Satellites of Infinite Rank in the Smooth Concordance Group

Abstract

We conjecture that satellite operations are either constant or have infinite rank in the concordance group. We reduce this to the difficult case of winding number zero satellites, and use SO(3) gauge theory to provide a general criterion sufficient for the image of a satellite operation to generate an infinite rank subgroup of the smooth concordance group C. Our criterion applies widely, notably to scores of unknotted patterns for which the corresponding operators on the topological concordance group are zero. We raise some questions and conjectures regarding satellite operators and their interaction with concordance.

On the Upsilon invariant of fibered knots and right-veering open books

“…Also recall that cabling induces a map on the concordance group (see e.g. [HPC18]). Thus, the (p, 1)−cable K p,1 of K is also slice and fibered.…”

“…The knots K and J have the same genus and are both slice since cabling is an operator on the concordance group (see e.g. [HPC18]). Thus, the knot K with right-veering monodromy is concordant to the knot J with non-right-veering monodromy.…”

An infinite-rank summand from iterated Mazur pattern satellite knots

“…Are the iterated Whitehead doubles independent [23]? For any winding-number-zero pattern P which induces a non-constant map on C, are the associated graded groups of the P -filtration C ⊇ ImP ⊇ ImP 2 ⊇ • • • always of infinite rank [13]? Perhaps more classically, if we allow the companion and pattern knots to vary, are algebraic knots linearly independent [26]?…”

Homomorphism obstructions for satellite maps

“…To sample just a few results, the satellite construction features prominently in the first evidence for a fractal structure on concordance [CHL11]; the first examples of non-smoothly concordant knots with homeomorphic 0-surgeries [Yas15]; and the first knots in homology spheres which do not bound PL discs in any contractible 4-manifold [Lev16]. As a result, satellite operations have become an object of study in their own right, with recent work in the area focusing on the existence of interesting bijective satellite maps [GM95,MP18], the behavior of the 4-genera of knots under satelliting [CH18,Pic18,Mil19,FMPC19], and on satellite maps with image of infinite rank [HPC16].…”