arXiv subjects

Komal Negi

Publications and source records attributed to Komal Negi.

10 recordsLinked to original sources

arXiv2025

Alexander-Markov correspondence for doodles on closed surfaces

In this paper, we introduce twisted virtual doodles, defined as stable equivalence classes of immersed circles on closed surfaces that may be non-orientable. These objects admit planar representative diagrams, considered up to a suitable set of Reidemeister-type moves. To develop the associated braid-theoretic framework, we define twisted virtual twin groups as natural extensions of virtual twin groups, and establish Alexander- and Markov-type theorems in this set-up. This shows that twisted virtual doodles unify and extend both classical and virtual doodle theories. We further investigate the structure of the pure twisted virtual twin group, providing a presentation and deriving several structural and combinatorial properties. In particular, we obtain two interesting decompositions of the twisted virtual twin group and its pure subgroup, from which it follows that both groups have trivial center and are residually finite as well as Hopfian.

math.GT↗

arXiv2025

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

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↗

arXiv2025

The monoid structure of singular twisted virtual braids

In this paper, we examine specific submonoids within the singular twisted virtual braid monoid $STVB_n$. Notably, we establish that the singular twisted virtual pure braid monoid $STVP_n$ serves as the kernel of an epimorphism from $STVB_n$ onto the symmetric group $S_n$. We identify the generators and defining relations for $STVP_n$. Additionally, we construct other epimorphisms from $STVB_n$ onto $S_n$, whose kernels are analogous to $STVP_n$, and determine their respective generators and defining relations. Furthermore, we demonstrate the embedding of the monoid $STVB_n$ into a group. Also, we provide the extension of the representation of the twisted virtual braid group to the representation of the singular twisted virtual braid monoid.

math.GR↗

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

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↗