arXiv subjects

Deepanshi Saraf

Publications and source records attributed to Deepanshi Saraf.

6 recordsLinked to original sources

arXiv2026

Dehn quandles of surfaces and their bounded cohomology

We introduce new families of quandles that serve as invariants for classifying closed orientable surfaces. These families generalize the classical Dehn quandle and are defined, respectively, on isotopy classes of unoriented closed curves and on integral weighted multicurves. We establish their fundamental algebraic properties and construct a natural quandle covering that relates them. We then analyze their metric properties, showing that these quandles are unbounded with respect to the quandle metric. Next, we compute their second bounded quandle cohomology, proving it to be infinite-dimensional. We also establish a version of the Gromov Mapping Theorem, showing that the natural map from an abelian quandle extension onto the original quandle induces an injection on bounded quandle cohomology in every dimension. Finally, inspired by recent developments in quandle rings, we analyze idempotents in the integral quandle rings arising from the classical Dehn quandle of a surface.

math.GT↗

arXiv2025

Second bounded cohomology of knot quandles

In this paper, we explore the bounded cohomology of quandles and its applications to knot theory. We establish two key results that provide sufficient conditions for the infinite dimensionality of the second bounded cohomology of quandles. The first condition involves a subspace of homogeneous group quasimorphisms on the inner automorphism group of the quandle, whereas the second condition concerns the vanishing of the stable commutator length on a subgroup of this inner automorphism group. As topological applications, we show that the second bounded cohomology of the quandle of any non-split link whose link group is non-solvable as well as the quandle of any split link, is infinite dimensional. From these results, we conclude that the second bounded cohomology of the knot quandle detects the unknot. On the algebraic side, we prove that the second bounded cohomology of a free product of quandles is infinite dimensional if the inner automorphism group of at least one of the free factors is amenable. This leads to the result that the second bounded cohomology of free quandles of rank greater than one, as well as their canonical quotients, is infinite dimensional.

math.GT↗

arXiv2024

Automorphisms, cohomology and extensions of symmetric quandles

It is well-known that the cohomology of symmetric quandles generates robust cocycle invariants for unoriented classical and surface links. Expanding on the recently introduced module-theoretic generalized cohomology for symmetric quandles, we 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, some general aspects of dynamical cocycles and extensions are discussed.

math.QA↗

arXiv2024

Fundamental $n$-quandles of links are residually finite

In this paper, we investigate the residual finiteness and subquandle separability of quandles, properties that respectively imply the solvability of the word problem and the generalized word problem for quandles. From Winker's work, we know that fundamental $n$-quandles of oriented links, which are canonical quotients of their fundamental quandles, are closely associated with $n$-fold cyclic branched covers of the 3-sphere branched over these links. We prove that the fundamental $n$-quandle of any oriented link in the 3-sphere is residually finite for each $n\ge 2$. This supplements the recent result by Bardakov, Singh and the third author on residual finiteness of fundamental quandles of oriented links, and the classification by Hoste and Shanahan of links whose fundamental $n$-quandles are finite for some $n$. We also establish several general results on these finiteness properties and identify many families of quandles admitting them.

math.GT↗

arXiv2024

Generalized (co)homology of symmetric quandles over homogeneous Beck modules

A quandle equipped with a good involution is referred to as symmetric. It is known that the cohomology of symmetric quandles gives rise to strong cocycle invariants for classical and surface links, even when they are not necessarily oriented. In this paper, we introduce the category of symmetric quandle modules and prove that these modules completely determine the Beck modules in the category of symmetric quandles. Consequently, 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 symmetric quandle (co)homology developed by Kamada and Oshiro [Trans. Amer. Math. Soc. (2010)]. Our constructions also apply to symmetric racks. We conclude by establishing an explicit isomorphism between the second cohomology of a symmetric quandle and the first cohomology of its associated group.

math.QA↗

arXiv2023

Generalised Legendrian racks of Legendrian links

A generalised Legendrian rack is a rack equipped with a Legendrian structure, which is a pair of maps encoding the information of Legendrian Reidemeister moves together with up and down cusps in the front diagram of an oriented Legendrian link. Employing a purely rack theoretic approach, we associate a generalised Legendrian rack (or a GL-rack) to an oriented Legendrian link, and prove that it is an invariant under Legendrian isotopy. As immediate applications, we prove that this invariant distinguishes infinitely many oriented Legendrian unknots and oriented Legendrian trefoils. To comprehend their algebraic structure, we prove that every GL-rack admits a homogeneous representation. Further, using the idea of trunks, we define modules over GL-racks, and prove the equivalence of the category of GL-rack modules and the category of Beck modules over a fixed GL-rack.

math.GT↗