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Ric Ancel, PMA 12.166: Isotopies, semi-isotopies and thickenings of wild knots and links
Monday, September 26, 2022, 02:00pm - 03:00pm
Let I = [0,1]. Possibly wild links K and L in a 3-manifold M are (non-ambiently) isotopic if there is a level-preserving embedding e : K ? I → M ? I such that e(K ? &ob;0&cb;) = K ? &ob;0&cb; and e(K ? &ob;1&cb;) = L ? &ob;1&cb;. Motivating Conjecture: Every (wild) knot in S3 is isotopic to an unknot. It is not known whether the Bing sling ? a wild knot in S3 that pierces no disk ? is isotopic to an unknot. Proposition: Every knot in S3 that pierces a disk ? even a wild disk ? is isotopic to an unknot. Let K, L be links in a 3-manifold M. K is semi-isotopic to L if there is an annulus A ⊂ M3 ? I such that A ∩ (M3 ? ∂I) = ∂A = (K ? &ob;0&cb;) ∪ (L ? &ob;1&cb;) and there is a homeomorphism e : K ? [0,1) → A ? (L ? &ob;1&cb;) such that e(K ? &ob;t&cb;) ⊂ M3 ? &ob;t&cb; for every t ∈ [0,1). A thickening of K in M is a compact 3-manifold T ⊂ M such that K ⊂ int(T), the inclusion of K in T is a homotopy equivalence and T is a disk bundle over a link. A link J is a core of a thickening T if J is the 0-section of a disk bundle structure on T. K is thickenable if it has a thickening. We generalize a technique of C. Giffen called shift-spinning to prove: Theorem 1. If a link K in a 3-manifold M has a thickening T with core J, then K is semi-isotopic to J. Theorem 2. Every knot is S3 has a thickening which is an unknotted solid torus. Corollary 1. Every knot in S3 is semi-isotopic to an unknot. Using a result of S. Melikhov, we obtain: Corollary 2. There is a non-thickenable 2-component link in S3. We also recover the following unpublished result of C. Giffen: Corollary 3. F-isotopic links are I-equivalent. This is joint work with Sergey Melikhov.
Location: PMA 12.166

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