We present evidence supporting the conjecture that, in the topological category, the slice genus of
a satellite knot
is bounded above by the sum of the slice genera of
and
. Our
main result establishes this conjecture for a variant of the topological slice genus, the
–slice genus.
Notably, the conjectured upper bound does not involve the algebraic winding number of the pattern
. This stands in
stark contrast with the smooth category, where, for example, there are many genus 1 knots whose
–cables have arbitrarily large
smooth
–genera. As an application,
we show that the
–cable
of any knot of
–genus
1 (eg the figure-eight or trefoil knot) has topological slice genus at most 1, regardless of the
value of
.
Further, we show that the lower bounds on the slice genus coming from the
Tristram–Levine and Casson–Gordon signatures cannot be used to disprove the
conjecture.
Keywords
4–genus, concordance, satellite knot, algebraic genus