On the Robertello invariants of proper links (Q795376)

The author generalizes the Robertello invariant of a knot to two mutually related invariants \(\delta\) (L) and \(\delta_ 0(L)\) of a proper oriented link L in a \({\mathbb{Z}}/2\)-homology 3-sphere S. (A link \(L=l_ 1\cup...\cup l_ n\) is called proper if for each \(i=1,...,n\) the sum of rational linking numbers \(Link(l_ i,l_ j)\) with \(j\neq i\) has an even numerator.) Both \(\delta\) and \(\delta_ 0\) take values in \({\mathbb{Q}}/{\mathbb{Z}}\); the difference \(\delta(L)-\delta_ 0(L)\) equals \((1/8)\sum_{i>j}Link(l_ i,l_ j) (mod {\mathbb{Z}}).\) It is shown that \(\delta_ 0\) is invariant under cobordism and fusion and that \(\delta\) is invariant under arbitrary reorientation of the link. Formulas are given which relate \(\delta\) (L) and \(\delta_ 0(L)\) with the corresponding invariants of the link \(\tilde L\subset \tilde S\) obtained from \(L\subset S\) by taking a p-fold cyclic branched covering of S, branched along L, with prime \(p\geq 2\).