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Satellite operators on concordance classes
Definition
Let V=S¹×D² be the standard oriented solid torus, with oriented core, meridional disk and product longitude. A pattern is a smooth oriented knot P in the interior of V. Its winding number w(P) is its algebraic intersection with an oriented meridional disk, equivalently the coefficient of the core in [P]∈H₁(V;Z). For a smooth oriented knot K in S³, let e_K:V→S³ be a zero-framed tubular embedding carrying the core to K and carrying the meridian and product longitude to the positive meridian and preferred longitude of K. The satellite knot P(K) is e_K(P).
When concordance preservation has been established, the formula [K]↦[P(K)] defines a set map on each of C_sm, C_top, C_ex and C^{1/n}. The solid-torus meridian is strong winding number ±1 for P if, after V is embedded as the standard unknotted solid torus and P(U) denotes the resulting closure, that meridian normally generates π₁(S³\P(U)). Normal generation means that the smallest normal subgroup containing the element is the whole group. Strong winding number ±1 is a separate condition from ordinary winding number ±1. No additivity or homomorphism property is part of this definition.
Depends on
Used by
- A knotted closure can have strong winding number one Counterexample
- Worked cases: identity and cable operators Example
- Satellite operators preserve concordance Lemma
- Satellite substitution across connected sum Lemma
- Strong winding is preserved by companion insertion Lemma
- The CDR pattern zero-surgery homology cobordism Lemma
- Weak injectivity and auxiliary-pattern reduction Lemma
- Smooth satellite filtrations with topologically slice images Open problem
- Cochran–Davis–Ray satellite injectivity Theorem
- Infinite rank for iterated winding-zero satellites Theorem
Dependency tree · two levels
9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Tim D. Cochran, Christopher W. Davis, and Arunima Ray, “Injectivity of satellite operators in knot concordance” (standard reference, not scraped)
- Arunima Ray, “Slice Knots and Knot Concordance” (standard reference, not scraped)