In this paper we introduce the persistent magnitude, a new numerical invariant of (sufficiently nice) graded persistence modules. It is weighted and signed count bars module, in which bar form [a,b) degree d counted with weight (e−a−e−b) sign (−1)d. Persistent magnitude has good formal properties, such as additivity respect to exact sequences compatibility tensor products, interpretations terms...

1.1. Khovanov’s Homology and Conjecture. In his 1999 paper [K1] in Duke Mathematical Journal, Mikhail Khovanov showed that the Jones link polynomial is the graded Euler characteristic of a bigraded homology module H (D) over Z[c], associated to a diagram D of the link. H (D) is the homology of a bigraded chain complex C(D) with a (1, 0)-bigraded differential. He also explained how each link cob...

A new class of coefficients for the Hopf-cyclic homology of module algebras and coalgebras is introduced. These coefficients, termed stable anti-Yetter-Drinfeld contramodules, are both modules and contramodules of a Hopf algebra that satisfy certain compatibility conditions. 1. Introduction. It has been demonstrated in [8], [9] that the Hopf-cyclic homology developed by Connes and Moscovici [5]...

We extend Lawrence's representations of the braid groups to relative homology modules, and we show that they are free modules over a Laurent polynomials ring. define homological operators actually provide representation for an integral version $U_q \mathfrak{sl}(2)$. suggest isomorphism between given basis standard tensor products Verma it preserve ring coefficients, action \mathfrak{sl}(2)$, g...

We explain how Queffelec–Sartori’s construction of the HOMFLY-PT link polynomial can be interpreted in terms parabolic Verma modules for \({\mathfrak {gl}_{2n}}\). Lifting to world categorification, we use 2-Verma give a higher representation theory Khovanov–Rozansky’s triply graded homology.