arXiv subjects

Madeti Prabhakar

Publications and source records attributed to Madeti Prabhakar.

At least 19 recordsLinked to original sources

arXiv2026

B-index polynomial for twisted knots

A twisted link is a generalization of a virtual link associated with link diagrams on closed surfaces, which may be non-orientable. In this paper, we generalize the notion of the index value for twisted knots. Based on this generalization, we introduce a polynomial invariant for twisted knots, called the $\mathcal{B}$-Index polynomial. Furthermore, we construct a family of twisted knots $\{D_n\}_{n\geq 1}$ with arc shift number $n$ and determine their $\mathcal{B}$-Index polynomials explicitly in terms of $n$. These results demonstrate the effectiveness and sensitivity of the $\mathcal{B}$-Index polynomial as an invariant of twisted knots. We also study the behavior of this polynomial under mirror images and orientation reversal. Furthermore, we conclude this paper by investigating the cosmetic crossing change conjecture and establishing a condition under which a crossing does not admit cosmetic behavior.

math.GT↗

arXiv2026

Matrix representations of the twisted virtual braid group and its extensions

This paper classifies complex local representations of the twisted virtual braid group, $TVB_2$, into $\mathrm{GL}_3(\mathbb{C})$. It shows that such representations fall into eight types, all of which are unfaithful and reducible to a two-dimensional representation. Further reducibility to a one-dimensional representation is analyzed for specific types. The paper also examines complex homogeneous local representations of $TVB_n$ into $\mathrm{GL}_{n+1}(\mathbb{C})$ for $n \geq 3$, identifying seven unfaithful types. Additionally, complex local representations of the singular twisted virtual braid group, $STVB_2$, into $\mathrm{M}_3(\mathbb{C})$ are classified into thirteen unfaithful types. Finally, the paper demonstrates that not all complex local extensions of $TVB_2$ representations to $STVB_2$ conform to a $Φ$-type extension.

math.RT↗

arXiv2026

Presentations and Representations of the Multi-Virtual Twin Group and Associated Subgroups

Motivated by the notion of the multi-virtual braid group introduced by L. Kauffman and by the study of extensions of the well-known twin group T_n, n >= 2, we introduce a new group called the multi-virtual twin group M_kVT_n, where k >= 1 and n >= 2, together with two associated subgroups: the multi-virtual pure twin group M_kVPT_n and the multi-virtual semi-pure twin group M_kVHT_n.We classify all homogeneous 2-local representations of M_kVT_n into GL_n(C) for all k >= 1 and n >= 3, and show that they fall into exactly eight distinct types. We also investigate their main properties, including faithfulness and irreducibility, proving that they are generally unfaithful and providing necessary and sufficient conditions for their irreducibility.Furthermore, for certain values of k and n, we construct non-local representations of M_kVPT_n induced from those of M_kVT_n, and we determine the conditions under which these induced representations are irreducible. Finally, we present several problems for future research in this area.

math.GT↗

arXiv2026

Multi-virtual braid groups

L. Kauffman (2024) introduced multi-virtual and symmetric multi-virtual braid groups, which are generalizations of the virtual braid group. We introduce multi-virtual pure and multi-virtual semi-pure braid groups, which are normal subgroups of index $n!$. We give a set of generators and defining relations for these groups, show that multi-virtual (symmetric multi-virtual) braid group is a semi-direct products of multi-virtual pure (symmetric multi-virtual pure) braid group and symmetric group. Also, we introduce multi-welded and multi-unrestricted braid groups and examines structure of three-strand 2-virtual braid group and some its subgroups and quotients. The paper concludes by outlining open problems and suggesting avenues for future research in this area.

math.GR↗

arXiv2025

On representations of the multi-virtual braid group $M_kVB_n$ and the multi-welded braid group $M_kWB_n$

