In this article, we introduce a fixed parameter tractable algorithm for computing the Turaev-Viro invariants TV4,q , using the dimension of the first homology group of the manifold as parameter. This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parameter. The computation of TV4,q is known to be #P-hard in general; using a topol...

We define the contact homology for Legendrian submanifolds in standard contact (2n + 1)-space using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex n-space. This homology provides new invariants of Legendrian isotopy which indicate that the theory of Legendrian isotopy is very rich. Indeed, in [4] the homology is used to detect infinite families of pairwise non...

We develop a curved Koszul duality theory for algebras presented by quadratic-linear-constant relations over unital versions of binary quadratic operads. As an application, we study Poisson $n$-algebras given polynomial functions on standard shifted symplectic space. compute explicit resolutions these using duality. use to derived enveloping and factorization homology parallelized simply connec...

Three-taxon statement analysis (3TA) and standard cladistic analysis (SCA) were evaluated relative to propositions of taxic homology. There are definite distinctions between complement relation homologs and paired homologs. The complement relation is discussed, relative to rooting, parsimony, and taxic propositions of homology. The complement relation, as implemented in SCA, makes sense only be...

In this article, we show that magnitude homology and path are closely related, give some applications. We define differentials MH k ℓ ( G ) ⟶ − 1 $\operatorname{MH}^{\ell }_k(G) \longrightarrow \operatorname{MH}^{\ell -1}_{k-1}(G)$ between homologies of a digraph $G$ , which make them chain complexes. Then its $\mathcal {MH}^{\ell }_k(G)$ is non-trivial homotopy invariant in the context ‘homoto...

The taxonomic relationship between some hypha-forming Cryptococcus neoformans strains and those that morphologically fit the standard description of this species was determined on the basis of DNA base ratios and DNA-DNA duplex formation. The average guanosine plus cytosine content, as estimated from melting temperature profiles, varied between 43.0 and 45.9% for all C. neoformans strains exami...

Contact homology for Legendrian submanifolds in standard contact (2n + 1)space is rigorously defined using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex n-space. The homology provides new invariants of Legendrian isotopy. These invariants show that the theory of Legendrian isotopy is very rich. For example, they detect infinite families of pairwise non-isotop...

While standard persistent homology has been successful in extracting information from metric datasets, its applicability to more general data, e.g. directed networks, is hindered by its natural insensitivity to asymmetry. We study a construction of homology of digraphs due to Grigor’yan, Lin, Muranov and Yau, and extend this construction to the persistent framework. The result, which we call pe...