Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFundamental Heaps for Surface Ribbons and Cocycle Invariants
175
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We introduce the notion of fundamental heap for compact orientable surfaces with boundary embedded in $3$-space, which is an isotopy invariant of the embedding. It is a group, endowed with a ternary heap operation, defined using diagrams of surfaces in a form of thickened trivalent graphs called surface ribbons. We prove that the fundamental heap has a free part whose rank is given by the number of connected components of the surface. We study the behavior of the invariant under boundary connected sum, as well as addition/deletion of twisted bands, and provide formulas relating the number of generators of the fundamental heap to the Euler characteristics. We describe in detail the effect of stabilization on the fundamental heap, and determine that for each given finitely presented group there exists a surface ribbon whose fundamental heap is isomorphic to it, up to extra free factors. A relation between the fundamental heap and the Wirtinger presentation is also described. Moreover, we introduce cocycle invariants for surface ribbons using the notion of mutually distributive cohomology and heap colorings. Explicit computations of fundamental heap and cocycle invariants are presented.
rate research
Read More
A heap is a set with a certain ternary operation that is self-distributive (TSD) and exemplified by a group with the operation $(x,y,z)mapsto xy^{-1}z$. We introduce and investigate framed link invariants using heaps. In analogy with the knot group, we define the fundamental heap of framed links using group presentations. The fundamental heap is determined for some classes of links such as certain families of torus and pretzel links. We show that for these families of links there exist epimorphisms from fundamental heaps to Vinberg and Coxeter groups, implying that corresponding groups are infinite. A relation to the Wirtinger presentation is also described. The cocycle invariant is defined using ternary self-distributive (TSD) cohomology, by means of a state sum that uses ternary heap $2$-cocycles as weights. It is shown that the cohomology splits into two types, called degenerate and nondegenerate, and that the degenerate part is one dimensional. Subcomplexes are constructed based on group cosets, that allow computations of the nondegenerate part. We apply colorings inferred from fundamental heaps to compute cocycle invariants, and prove that the invariant values can be used to derive algebraic properties of the cohomology.
We extend the quandle cocycle invariant to oriented singular knots and links using algebraic structures called emph{oriented singquandles} and assigning weight functions at both regular and singular crossings. This invariant coincides with the classical cocycle invariant for classical knots but provides extra information about singular knots and links. The new invariant distinguishes the singular granny knot from the singular square knot.
We bring cocycle enhancement theory to the case of psyquandles. Analogously to our previous work on virtual biquandle cocycle enhancements, we define enhancements of the psyquandle counting invariant via pairs of a biquandle 2-cocycle and a new function satisfying some conditions. As an application we define new single-variable and two-variable polynomial invariants of oriented pseudoknots and singular knots and links. We provide examples to show that the new invariants are proper enhancements of the counting invariant are are not determined by the Jablan polynomial.
Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Gra~na. We specialize that theory to the case when there is a group action on the coefficients. First, quandle modules are used to generalize Burau representations and Alexander modules for classical knots. Second, 2-cocycles valued in non-abelian groups are used in a way similar to Hopf algebra invariants of classical knots. These invariants are shown to be of quantum type. Third, cocycles with group actions on coefficient groups are used to define quandle cocycle invariants for both classical knots and knotted surfaces. Concrete computational methods are provided and used to prove non-invertibility for a large family of knotted surfaces. In the classical case, the invariant can detect the chirality of 3-colorable knots in a number of cases.
This paper is a brief overview of some of our recent results in collaboration with other authors. The cocycle invariants of classical knots and knotted surfaces are summarized, and some applications are presented.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
