紀要論文

紀要論文

text

Osaka Journal of Mathematics. 2025, 62(3), p. 351-379

CHARACTERIZATION AND FURTHER APPLICATIONS OF THE BAR-NATAN Ж-CONSTRUCTION

Departments of Mathematics, The University of Osaka and Osaka Metropolitan University

英語

text

2025-07

Bar-Natan’s Ж-construction associates to each n component virtual link diagram L an (n + 1) component virtual link diagram Ж(L). If L₀, L₁ are equivalent virtual link diagrams, then Ж(L₀),Ж(L₁) are equivalent as semi-welded links. The importance of the Ж-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 multivariable Alexander polynomial of Ж(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 Ж-construction in terms of almost classical links. Several consequences of this characterization are explored. First, we give a purely geometric description of the Ж-construction. Secondly, the Ж-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 Ж-construction.

10.18910/102429

102429

CHARACTERIZATION AND FURTHER APPLICATIONS OF THE BAR-NATAN Ж-CONSTRUCTION

Departments of Mathematics, The University of Osaka and Osaka Metropolitan University

英語

2025-07

Osaka Journal of Mathematics

62

3

351

379

https://hdl.handle.net/11094/102429

10.18910/102429

https://doi.org/10.18910/102429