Journal of Knot Theory and Its Ramifications

In this short note, we prove that the sphere and 4-tube relations “almost” imply the torus relation in Bar-Natan’s geometric formalism of Khovanov homology theory. As an application, we point out that the invariance and the functoriality can be proved without using the torus relation.

In this paper, we construct a categorification of the two-variable Dye–Kauffman–Miyazawa polynomial by utilizing Bar-Natan’s construction of the Khovanov homology and homotopy quantum field theories (HQFTs) given by Turaev. In particular, for any stable equivalence class, we construct a $Z\oplus Z\oplus Z$-graded link homology over $F_2$ whose graded Euler characteristic is the two-variable Dye–Kauffman–Miyazawa polynomial. Moreover, we show that it is isomorphic to a special case of Dye–Kauffman–Manturov’s categorification. In this sense, we explain the special case of Dye–Kauffman–Manturov’s homology in terms of Bar-Natan’s construction.

In 2013, Cheng and Gao introduced the writhe polynomial of virtual knots and Kauffman introduced the affine index polynomial of virtual knots. We introduce a zero polynomial of virtual knots of a similar type by considering weights of a suitable collection of crossings of a virtual knot diagram. We show that the zero polynomial gives a Vassiliev invariant of degree 1. It distinguishes a pair of virtual knots that cannot be distinguished by the affine index polynomial and the writhe polynomial.

Let $M$ be a closed, connected and oriented surface of genus $g\ge 1$. In this work, we prove that the homotopy string links over $M$ , namely, ${\widehat{PB}}_n(M)$, are bi-orderable, i.e. there is an (explicit) left and right order for such group.

The elliptic associator of Enriquez can be used to define an invariant of tangles embedded in the thickened torus, which extends the Kontsevich integral. This construction by Humbert uses the formulation of categories with elliptic structures. In this work we show that an extension of the LMO functor also leads to an elliptic structure on the category of Jacobi diagrams which is used by the Kontsevich integral, and find the relation between the two structures. We use this relation to give an alternative proof for the properties of the elliptic associator of Enriquez. Those results can lead the way to finding associators for higher genera.

We show that the knot lattice homology of a knot in an $L$-space is chain homotopy equivalent to the knot Floer homology of the same knot (viewed these invariants as filtered chain complexes over the polynomial ring $\mathbb{Z}/2\mathbb{Z}[U]$). Suppose that $G$ is a negative definite plumbing tree which contains a vertex $w$ such that $G-w$ is a union of rational graphs. Using the identification of knot homologies we show that for such graphs the lattice homology ${ℍ𝔽}^{-}(G)$ is isomorphic to the Heegaard Floer homology ${\mathrm{\text{HF}}}^{-}(Y_G)$ of the corresponding rational homology sphere $Y_G$.

Simple crystallizations are edge-colored graphs representing piecewise linear (PL) 4-manifolds with the property that the 1-skeleton of the associated triangulation equals the 1-skeleton of a 4-simplex. In this paper, we prove that any (simply-connected) PL 4-manifold $M$ admitting a simple crystallization admits a special handlebody decomposition, too; equivalently, $M$ may be represented by a framed link yielding ${𝕊}^3$, with exactly ${\beta }_2(M)$ components (${\beta }_2(M)$ being the second Betti number of $M$). As a consequence, the regular genus of $M$ is proved to be the double of ${\beta }_2(M)$. Moreover, the characterization of any such PL 4-manifold by $k(M)=3{\beta }_2(M)$, where $k(M)$ is the gem-complexity of $M$ (i.e. the non-negative number $p-1$, $2p$ being the minimum order of a crystallization of $M$), implies that both PL invariants gem-complexity and regular genus turn out to be additive within the class of all PL 4-manifolds admitting simple crystallizations (in particular, within the class of all “standard” simply-connected PL 4-manifolds).