Journal of Knot Theory and Its Ramifications

In the present paper, we develop a picture formalism which gives rise to an invariant that dominates several known invariants of classical and virtual knots: the Jones polynomial ${𝒳}\left(\cdot \right)$, the Kuperberg bracket ${\left|\cdot \right.〉}_{A_2}$, and the normalized arrow polynomial ${𝒲}\left[\cdot \right]$.

In this paper, we prove an algorithmical solvability of exponential-Diophantine equations in rings represented by matrices over fields of positive characteristic. Consider the system of exponential-Diophantine equations

We provide a physical definition of new homological invariants ${{\mathscr{H}}}_a(M_3)$ of 3-manifolds (possibly, with knots) labeled by abelian flat connections. The physical system in question involves a 6d fivebrane theory on $M_3$ times a 2-disk, $D^2$, whose Hilbert space of BPS states plays the role of a basic building block in categorification of various partition functions of 3d ${𝒩}=2$ theory $T[M_3]$: $D^2\times S^1$ half-index, $S^2\times S^1$ superconformal index, and $S^2\times S^1$ topologically twisted index. The first partition function is labeled by a choice of boundary condition and provides a refinement of Chern–Simons (WRT) invariant. A linear combination of them in the unrefined limit gives the analytically continued WRT invariant of $M_3$. The last two can be factorized into the product of half-indices. We show how this works explicitly for many examples, including Lens spaces, circle fibrations over Riemann surfaces, and plumbed 3-manifolds.

In V. O. Manturov, On free knots, preprint (2009), arXiv:math.GT/0901.2214], the second named author constructed the bracket invariant $[\cdot ]$ of virtual knots valued in pictures (linear combinations of virtual knot diagrams with some crossing information omitted), such that for many diagrams $K$, the following formula holds: $[K]=\tilde{K}$, where $\tilde{K}$ is the underlying graph of the diagram, i.e. the value of the invariant on a diagram equals the diagram itself with some crossing information omitted. This phenomenon allows one to reduce many questions about virtual knots to questions about their diagrams. In [S. Nelson, M. E. Orrison and V. Rivera, Quantum enhancements and biquandle brackets, preprint (2015), arXiv:math.GT/1508.06573], the authors discovered the following phenomenon: having a biquandle coloring of a certain knot, one can enhance various state-sum invariants (say, Kauffman bracket) by using various coefficients depending on colors. Taking into account that the parity can be treated in terms of biquandles, we bring together the two ideas from these papers and construct the picture-valued parity-biquandle bracket for classical and virtual knots. This is an invariant of virtual knots valued in pictures. Both the parity bracket and Nelson–Orrison–Rivera invariants are partial cases of this invariant, hence this invariant enjoys many properties of various kinds. Recently, the authors together with E. Horvat and S. Kim have found that the picture-valued phenomenon works in the classical case.

We prove that the equation $tan(x)-x=a$ is unsolvable in elementary functions.

In this paper, Rubik’s cube group in $n$-dimensional case is described. In that case, some interesting effects arise, which is related with angular rotations. For $n=4$, the group ${\mathbb{Z}}_3$ arises, which can be realized as a factor group of $A_4$ by commutation $[A_4,A_4]=V_4$. For $n\ge 5$ there are no such effects, because corresponding factor is trivial. Rubik’s cube is an important model to study effects of groups related with, in particular, holonomy groups from differential geometry.

I point out a conceptually simple way of establishing one of the main lemmas in the paper “Solvability of equations in elementary functions” by A. Kanel-Belov, A. Malistov and R. Zaytsev.

We investigate the relationship between the quandle and biquandle coloring invariant and obtain an enhancement of the quandle and biquandle coloring invariants using biquandle structures.

We also continue the study of biquandle homomorphisms into a medial biquandle begun in [Hom quandles, J. Knot Theory Ramifications 23(2) (2014)], finding biquandle analogs of results therein. We describe the biquandle structure of the Hom-biquandle, and consider the relationship between the Hom-quandle and Hom-biquandle.