Two homologies for handlebody-knots (Q6190606)

A handlebody-link is a disjoint union of handlebodies embedded in \(\mathbb{R}^3\). An \(S^1\)-orientation of a handlebody-link is an orientation on all genus one components of the handlebody-link, where an orientation of a genus one component \(H\) is an orientation of the core of \(H\). A handlebody-knot is a handlebody-link with one component. \newline The main results are as follow. The author gives differential graded algebras \((C^-(D), \partial)\) and \((C^+(D), \partial)\) associated with a diagram \(D\) of a handlebody-link \(H\). Together with the result due to \textit{A. Ishii} [Int. J. Math. 26, No. 14, Article ID 1550116, 23 p. (2015; Zbl 1337.57004)], the homologies \(H^-_*(H)\) of \((C^-(D), \partial)\) and \(H^+_*(H)\) of \((C^+(D), \partial)\) are invariants of an \(S^1\)-oriented handlebody-link. Further, the author calculates the 0-dimensional homologies for handlebody-knots up to 6 crossings. The construction of the differential graded algebras is given by modifying those for knots and other knotted objects due to Ng.