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(4). Full twists and stability of knots and quivers
We relate the stability of knot invariants under twisting a pair of strands to the stability of symmetric quivers under unlinking (or linking) operation. Starting from the HOMFLY-PT skein relations, we confirm the stable growth of $Sym^r$-coloured HOMFLY-PT polynomials under the addition of a full twist to the knot. On the other hand, we show that symmetric quivers exhibit analogous stable growth under unlinking or linking of the quiver augmented with the extra node; in some cases this augmented quiver captures the spectrum of motivic Donaldson-Thomas invariants of all quivers in the sequence. Combining these two versions of the stable growth, we conjecture that performing a full twist on any knot corresponds to appropriate unlinking or linking of the corresponding augmented quiver -- this statement is an important step towards a direct definition of the knot-quiver correspondence based on the knot diagram. We confirm the conjecture for all twist knots, $(2,2p+1)$ torus knots, and all pretzel knots up to 15 crossings with an odd number of twists in each twist region.
With Piotr Kucharski, Dmitry Noshchenko, Ramadevi Pichai, Vivek Kumar Singh, Marko Stošić
(3). Gukov-Pei-Putrov-Vafa conjecture for SU(N)/Z_m
In this work, we studied this conjecture for non-simply connected gauge group SU(N)/Z_m, where m is some divisor of N. This conjecture relates the Witten-Reshetikhin-Turaev invariant to q-series invariant of three-manifold (widely referred to as Z-hat invariant). We found that the Z-hat invariant does not depend on the quotienting of SU(N) by Z_m.
With Pichai Ramadevi
(2). Knot-quiver correspondence for double twist knots
In this project, we studied the knot-quiver correspondence for a class of double-twist knots. We used the reverse engineering of Melvin-Morton-Rojansky formalism to deduce the quiver matrix for K(p,-2) and K(p,-3) knot families with p>-1 and p>-2, respectively.
With Pichai Ramadevi, Vivek Kumar Singh, A. Dwivedi, S. Dwivedi, B.P. Mandal
(1). Z-hat invariants for SO(3) and OSp(1|2) Groups
Three-manifold invariants "Z-hat", also known as homological blocks, are q-series with integer coefficients. Explicit q-series form for Z-hat is known for SU(2) group, supergroup SU(2|1) and ortho-symplectic supergroup OSp(2|2). We focus on Z-hat for SO(3) group and orthosymplectic supergroup OSp(1|2) in this paper. Particularly, the change of variable relating SU(2) link invariants to the SO(3) and OSp(1|2) link invariants plays a crucial role in explicitly writing the q-series.
With Pichai Ramadevi
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