Milnor’s
–invariants
of links in the
–sphere
vanish
on any link concordant to a boundary link. In particular, they are trivial on any knot in
. Here we consider knots
in thickened surfaces
,
where
is closed
and oriented. We construct new concordance invariants by adapting the Chen–Milnor theory
of links in
to an extension of the group of a virtual knot. A key ingredient is the Bar-Natan
map,
which allows for a geometric interpretation of the group extension. The group
extension itself was originally defined by Silver and Williams. Our extended
–invariants
obstruct concordance to homologically trivial knots in thickened surfaces.
We use them to give new examples of nonslice virtual knots having trivial
Rasmussen invariant, graded genus, affine index (or writhe) polynomial, and
generalized Alexander polynomial. Furthermore, we complete the slice status
classification of all virtual knots up to five classical crossings and reduce to
(out
of
)
the number of virtual knots up to six classical crossings having unknown slice
status.
Our main application is to Turaev’s concordance group
of
long knots on surfaces. Boden and Nagel proved that the concordance group
of classical knots
in
embeds into
the center of
.
In contrast to the classical knot concordance group, we show
is not
abelian, answering a question posed by Turaev.
