arXiv subjects

Vaibhav Keshari

Publications and source records attributed to Vaibhav Keshari.

4 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↗

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↗