Virtual Legendrian knots and possible relations to causality in Borde–Sorkin spacetimes

We review our results with Stefan Nemirovski on the relation of Legendrian linking and causal structure in globally hyperbolic spacetimes. After that we discuss virtual Legendrian knots defined as Legendrian knots in $ST^*M$ studied up to isotopy and modification of $ST^*M$ induced by surgery on $M$ that were introduced by Patricia Cahn and Asa Levi. We formulate a few conjectures about them and conjecturally relate them to causality in Borde–Sorkin spacetimes that are manifolds with a Lorentz metric that is degenerate at critical points of a Morse "timelike" function and that consist of globally hyperbolic pieces between critical levels. We prove these conjectures for $2+1$ dimensional Borde–Sorkin spacetimes.
The talk is based on the joint work with Rustam Sadykov, Kansas State University.

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