Invariants of handlebody-knots via Yokota's invariants (Q2872789)

The paper under review constructs quantum \(\mathcal{U}_q(\mathfrak{sl}_2)\) type invariants for handlebody knots via Yokota's spatial graph invariants. Handlebody knots differ from spatial graphs in that they admit one extra Reidemeister move known as the IH move [\textit{A. Ishii}, Algebr. Geom. Topol. 8, No. 3, 1403--1418 (2008; Zbl 1151.57007)]. Certain linear sums of Yokota's spatial graph invariants are invariant under the IH move, and are therefore invariants of handlebody knots. These are the first quantum type invariants for handlebody knots to appear in print.NEWLINENEWLINESection 4 gives a table of computations of these invariants for handlebody knots of up to six crossings, showing in particular that they are non-trivial. Section 5 identifies these invariants with certain Witten--Reshetikhin--Turaev invariants of an associated \(3\)--manifold.