Milnor’s -invariants for links in the 3-sphere vanish on any link concordant to a boundary link. In particular, they are are trivial for any classical knot. Here we define an analogue of Milnor’s concordance invariants for knots in thickened surfaces , where is closed and oriented. These invariants vanish on any knot concordant to a homologically trivial knot in . We use them to give new examples of non-slice virtual knots having trivial Rasmussen invariant, graded genus, affine index polynomial, and generalized Alexander polynomial. Moreover, we complete the slice status classification of the 2564 virtual knots having at most five classical crossings and reduce to four (of 90235) the number of virtual knots with six classical crossings having unknown slice status. Furthermore, we prove that in contrast to the classical knot concordance group, the virtual knot concordance group is not abelian. As part of the construction of the extended -invariants, we also obtain a generalization of the -invariants of classical links in to ribbon torus links in and welded links.
2 thoughts on “Extending Milnor’s concordance invariants to virtual knots and welded links”
Do you think there is a way to compute Milnor’s invariants in this welded setting via iterated intersections of certain surfaces (analogous to Cochran’s perspective for Milnor’s invariants in the classics setting)?
Good question! If each of the components of the welded link is almost classical (so that they bound Seifert surfaces), I suspect that the answer is “yes”. I don’t know what the analogue would be when some of the components are not almost classical. It would be nice if one could express the first non-vanishing Milnor invariants as certain linear combinations of linking numbers.
![\Sigma \times [0,1]](https://s0.wp.com/latex.php?latex=%5CSigma+%5Ctimes+%5B0%2C1%5D&bg=ffffff&fg=000000&s=0 "\Sigma \times [0,1]")
Do you think there is a way to compute Milnor’s invariants in this welded setting via iterated intersections of certain surfaces (analogous to Cochran’s perspective for Milnor’s invariants in the classics setting)?
Good question! If each of the components of the welded link is almost classical (so that they bound Seifert surfaces), I suspect that the answer is “yes”. I don’t know what the analogue would be when some of the components are not almost classical. It would be nice if one could express the first non-vanishing Milnor invariants as certain linear combinations of linking numbers.