Abstract
On behalf of the Department of Mathematical Sciences, you are cordially invited to attend the open PhD thesis defence, the details of which are as follows:
Student Name: Mr. Biswadeep Karmakar (PH20022)
Date: 09 July 2025 (Wednesday)
Time : 05:00 PMVenue: Room 5A, AB-II
PhD Thesis Title: Cohomological aspects of symmetric quandles.
Abstract: In classical knot theory, the existence of an orientation often becomes a crucial prerequisite, particularly in situations where one intends to distinguish knots using the quandle or rack-theoretic cocycle invariants. Kamada and Oshiro addressed the challenge of eliminating this need for orientation by introducing symmetric quandles and their (co)homology groups. They defined a quandle cocycle invariant for an unoriented link using a given symmetric quandle 2-cocycle, and showed that it aligns with the quandle cocycle invariant of an oriented link when an orientation is arbitrarily assigned. Similarly, an invariant for unoriented surface links was defined using symmetric quandle 3-cocycles, yielding comparable results. A notable outcome of this is that it allows us to estimate the minimal triple point numbers of non-orientable surface links. Moreover, symmetric quandles have also been employed to define invariants for spatial graphs and handlebody-links. These results, along with further refinements, underscore the significance of symmetric quandles in low-dimensional topology. The aim of this thesis is to further probe symmetric quandles from a categorical and (co)homological perspective.
Given a category C and an object X in C, a Beck module over X is an abelian group object in the slice category C|X. Beck modules provide a very general notion of a coefficient module for an associated (co)homology theory. Their strength is exemplified by the fact that they simultaneously generalize the concept of coefficient modules in group cohomology, Lie algebra cohomology, and Hochschild cohomology of associative algebras. Pursuing this point of view, we introduce the category of symmetric quandle modules and prove that these modules completely determine the Beck modules in this category. This establishes suitable coefficient objects for constructing appropriate (co)homology theories. We develop an extension theory of modules over symmetric quandles, and propose a generalized (co)homology theory for symmetric quandles with coefficients in a homogeneous Beck module, which also recovers the (co)homology theory developed by Kamada and Oshiro. Our constructions also apply to symmetric racks as well. The significance of this (co)homology theory lies in its potential to yield invariants (for example, for spatial graphs and low dimensional manifolds) that are more comprehensive and capable of capturing additional information compared to the existing (co)homology, which is a specific case when the action is trivial.
In order to better understand this generalized (co)homology, the next natural step is to perform some computations and find its relations to existing (co)homologies. In this direction, we establish an explicit isomorphism between the second cohomology of a symmetric rack and the first cohomology of its associated group. We also derive a four-term exact sequence that relates 1-cocycles, second cohomology, and a specific group of automorphisms associated with the extensions of symmetric quandles. This exact sequence shows that the obstruction to lifting and extending automorphisms is found in the second symmetric quandle cohomology. Additionally, we introduce dynamical cocycles and extensions of symmetric quandles in the spirit of Andruskiewitsch and Grana, and observe that group extensions naturally lead to their dynamical extensions. Again, these constructions apply to symmetric racks as well.
Tea will be served at 04:45 PM.
Best regards,
Mahender.