New invariants for virtual knots via spanning surfaces (Q6569886)

A knot \( K \subset \Sigma \times [0,1] \), for \( \Sigma \) a closed orientable surface, does not generically possess a Seifert surface, so that there is no immediate extension of Alexander invariants to this setting. There are a number of ways of overcoming this (see, for example, [\textit{H. U. Boden} et al., J. Knot Theory Ramifications 24, No. 3, Article ID 1550009, 62 p. (2015; Zbl 1364.57005)] and references therein), to which the paper under review adds various new constructions.\N\NLet \( - K \), \( K^{\ast} \) denote the orientation-reverse and mirror image of \(K\), respectively. The main results of the paper concern Alexander invariants defined using the fact that \( K \sqcup - K \) and \( K \sqcup -K^{\ast} \) are necessarily null-homologous in \( \Sigma \times [0,1] \). Using this the authors introduce a class of Seifert surfaces known as \textit{twin Seifert surfaces}, and define new polynomial and signature invariants.\N\NSome of these invariants descend to invariants of virtual knots, and are strong enough to obstruct classicality and distinguish virtual knots from their mirror image.\N\NThe paper also contains several constructions of Floer homologies for knots in thickened surfaces and virtual knots, and considers their properties and applications.