The geometry of the knot concordance space

Abstract

Most of the 50-years of study of the set of knot concordance classes, C, has focused on its structure as an abelian group. Here we take a different approach, namely we study C as a metric space admitting many natural geometric operators, especially satellite operators. We consider several knot concordance spaces, corresponding to different categories of concordance, and two different metrics. We establish the existence of quasi-n-flats for every n, implying that C admits no quasi-isometric embedding into a finite product of (Gromov) hyperbolic spaces. We show that every satellite operator is a quasi-homomorphism P : C → C. We show that winding number one satellite operators induce quasi-isometries with respect to the metric induced by slice genus. We prove that strong winding number one patterns induce isometric embeddings for certain metrics. By contrast, winding number zero satellite operators are bounded functions and hence quasi-contractions. These results contribute to the suggestion that C is a fractal space. We establish various other results about the large-scale geometry of arbitrary satellite operators. 1 n . Here C top is the (usual) set of topological knot concordance classes wherein two knots are equivalent if there exists a collared proper topological embedding of an annulus into a topological manifold homeomorphic to S 3 × [0, 1] which restricts on its boundary to the given knots. C ex , short for C exotic , is the set of equivalence classes of knots where two are equivalent if they cobound a properly, smoothly embedded annulus in a smooth manifold homeomorphic to S 3 × [0, 1]; that is to say they are concordant in S 3 × [0, 1] equipped with a possibly exotic smooth structure (see [4][25,

Grope metrics on the knot concordance set

“…Let be the metric on defined by the slice genus, which was studied in detail in . Let be the associated norm.…”

“…for any x, y ∈ X, and if there is a constant C 0 such that for any z ∈ Y there exists an x ∈ X such that d Y (z, f (x)) C. Let d s be the metric on C defined by the slice genus, which was studied in detail in [10]. Let s be the associated norm.…”

“…An explicit study of metrics on the set of concordance classes of knots was begun by the first and second authors . Metrics were studied arising from the slice genus, and the homology norm, which measures the second Betti number of a 4‐manifold in which two knots and cobound an annulus.…”

“…10. A map f : (X, d) → (Y, d ) between pseudo-metric spaces is called a contraction mapping if there exists some 0 δ < 1 such that d (f (x), f(w)) δd(x, w) for all x, w ∈ X.Proposition 5.11.…”

Winding number $m$ and $-m$ patterns acting on concordance

“…While most of the investigations of C, the collection of knots modulo concordance, have focused on its group structure, it is also natural to consider it as a metric space with metric d(K, J) := g 4 (K# − J). Cochran and Harvey [CH17] considered this geometric structure, focusing on the metric properties of maps induced by patterns in solid tori. Following their work, we consider the distance between two patterns, defined as d(P, Q) = sup K∈C d(P (K), Q(K)) ∈ {0, 1, 2, .…”