This paper classifies complex homogeneous $2$-local representations of the multiple virtual braid group $M_kVB_n$ into $\mathrm{GL}_n(\mathbb{C})$ for $n\geq3$ and $k >1$, showing that such representations fall into exactly $2^{k+1}+1$ distinct types, out of which except three all are unfaithful. In addition, this paper investigates complex homogeneous $2$-local representations of the multiple welded braid group $M_kWB_n$ into $\mathrm{GL}_n(\mathbb{C})$ for $n\geq3$ and $k >1$, identifying $3 \cdot 2^{k-1} +1$ representations. Moreover, the article includes a construction of a non-local representation of $M_2WB_3$ that extends the known LKB representation of the braid group on $3$ strands, namely $B_3$, making a path towards constructing non-local representations of $M_kWB_n$ in general.

math.RT↗

arXiv2025

Classification of virtual links by arc shift move

In this paper, we establish that the arc shift operation on a $n$-component virtual link diagram acts as an unknotting operation when the virtual link is $n$-homogeneous proper, aiding in the classification of $ n $-component virtual links up to arc shift equivalence. We explore the connection between the arc shift number and the odd writhe of virtual links which are homogeneous proper. Additionally, we identify sequences of virtual link diagrams $ L_n $ for which the upper bound of the arc shift number is exactly $ n $.

math.GT↗

arXiv2024

Orbits by the up-down action of braid diagrams

The set of all virtual or classical braid diagrams forms a monoid and gives a natural monoid action on a direct product of ${\mathbb Z}$ called the up-down action. In this paper, we determine the orbit of every tuple of ${\mathbb Z}$ under the up-down action of virtual or classical braid diagrams. Moreover, we determine the orbit for irreducible braid diagrams. We also consider the isotropy submonoid and give a condition for a braid diagram to admit an up-down coloring to its closure.

math.GT↗

arXiv2024

Warping labeling for twisted knots and twisted virtual braids

In this paper, we introduce the concept of the warping degree for twisted knots, construct an invariant for them, and utilize it to establish a labeling scheme for these knots, known as ``warping labeling". We have identified that a warping labeling can be extended to twisted virtual braids, enabling the creation of a function that remains invariant under all R-moves except the R2 move. By limiting the labeling set to $\mathbb{Z}_2$, we can develop invariants for twisted virtual braids.

math.GT↗

arXiv2024

Singular twisted links and singular twisted virtual braids

The concepts of twisted knot theory and singular knot theory inspire the introduction of singular twisted knot theory. This study showcases similar findings for singular twisted links, including the Alexander theorem and the Markov theorem derived from knot theory. Moreover, in this paper we define singular twisted virtual braids and their monoid structure. Additionally, we provide both a monoid and a reduced monoid presentation for singular twisted virtual braids.

math.GT↗

arXiv2023

Twisted Virtual Braid Group

In this paper we study some subgroups and their decompositions in semi-direct product of the twisted virtual braid group $TVB_n$. In particular, the twisted virtual pure braid group $TVP_n$ is the kernel of an epimorphism of $TVB_n$ onto the symmetric group $S_n$. We find the set of generators and defining relations for $TVP_n$ and show that $TVB_n = TVP_n \rtimes S_n$. Further we prove that $TVP_n$ is a semi-direct product of some subgroup and abelian group $\mathbb{Z}_2^n$. As corollary we get that the virtual pure braid group $VP_n$ is a subgroup of $TVP_n$. Also, we construct some other epimorphism of $TVB_n$ onto $S_n$. Its kernel, $TVH_n$ is an analogous of $TVP_n$. We find its set of generators and defining relations and construct its decomposition in a semi-direct product.

math.GR↗

arXiv2023

Twisted virtual braids and twisted links

Twisted knot theory introduced by M. Bourgoin is a generalization of knot theory. It leads us to the notion of twisted virtual braids. In this paper we show theorems for twisted links corresponding to the Alexander theorem and the Markov theorem in knot theory. We also provide a group presentation and a reduced group presentation of the twisted virtual braid group.

