On chromatic symmetric homology and planarity of graphs
Azzurra Ciliberti, Luca Moci
备注：
11 pages. Some changes have been made in Section 3 in order to improve readability. arXiv admin note: substantial text overlap with arXiv:1911.13297 by other authors
Sazdanovic 和 Yip 定义了 Stanley 色函数的一种范畴化，称为色对称同调。本文证明：若图 G 非平面，则其在双次数 (1,0) 下的色对称同调包含 Z_2 挠率，这正如 Chandler、Sazdanovic、Stella 和 Yip 所猜想。证明采用基于 Kuratowski 定理的递归论证。
英文摘要：
Sazdanovic and Yip defined a categorification of Stanley's chromatic function called the chromatic symmetric homology. In this paper we prove that (as conjectured by Chandler, Sazdanovic, Stella and Yip), if a graph $G$ is non-planar, then its chromatic symmetric homology in bidegree (1,0) contains $\mathbb{Z}_2$-torsion. Our proof follows a recursive argument based on Kuratowsky's theorem.
On the first group of the chromatic cohomology of graphs
The algebra of truncated polynomials A_m=Z[x]/(x^m) plays an important role in the theory of Khovanov and Khovanov-Rozansky homology of links. We have demonstrated that Hochschild homology is closely related to Khovanov homology via comultiplication free graph cohomology. It is not difficult to compute Hochschild homology of A_m and the only torsion, equal to Z_m, appears in gradings (i,m(i+1)/2) for any positive odd i. We analyze here the grading of graph cohomology which is producing torsion for a polygon. We find completely the cohomology H^{1,v-1}_{A_2}(G) and H^{1,2v-3}_{A_3}(G). The grou…
