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On the Homology Theory for the Chromatic Polynomials

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On the homology theory for the chromatic polynomials Zipei Zhuang July 9, 2021 Abstract In [9], Rong and Helme-Guizon defined a categorification for the chro- ∗ matic polynomial PG(x) of graphs G, i.e. a homology theory H (G) whose Euler characteristic equals PG(x). In this paper, we showed that the ratio- ∗ nal homoology H (G; Q) is supported in two lines, and develop an analogy of Lee's theory for Khovanov homology. In particular, we develop a new homology theory HLee(G), and showed that there is a spectral sequence ∗ whose E2 -term is isomorphic to H (G) converges to HLee(G). 1 Introduction There has been a long history for the interplay between knot theory and graph theory. For example, the Tait graph of a knot diagram relates the Jones polynomial to the Tutte polynomial in graph theory, see [14] [3]. In particular, it gives a spanning tree model for the Jones polynomial, which gives new proofs on some old conjectures in knot theory(see [12]). In[10], he defined a homology theory for knots whose Euler characteristic is the Jones polynomial. This generated interest on categorifying some polynomial invariants in graph theory, which have similar "state sum" expansions as the Jones polynomial. In particular, a homology theory for graphs whose Euler characteristic is theh chromatic polynomial was introduced in [9]. In [11], Lee replaced the differential d in the Khovanov chain complex (C∗(D); d) (with rational coefficients) by a new differential Φ+d. The homology ∗ arXiv:2107.03671v1 [math.GT] 8 Jul 2021 HLee(D) of (C (D); Φ + d) can be calculated explictly. Furthermore, he showed that there is a spectral sequence converging to HLee(D), whose E2 term is isomorphic to the Khovanov homology H(D). Using this, he showed that the Khovanov homology of an alternating knot of degree difference (1,4) are paired except in the 0-th cohomology group. Together with the fact that the homology of an alternating knot is supported in two lines he showed HLee(D; Q) that is determined by its Jones polynomial and signature. In [13] the author defined an invariant using Lee's theory. It provides a lower bound for the smooth slice genus of K, and leads to a purely combinatorial proof of the Milnor conjecture. 1 The idea to use a spanning tree model for calculating khovanov homology was considered in [16] [4]. It leads to a new proof on Lee's theorem on the Khovanov homology for alternating knots. This paper was an attempt to develop an analogus Lee's theory for Helme- Guizon and Rong's categorification H∗(G; Q) for the chromatic polynomial. In Section 2, we review the categorification for the chromatic polynomial defined in [9]; In Section 3, we construced a spanning tree model for H∗(G; Q). As a corollary, we showed that it is supported in 2 lines; In Section 4, we develop a \Lee theory" for chromatic homology. We showed ∗ that our Lee homology HLee(G) has a basis in 1-1 correspndence to the colorings of the vertices using the colors f1; xg, and there is a spectral sequence converging ∗ ∗ to HLee(G) whose E2-term is isomorphic to H (G) as a Q-module. Invariants from the filtration on the homology group are also discussed. 2 Review of the chromatic homology Let G be a graph with vertex set V = V (G) and edge set E = E(G). Denote by k(G) the number of components of G. 2.1 The chromatic polynomial For any subset s ⊂ E, we denote by hsi the spanning subgraph of G generated by s. The chromatic polynomial PG(x) of G is characterized by the properties: ( 0 if G has a loop · PG(x) = (1) PG−e(x) − PG=e(x) for any e 2 E(G) not a loop k · PNk (x) = x where Nk is the graph with k vertices and no edges (2) For each positive integer n, PG(n) euqals the number of cloring the vertices of G so that adjacent vertices have different colors. There is a state-sum expansion for the chromatic polynomial: X jsj khsi X i X khsi PG(x) = (−1) x = (−1) x (3) s⊂E i≥0 jsj=i here jsj denotes the number of edges in s. An enhanced state of a graph G is a pair S = (s; c), where s is a state of G, and c is an assignment of colors in the set f1; xg to each component of the spanning subgraph hsi. Define i(S)= the number of edges in s, j(S)= the number of x in c. The chromatic polynomial can be written as: X jsj khsi X i(S) j(S) PG(1 + q) = (−1) (1 + q) = (−1) q (4) s⊂E S enhanced state We call it the enhanced state expansion of the chromatic polynomial. 