Characterization and Further Applications of the Bar-Natan Zh-Construction

TL;DR

This paper explores the properties and applications of the Bar-Natan Zh-construction, linking virtual and classical knot invariants, and providing new insights into knot concordance and invariants extension.

Contribution

It offers a characterization of the Zh-construction for almost classical links and demonstrates its use in deriving invariants and understanding knot properties.

Findings

Zh-construction unifies classical and virtual knot invariants

Generalized Alexander polynomial acts as a slice obstruction

Extension of quandle invariants via Zh-construction

Abstract

Bar-Natan's Zh-construction associates to each $n$ component virtual link diagram $L$ an $(n+1)$ component virtual link diagram $Zh(L)$. If $L_0,L_1$ are equivalent virtual link diagrams, then $Zh(L_0),Zh(L_1)$ are equivalent as semi-welded links. The importance of the $Zh$-construction is that it unifies several classical knot invariants with virtual knot invariants. For example, the generalized Alexander polynomial of a virtual link diagram $L$ is identical to the usual multi-variable Alexander polynomial of $Zh(L)$. From this it follows that the generalized Alexander polynomial is a slice obstruction: it vanishes on any knot concordant to an almost classical knot. Our main result is a characterization theorem for the $Zh$-construction in terms of almost classical links. Several consequences of this characterization are explored. First, we give a purely geometric description of the $Zh$-construction. Secondly, the $Zh$-construction is used to obtain a simple derivation of the Dye-Kauffman-Miyazawa polynomial. Lastly, we show that every quandle coloring invariant and quandle 2-cocycle coloring invariant can be extended to a new invariant using the $Zh$-construction.