The article introduces a new non‑archimedean metric, the cophenetic metric, defined on persistent homology classes of all degrees. It demonstrates that zeroth persistent homology combined with this metric and various hierarchical clustering algorithms yields statistically verifiable, commensurate topological information on multiple datasets. The resulting clusters, evaluated by silhouette score and Rand index, perform well, and the metric enables visualization of inter‑relations among persistent homology classes across all degrees via rooted trees.
By \.Ismail G\"uzel, Atabey Kaygun
arXiv:2609. 17899v1 Announce Type: new Abstract: We extend the discrete complex complement quotient (DCCQ) framework from binary Bernoulli counts to multinomial count compositions.
By Y. Kenan Y{\i}lmaz
arXiv:2607. 05835v1 Announce Type: cross Abstract: For every loopless matroid $M$ and every Feichtner--Yuzvinsky building set $\mathcal{G}$ containing the top flat, we construct an integral tangent class $T_{M,\mathcal{G}}^{\mathbb{Z}}\in K_{\mathbb{Z}}(M,\mathcal{G})$; in the realizable case it specializes to the class of the tangent bundle of the corresponding wonderful compactification, it recovers the Hilbert series of the Chow ring through Hirzebruch--Riemann--Roch, and it satisfies the expected Chern-alpha lower bounds.
By Ronnie Cheng, Shurui Liu, Guoxiong Gao
arXiv:2607. 23732v1 Announce Type: cross Abstract: In Question~3.
By Xinan Dai, Wenhao Deng, Yingdong Shi, Tailin Wu, Yuchen Yang
arXiv:2512.23348v3 Announce Type: replace-cross
Abstract: We introduce a data-analysis framework based on filtrations of finite topological spaces. Starting from a finite metric data set, we construc...
By Sel\c{c}uk Kayacan
arXiv:2608. 18570v1 Announce Type: cross Abstract: Machine learning and data analysis techniques have recently emerged as powerful tools for identifying patterns and formulating conjectures in mathematical research, most notably in the field of low-dimensional topology.
By Brandon Robinson, Shimal Harichurn, Fabian Ruehle, Sergei Gukov, Rak-Kyeong Seong, Miranda C. N. Cheng