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Strong winding is preserved by companion insertion
Statement
Let P be a pattern of strong winding number ±1. For every oriented knot K, regard P(K) as the pattern obtained by tying K into each strand of P in the companion solid torus. Then this new pattern P(K) also has strong winding number ±1. A pattern obtained from any strong-winding pattern by taking connected sum with a local knot also has strong winding number ±1.
Statement references
Facts & Assumptions
Given: A pattern P, its unknotted-solid-torus closure P(U), strong-winding meridian η, and a knot K inserted as companion.
Strong winding means the axis meridian normally generates the closure complement group (Satellite operators on concordance classes).
The knot exterior has meridian μ_K and preferred longitude λ_K on its boundary torus (Knot exterior, Meridian and preferred longitude).
The knot-group meridian normally generates π₁(S³\K): meridional filling gives S³, and van Kampen quotients by the normal closure of μ_K to π₁(S³)=1 (Knot exterior, Meridian and preferred longitude).
Connected sum is formed by joining the two knot exteriors along a meridional annulus (Oriented connected sum and prime knot).
Proof
Let λ denote the longitude of the pattern solid torus. Filling the outer boundary of the pattern exterior along λ produces S³∖P(U), so the induced epimorphism π₁(ST∖P)→π₁(S³∖P(U)) has kernel normally generated by λ. By [F1], the target is normally generated by the image of η; hence π₁(ST∖P) is normally generated by η and λ.
For a local connected sum with any knot J, the closure-complement group is the amalgamated product of the two knot groups along their common meridian, by cutting at the connected-sum sphere and applying van Kampen. Each knot-group factor is normally generated by its meridian.
Decompose the complement of P(K) along the companion torus into ST∖P and S³∖K. Van Kampen gives a generating set normally contained in the images of these two groups. The first is normally generated by η and λ by step 1.1; the second is normally generated by μ_K by [F3].
Under the torus gluing, λ is identified with λ_K and η with μ_K. Since μ_K normally generates the companion-knot group, λ becomes redundant, and η normally generates the entire satellite-complement group. The outer solid-torus meridian for the pattern P(K) is isotopic in the complement to η, so it too normally generates.
The solid-torus meridian normally generates the strong-winding factor by hypothesis; quotienting the amalgamated product by that meridian kills this factor and the common meridian. The other knot-group factor is then killed because its meridian normally generates it. Thus the whole closure-complement group is normally generated by the solid-torus meridian.
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Sources
- Tim D. Cochran, Christopher W. Davis, and Arunima Ray, "Injectivity of satellite operators in knot concordance" (standard reference, not scraped)