2 2.2 The categorification for the chromatic polynomial Definition 1. Let M = ⊕jMj be a graded Q-module. The graded dimension of M is X j qdimM = dimQ(Mj ⊗ Q) · q (5) j Define Mflgj = Mj−l so that qdimMflg = ql · qdimM (6) The graded dimension has the property qdim(M ⊕ N) = qdim(M) + qdim(N) (7) qdim(M ⊗ N ) = qdimM · qdimN Define M to be a graded free Q-module with basis 1; x whose degree are 0 and 1 respectively, i.e. M = Q ⊕ Qx , qdimM = 1 + q. Let n = jE(G)j. Order the elements of E(G), so that a state corresponds n to a sequence (1; 2; ··· ; n) 2 f0; 1g (i = 1 if and only if the i-th edge is contained in the state). Denote by ks the number of connected components of the suubgraph hsi. Define ⊗ks Ms(G) = M (8) where each M corresonds to a component of hsi. Define the chromatic chain complex i M C (G) = Ms(G) (9) jsj=i Let s; s0 be two states, such that s0 is obtained from s by adding an edge. Then there is a pre-edge map : + ds : Ms(G) −! Ms0 (G) (10) defined as follows: · If adding the edge e doesn't change the number of components, then the pre-edge map is the identity; + · If e connects two components of hsi, then ds is defined using m : M ⊗ M −! M (11) m(1 ⊗ 1) = 1; m(1 ⊗ x) = m(x ⊗ 1) = x; m(x ⊗ x) = 0 (12) Finally the differential di : Ci(G) −! Ci+1(G) is defined by i X jsj + d = (−1) ds (13) jsj=i 3 (C∗∗(G); d) is a chain complex: X n(e) X X n(e) d ◦ d(S) = d( (−1) S ) = (−1) (S ) 0 e e e (14) e2E(G)−S e0 2E(G−s−e) e2E(G)−s Note that the summand corresponding to (Se)e0 and (Se0 )e cancel, the above summation equals to 0. Definition 2. [9] We call (C∗(G); d) defined above the chromatic complex of G. There is an equivalent definition using enhanced states. Define an operation ∗ on the two-variable set f1; xg 1 ∗ 1 = 1; 1 ∗ x = x ∗ 1 = x; x ∗ x = 0 (15) Let G = (V; E) be a graph with an order on E. The bigraded chain complex is defined as follows: i;j · C (G) = the free abelian group generated by all enhanced states S with i(S) = i and j(S) = j. i;j i+1;j · The differential d : C (G) −! C (G) is X n(e) S = (s; c) −! (−1) Se (16) e2E(G)−S where n(e) is the number of edges in s whose order are smaller than e, and Se = (s [ feg; ce). Here ce is given as follows: · If e does not connect two components in s, then the components of s are in 1-1 correspondence with the components of S [ e, and c induces the coloring ce of s [ feg; ·If e connects two components, say E1;E2 of s, then ce(E1 [ E2 [ feg) = c(E1) ∗ c(E2) (17) ce(Ei) = c(Ei) for i= 3, 4, ··· (If c(E1) = c(E2) = x, then Se is defined to be 0). In [9], it is proved that the above two chain complex are isomorphic and are independent of the choice of the ordering on edges, hence is an invariant for the graph. In particular, its homology and Euler characteristic are graph invariants. P i Indeed, the Euler characteristic i(−1) #fenhanced states S with i(S) = ig equals to the chromatic polynomial by Eq.(4) 3 The spanning tree model Let G be a graph with an order on its edge set: E(G) = fe1; e2; :::eng. Let F be a spanning forest of G. For an edge e2 = E(F ), define the cycle cyc(e; F ) to be unique cycle of F + e; For e 2 E(F ), define the cut cut(e; F ) = fe0 2 E(G) F − e + e0 is a spanning forestg. 4 Call an edge e 2 E(F ) internally active if it is the largest edge in cut(e; F ), i.e. ei 2 E(F ) is internally active if i > j whenever ej 2 cut(ei;F ). Similarly, call e 2 E(G) − E(F ) externally active if it is the largest edge in cyc(e; F ). There is a spanning tree expansion for the chromatic polynomial: r(G) k(G) X i(T ) PG(x) = (−1) x (1 − x) (18) T spanning trees here i(T ) = the number of internally active edges of T .
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