arXiv research

Mutant knots, in the sense of Conway, are known to share the same Homfly polynomial. Their 2-string satellites also share the same Homfly polynomial, but in general their m-string satellites can have different Homfly polynomials for m>2. We show that, under conditions of extra symmetry on the constituent 2-tangles, the…

Study of $G_2$ compactifications in M- / string theory.

We study the boundary conditions in the topologically twisted Chern-Simons matter theories with the Lie 3-algebraic structure. We find that the supersymmetric boundary conditions and the gauge invariant boundary conditions can be unified as the complexified gauge invariant boundary conditions which lead to the supergro…

Generalising previous results on classical braid groups by Artin and Lin, we determine the values of m, n $\in$ N for which there exists a surjection between the n-and m-string braid groups of an orientable surface without boundary. This result is essentially based on specific properties of their lower central series, …

Study on coloring virtual tangles with integer and modular arithmetic.

The $C_k$-equivalence is an equivalence relation generated by $C_k$-moves defined by Habiro. Habiro showed that the set of $C_k$-equivalence classes of the knots forms an abelian group under the connected sum and it can be classified by the additive Vassiliev invariant of order $\leq k-1$. We see that the set of $C_k$-…

Let M be a compact, connected non-orientable surface without boundary and of genus g greater than or equal to 3. We investigate the pure braid groups P_n(M) of M, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence 1 --> P_m(M {x_1,...,x_n}) --> P_{n+m}(M) --> P_n(M) --> 1, where m,n ar…

We examine the action of the fundamental group $Γ$ of a Riemann surface with $m$ punctures on the middle dimensional homology of a regular fiber in a Lefschetz fibration, and describe to what extent this action can be recovered from the intersection numbers of vanishing cycles. Basis changes for the vanishing cycles re…

We study the pure braid groups $P_n(RP^2)$ of the real projective plane $RP^2$, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence $1 \to P_m(RP^2 {x_1,...,x_n} \to P_{n+m}(RP^2) \stackrel{p_{\ast}}{\to} P_n(RP^2) \to 1$, where $n\geq 2$ and $m\geq 1$, and $p_{\ast}$ is the homomorphis…

Abstract M5 branes on ADE singularities yields BPS spectrum and partition functions.

Our aim is to determine the lower central series (LCS) and derived series (DS) for the braid groups of the sphere and of the finitely-punctured sphere. We show that for all n (resp. all n\geq 5), the LCS (resp. DS) of the n-string braid group B\_n(S^2) is constant from the commutator subgroup onwards, and that Γ\_2(B\_…