is a product of Gaussian polnomials and (ii) P (q) = [N] q! [b ] q [b 2] q... [b k ] q where 2 b b 2... b k < N. 5 Applications In this section we consider the case of a realizable Catalan state C Cat (m n) which admits a horizontal cross-section of size n or a vertical one of size m. Recall for a Catalan state C Cat (m n) we denote b C the Catalan state obtained from C b reflecting C about the -ais (see Figure 5.). Conditions for the strict unimodalit of coefficients are given in [4]. 2

13 2 n-2 n- n 2 n-2 n- n -2 C -2-2 C m-2 Catalan state C m- m Catalan state C 2 2 n-2 n- n 2 n-2 n- n Figure 5. Theorem 5. Let C Cat (m n) be a realizable Catalan state with a cross-section of size m that is the product 2 C v C 2 = C and the common boundar of C and C 2 is intersected b arcs of C eactl n times (see Figure 5.2 and an eample shown in Figure 5.). Then C and C 2 are in L F (m n) and C (A) = C (A) C 2 (A) 2 n-2 n- n k C n inrersection points... k 5 6 C l C C 2 4 k+... C2... k C C v C n-2 n- n Figure C Figure 5. Eample 5.2 Consider Catalan states C = C (m) (with no nesting) shown in Figure 5.4 we have { A (A 2 +A 2 ) m+ C (m) (A (A) = 2 +A 2 ) 2 if m is odd (A 2 + A 2 ) (A2 +A 2 ) m (A 2 +A 2 ) 2 if m is even. The formula follows b induction C (2k+) (A) = A( + (A 2 + A 2 )C (2k) ) and C (2k) = A (A 2 + A 2 )C (2k ). We observe that not all roots of C (m) (A) are roots of unit. smmetric but it is not a product of cclotomic polnomials. Furthermore the polnomial C (A) is Eample 5. Consider the Catalan state C as in Figure 5.5. B Theorem 5. C(A) = A( + A 4 + A 8 ) = A[] A 4. Thus for C k = C v C v... v C we have C k (A) = A k ([] A 4) k. Notice that for k > it does not satisf condition (ii) of Theorem 4.5. Corollar 5.4 For a realizable C Cat (m ) the coefficients C (A) (up to a power of A) is a product of [2] A 4 [] A 4 and (A2 +A 2 ) 2k (A 2 +A 2 ) 2. In particular C(A) is smmetric (palindromic) its highest and the lowest coefficients are equal to and there are no gaps. Furthermore if C has a horizontal line cutting it in eactl points then C (A) (up to a power of A) is a product of powers of [2] A 4 [] A 4. 2 The operation v corresponds to the product in Temper-Lieb algebra as described b Louis Kauffman [0].

14 m - odd m - even Figure 5.5 Figure 5.4 Proof. Our proof follows b a careful case analsis using Theorem 5.. We note that if n = 4 (that is C Cat (m 4)) then C (A) is not necessaril smmetric. For eample for the Catalan state in Figure 2.9 we have 6 Acknowledgments C (A) = + 2A 4 + A 8 + A 2. J. H. Prztcki was partiall supported b the Simons Collaboration Grant-6446 and CCAS Dean s Research Chair award. Authors would like to thank Ivan Dnnikov and Krzsztof Putra for man useful computations/discussions. References [] D. Bullock Rings of SL 2 (C) characters and the Kauffman bracket skein module Comment. Math. Helv. 72(4) 997 pp ; E-print: [2] D. Bullock J.H. Prztcki Multiplicative Structure of Kauffman Bracket Skein Module Quantizations Proceedings of the American Mathematical Societ28() 2000 pp. 92-9; E-print: [] Z. Cheng S. Mukherjee J.H. Prztcki X. Wang S.Y. Yang Realization of plucking polnomials Journal of Knot Theor and Its Ramifications 26(2) (9 pages). [4] Z. Cheng S. Mukherjee J.H. Prztcki X. Wang S.Y. Yang Strict unimodalit of q-polnomials of rooted trees Journal of Knot Theor and Its Ramifications to appear E-print: arxiv: [5] Z. Cheng S. Mukherjee J.H. Prztcki X. Wang S.Y. Yang Rooted Trees with the Same Plucking Polnomial; E-print: arxiv: [6] M.K. Dabkowski C. Li J.H. Prztcki Catalan states of lattice crossing Topolog and its Applications 82 (205) pp -5; E-print: arxiv: [math.gt]. 4

15 [7] C. Frohman R. Gelca Skein Modules and the Noncommutative Torus Transactions of the American Mathematical Societ52(0) 2000 pp [8] M. Hajij The colored Kauffman skein relation and the head and tail of the colored Jones polnomial Volume 26() (4 pages); E-print: arxiv: [9] J. Hoste J. H. Prztcki The skein module of genus Whitehead tpe manifolds Journal of Knot Theor and Its Ramifications 4() 995 pp [0] L.H. Kauffman On knots Annals of Math. Studies 5 Princeton Universit Press 987. [] J.H. Prztcki Skein modules of -manifolds Bull. Ac. Pol.: Math. 9(-2) 99 pp. 9-00; E-print: arxiv:math/06797 [math.gt]. [2] J.H. Prztcki Fundamentals of Kauffman bracket skein modules Kobe Math. J. 6() 999 pp ; E-print: arxiv:math/9809 [math.gt]. [] J.H. Prztcki Teoria wȩz lów i zwi azanch z nimi struktur dstrbutwnch (in Polish) (Knot Theor and distributive structures) Wdawnictwo Uniwerstetu Gdańskiego Second Edition Gdańsk 206. [4] J.H. Prztckiq-polnomial invariant of rooted trees; Arnold Mathematical Journal 2(4) ; E-print: arxiv: [math.co]. [5] J.H. Prztcki A.S. Sikora On skein algebras and Sl 2 (C)-character varieties Topolog9() 2000 pp. 5-48; E-print: [6] S. Yamada A topological invariant of spatial regular graphs Proc. Knots 90 De Gruter Berlin 992 pp Mieczslaw K. Dabkowski Jozef H. Prztcki Department of Mathematical Sciences Department of Mathematics Universit of Teas at Dallas The George Washington Universit Richardson TX Washington DC mdab@utdallas.edu prztck@gwu.edu and Universit of Gdańsk 5