Research

My research revolves primarily around low dimensional topology and algebraic structures related to and motivated by knot theory. Additionally, I am highly interested in the interactions between knot theory and graph theory. The title of my dissertation is ‘On Skein Modules and Homology Theories Related to Knot Theory’ and it is available here. Given below is a list of my projects.

Algebraic Structures Motivated by Knot Theory

Post-modern algebra deals primarily with the study of non-associative algebraic structures. Quandles are examples of such algebraic structures. The three axioms of a quandle correspond to the three Reidemeister moves in knot theory. Racks, spindles, and shelves are obtained by dropping some of the axioms of a quandle. The primary objective of this project is to study the multi-term distributive homology of these algebraic structures.

  • R. P. Bakshi, D. Ibarra, S. Mukherjee, T. Nosaka, and J. H. Przytycki, Schur multipliers and second quandle homology. ArXiV: https://arxiv.org/abs/1812.04704. (submitted)
  • S. Mukherjee and J. H. Przytycki, On the rack homology of graphic quandles. Nonassociative mathematics and its applications, 183-197, Contemp. Math., 721, 2019.

Homology of Quasibands

Quasibands are semigroups satisfying a weaker form of idempotence. Bands and associative shelves are some of the examples of quasibands. The well-known homology theories (rack, quandle, and one-term) for shelves are not good tools for studying associative shelves. The homology of quasibands was developed to address this issue. The main goal in this project is to perform a comparative study of this homology theory alongside group homology, Hochschild homology, and multi-term distributive homology for associative shelves.

  • S. Mukherjee, A homology theory for a special family of semi-groups. J. Knot theory Ramifications, 27 (2018), no. 3, 1840005, 27 pp.
  • A. S. Crans, S. Mukherjee, and J. H. Przytycki, On homology of associative shelves. J. Homotopy Relat. Struct. 12 (2017), no. 3, 741-763.

Khovanov homology

Khovanov homology is a categorification of the Jones polynomial. A bigraded chain complex is associated to a link whose homology is an invariant of the link itself. Additionally, the Euler characteristic of this chain complex, when interpreted appropriately, is the Jones polynomial. A wealth of information, in the form of torsion subgroups, is obtained from the Khovanov homology of a link which has no contribution towards the Euler characteristic of the chain complex and hence cannot be obtained from just the Jones polynomial in general. It is this arena which witnesses most of my research activities related to the topic.

  • S. Mukherjee, On odd torsion in even Khovanov homology. (in preparation)
  • S. Mukherjee, J. H. Przytycki, M. Silvero, X. Wang, and S. Y. Yang, Search for torsion in Khovanov homology. Experimental Mathematics, (27) 2018, no. 4, 488-497.

Skein Modules

The purpose of the theory of skein modules is to study oriented 3-manifolds based on the properties of an associated module (called the skein module). The module is generated by the collection of links along with the empty link. It is then divided by the submodule generated by the skein relations often corresponding to some knot polynomial. For instance, one may define the Kauffman bracket skein module for the Kauffman bracket polynomial or the HOMFLYPT skein module for the HOMFLYPT polynomial. In this project, the main focus is the study of Kauffman bracket skein modules of thickened surfaces.

  • R. P. Bakshi, S. Mukherjee, J. H. Przytycki, M. Silvero, and X. Wang, On multiplying curves in the Kauffman bracket skein algebra of the thickened four-holed sphere. ArXiV: https://arxiv.org/abs/1805.06062. (submitted)

The Przytycki Invariant

The Przytycki invariant, also known as the plucking polynomial, is an invariant of plane rooted trees. The initial motivation behind this theory is the Kauffman bracket skein module of lattice crossings. The diagrams of lattice crossings are resolved using the Kauffman bracket skein relation to obtain Kauffman states. The coefficients of some of these states can be realized as plucking polynomials of plane rooted trees. The most important objective of this project is the categorification of the Przytycki invariant.

  • Z. Cheng, S. Mukherjee, J. H. Przytycki, X. Wang, and S. Y. Yang, Rooted trees with the same plucking polynomial. Osaka J. Math, 56 (2019), no. 3, 14 pp.
  • Z. Cheng, S. Mukherjee, J. H. Przytycki, X. Wang, and S. Y. Yang, Strict unimodality of plucking polynomials of rooted trees. J. Knot theory Ramifications, 27 (2018), no. 7, 1841009, 19 pp.
  • Z. Cheng, S. Mukherjee, J. H. Przytycki, X.Wang, and S. Y. Yang, Realization of plucking polynomials. J. Knot theory Ramifications, 26 (2017), no. 2, 1740016, 9 pp.

Gram Determinants in Knot Theory

Modern work on Gram determinants in knot theory began with the construction of the Witten-Reshetikhin-Turaev invariants of 3-manifolds by Lickorish. The Gram determinant of type A is obtained from a matrix formed by evaluating, in the sphere, a bilinear form of non-intersecting connections in the disc. The present goal of this project is the study of the Gram determinant of type Mb (Mobius band), evaluated in the Klein bottle.

  • R. P. Bakshi, D. Ibarra, S. Mukherjee, and J. H. Przytycki, A generalization of the Gram determinant of type A. ArXiV: https://arxiv.org/abs/1905.07834. (submitted)