Papers › Characterization and Further Applications of the Bar-Natan Zh-Construction
Characterization and Further Applications of the Bar-Natan Zh-Construction
Micah Chrisman, Robert G. Todd
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Bar-Natan's Zh-construction associates to each n component virtual link diagram L an (n+1) component virtual link diagram Zh(L). If L₀,L₁ are equivalent virtual link diagrams, then Zh(L₀),Zh(L₁) 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.
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