math.GT↗

arXiv2022

Determinants of twisted generalized hybrid weaving knots

This article presents a formula for the determinant of the twisted generalized hybrid weaving knot $\hat{Q}_3(m_1,m_2,n,l)$, which is a closed 3-braid. As a corollary, we prove Conjecture 2 of Singh & Chbili [Nuclear Physics B, 980:115800 (2022)].

math.GT↗

arXiv2022

Some evaluations of Jones polynomials for certain families of weaving knots

In this paper, we derive formulae for the determinant of weaving knots $W(3,n)$ and $W(p,2)$. We calculate the dimension of the first homology group with coefficients in $\mathbb{Z}_3$ of the double cyclic cover of the $3$-sphere $S^3$ branched over $W(3,n)$ and $W(p,2)$ respectively. As a consequence, we obtain a lower bound of the unknotting number of $W(3,n)$ for certain values of $n$.

math.GT↗

arXiv2021

Recurrent Generalization of F-Polynomials for Virtual Knots and Links

F-polynomials for virtual knots were defined by Kaur, Prabhakar and Vesnin in 2018 using flat virtual knot invariants. These polynomials naturally generalize Kauffman's affine index polynomial and use smoothing in classical crossing of a virtual knot diagram. In this paper we introduce weight functions for ordered orientable virtual and flat virtual link. A flat virtual link is an equivalence class of virtual links in respect to a local symmetry changing type of classical crossing in a diagram. By considering three types of smoothings in classical crossings of a virtual link diagram and suitable weight functions, we provide a recurrent construction for new invariants. We demonstrate by providing explicit examples, that newly defined polynomial invariants are stronger than F-polynomials.

math.GT↗

arXiv2019

Gordian complexes of knots and virtual knots given by region crossing changes and arc shift moves

Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in $\mathbb{S}^3$. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using $v$-move and forbidden moves. In this paper we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high dimensional simplex in both the Gordian complexes, i.e., by region crossing change and by the arc shift move. For any given knot (respectively, virtual knot) diagram we construct an infinite family of knots (respectively, virtual knots) such that any two distinct members of the family have distance one by region crossing change (respectively, arc shift move). We show that that the constructed virtual knots have the same affine index polynomial.

math.GT↗

arXiv2018

An Unknotting Index for Virtual Links

Given a virtual link diagram $D$, we define its unknotting index $U(D)$ to be minimum among $(m, n)$ tuples, where $m$ stands for the number of crossings virtualized and $n$ stands for the number of classical crossing changes, to obtain a trivial link diagram. By using span of a diagram and linking number of a diagram we provide a lower bound for unknotting index of a virtual link. Then using warping degree of a diagram, we obtain an upper bound. Both these bounds are applied to find unknotting index for virtual links obtained from pretzel links by virtualizing some crossings

math.GT↗

arXiv2018

Two-variable polynomial invariants of virtual knots arising from flat virtual knot invariants

We introduce two sequences of two-variable polynomials $\{ L^n_K (t, \ell)\}_{n=1}^{\infty}$ and $\{ F^n_K (t, \ell)\}_{n=1}^{\infty}$, expressed in terms of index value of a crossing and $n$-dwrithe value of a virtual knot $K$, where $t$ and $\ell$ are variables. Basing on the fact that $n$-dwrithe is a flat virtual knot invariant we prove that $L^n_K$ and $F^n_K$ are virtual knot invariants containing Kauffman affine index polynomial as a particular case. Using $L^n_K$ we give sufficient conditions when virtual knot does not admit cosmetic crossing change.

math.GT↗

arXiv2014

An Unknotting Sequence for Torus Knots

In this paper, the authors give an unknotting sequence for torus knots and also provide unknotting numbers of $_n14_{17191}, \ _n14_{14274}, \ _n14_{18351}, \ _n14_{24498}$ and some other knots from the knot table of Hoste-Thistlethwite.

math.GT↗