arXiv subjects

Hitesh Raundal

Publications and source records attributed to Hitesh Raundal.

8 recordsLinked to original sources

arXiv2025

Biorderability of knot quandles of knots up to eight crossings

The paper investigates biorderability of knot quandles of prime knots up to eight crossings. We prove that knot quandles of knots $6_3$, $8_7$, $8_8$, $8_{10}$ and $8_{16}$ can not be biorderable. However, we see that knot quandles of knots $4_1$, $6_1$, $6_2$, $7_6$, $7_7$, $8_1$, $8_2$, $8_3$, $8_4$, $8_5$, $8_6$, $8_9$, $8_{11}$, $8_{12}$, $8_{13}$, $8_{14}$, $8_{17}$, $8_{18}$, $8_{20}$ and $8_{21}$ could be biorderable. We also give linear orders on the generating set of the knot quandle of a knot (among these knots) that could be extendable to biorders on the quandle.

math.GT↗

arXiv2022

Dehn quandles of groups and orientable surfaces

Unifying various constructions of quandles including Coxeter quandles, free quandles, knot quandles of prime knots and Dehn quandles of orientable surfaces, we introduce Dehn quandles of groups with respect to their subsets. It turns out that Dehn quandles are precisely the ones that embed naturally into their enveloping groups. We prove that the enveloping group of the Dehn quandle of a given group with respect to its generating set is a central extension of that group, and that enveloping groups of Dehn quandles of Artin groups and link groups with respect to their standard generating sets are the groups themselves. We discuss orderability of Dehn quandles and prove that free involutory quandles are left orderable, whereas certain generalised Alexander quandles are bi-orderable. Specialising to surfaces, we give generating sets for Dehn quandles of orientable surfaces with punctures and compute their automorphism groups. As applications, we recover a result of Niebrzydowski and Przytycki proving that the knot quandle of the trefoil knot is isomorphic to the Dehn quandle of the torus and also extend a result of Yetter on epimorphisms of Dehn quandles of orientable surfaces onto certain involutory homological quandles.

math.GT↗

arXiv2022

Presentations of Dehn quandles

The paper gives two approaches to write explicit presentations for the class of Dehn quandles using presentations of their underlying groups. The first approach gives finite presentations for Dehn quandles of a class of Garside groups and Gaussian groups. The second approach is for general Dehn quandles when the centralisers of generators of their underlying groups are known. Several examples including Dehn quandles of spherical Artin groups, surface groups and mapping class groups of orientable surfaces are given to illustrate the results.

math.GR↗

arXiv2021

Orderability of link quandles

The paper develops a general theory of orderability of quandles with a focus on link quandles of tame links and gives some general constructions of orderable quandles. We prove that knot quandles of many fibered prime knots are right-orderable, whereas link quandles of most non-trivial torus links are not right-orderable. As a consequence, we deduce that the knot quandle of the trefoil is neither left nor right orderable. Further, it is proved that link quandles of certain non-trivial positive (or negative) links are not bi-orderable, which includes some alternating knots of prime determinant and alternating Montesinos links. The paper also explores interconnections between orderability of quandles and that of their enveloping groups. The results establish that orderability of link quandles behave quite differently than that of corresponding link groups.

math.GT↗

arXiv2021

Spaces of polynomial knots in low degree

We show that all knots up to $6$ crossings can be represented by polynomial knots of degree at most $7$, among which except for $5_2, 5_2^*, 6_1, 6_1^*, 6_2, 6_2^*$ and $6_3$ all are in their minimal degree representation. We provide concrete polynomial representation of all these knots. Durfee and O'Shea had asked a question: Is there any $5$ crossing knot in degree $6$? In this paper we try to partially answer this question. For an integer $d\geq2$, we define a set $\mathcal{\tilde{P}}_d$ to be the set of all polynomial knots given by $t\mapsto\big(f(t),g(t),h(t)\big)$ such that $\text{deg}(f)=d-2$, $\text{deg}(g)=d-1$ and $\text{deg}(h)=d$. This set can be identified with a subset of $\mathbb{R}^{3d}$ and thus it is equipped with the natural topology which comes from the usual topology $\mathbb{R}^{3d}$. In this paper we determine a lower bound on the number of path components of $\mathcal{\tilde{P}}_d$ for $d\leq 7$. We define a path equivalence for polynomial knots in the space $\mathcal{\tilde{P}}_d$ and show that it is stronger than the topological equivalence.

math.GT↗

arXiv2021

Some spaces of polynomial knots

In this paper we study the topology of three different kinds of spaces associated to polynomial knots of degree at most $d$, for $d\geq2$. We denote these spaces by $\mathcal{O}_d$, $\mathcal{P}_d$ and $\mathcal{Q}_d$. For $d\geq3$, we show that the spaces $\mathcal{O}_d$ and $\mathcal{P}_d$ are path connected and the space $\mathcal{O}_d$ has the same homotopy type as $S^2$. Considering the space $\mathcal{P}=\bigcup_{d\geq2}\mathcal{O}_d$ of all polynomial knots with the inductive limit topology, we prove that it too has the same homotopy type as $S^2$. We also show that if two polynomial knots are path equivalent in $\mathcal{Q}_d$, then they are topologically equivalent. Furthermore, the number of path components in $\mathcal{Q}_d$ are in multiples of eight.

math.GT↗

arXiv2021

Topologies on sets of polynomial knots and the homotopy types of the respective spaces

A polynomial knot in $\mathbb{R}^n$ is a smooth embedding of $\mathbb{R}$ in $\mathbb{R}^n$ such that the component functions are real polynomials. In the earlier paper with Mishra, we have studied the space $\mathcal{P}$ of polynomial knots in $\mathbb{R}^3$ with the inductive limit topology coming from the spaces $\mathcal{O}_d$ for $d\geq3$, where $\mathcal{O}_d$ is the space of polynomial knots in $\mathbb{R}^3$ with degree $d$ and having some conditions on the degrees of the component polynomials. In the same paper, we have proved that the space of polynomial knots in $\mathbb{R}^3$ has the same homotopy type as $S^2$. The homotopy type of the space is the mere consequence of the topology chosen. If we have another topology on $\mathcal{P}$, the homotopy type may change. With this in mind, we consider in general the set $\mathcal{L}^n$ of polynomial knots in $\mathbb{R}^n$ with various topologies on it and study the homotopy type of the respective spaces. Let $\mathcal{L}$ be the union of the sets $\mathcal{L}^n$ for $n\geq1$. We also explore the homotopy type of the space $\mathcal{L}$ with some natural topologies on it.

math.GN↗

arXiv2021

Hecke algebra trace algorithm and some conjectures on weaving knots

Computing polynomial invariants for knots and links using braid representations relies heavily on finding the trace of Hecke algebra elements. There is no easy method known for computing the trace and hence it becomes difficult to compute the known polynomial invariants of knots using their braid representations. In this paper, we provide an algorithm to compute the trace of the Hecke algebra representation of any braid. We simplify this algorithm and write a Mathematica program to compute the invariants such as Alexander polynomial, Jones polynomial, HOMFLY-PT polynomial and Khovanov homology of a very special family of knots and links $W(n,m)$ known as weaving knots by expressing them as closure of weaving braids. We also explore on the relationship between the topological and geometric invariants of this family of alternating and hyperbolic knots (links) by generating data for the subfamilies $W(3,m)$, $W(4,m)$, $W(5,m)$ and $W(6,m)$ of weaving knots.

math.GT↗