Recently, Ozsváth and Szabó introduced some algebraic constructions computing knot Floer homology in the spirit of bordered homology, including a family algebras $\mathcal B (n)$ and, for generator braid group on $n$ strands, certain t

This paper focuses on developing an efficient algorithm for analyzing a directed network (graph) from topological viewpoint. A prevalent technique such analysis involves computation of homology groups and their persistence. These concepts are well suited spaces that not directed. As result, one needs concept accommodates orientations in input space. Path-homology developed graphs by Grigoryan e...

We construct a stable homotopy refinement of quantum annular homology, link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq 2$ we associate to an $L$ naive $\mathbb{Z}/r\mathbb{Z}$-equivariant spectrum whose cohomology is isomorphic the as modules over $\mathbb{Z}[\mathbb{Z}/r\mathbb{Z}]$. The construction relies on equivariant version Burnside category approach Laws...

We give a new proof of the non-triviality wheel graph homology classes using higher operations on Lie and derived version Koszul duality for modular operads.

We compute the first two symplectic quadratic K-theory groups of integers, or equivalently, stable homology group integral matrices preserving standard refinement. The main novelty in our calculation lies its method, which is based on high-dimensional manifold theory.

Transversality phenomena are studied for homology manifolds. For homology manifolds X, Y and Z, with Z embedded in Y with a neighborhood ν(Z) which has a given bundle structure, we define a map f : X → Y to be transverse to Z, if f−1(Z) = Z′ is a homology manifold, the neighborhood f−1(ν(Z)) has a bundle structure given by f∗ν(Z) and f induces the bundle map. In the case where the range is a ma...

Inspired by work of Lipshitz-Sarkar, we show that the module structure on link Floer homology detects split links. Using results Ni, Alishahi-Lipshitz, and establish an analogous detection result for sutured homology.