Concordances to prime hyperbolic virtual knots
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Concordances to prime hyperbolic virtual knots Micah Chrisman1 Received: 28 March 2020 / Accepted: 11 August 2020 © Springer Nature B.V. 2020
Abstract Let 0 , 1 be closed oriented surfaces. Two oriented knots K 0 ⊂ 0 × [0, 1] and K 1 ⊂ 1 × [0, 1] are said to be (virtually) concordant if there is a compact oriented 3-manifold W and a smoothly and properly embedded annulus A in W × [0, 1] such that ∂ W = 1 −0 and ∂ A = K 1 −K 0 . This notion of concordance, due to Turaev, is equivalent to concordance of virtual knots, due to Kauffman. A prime virtual knot, in the sense of Matveev, is one for which no thickened surface representative K ⊂ × [0, 1] admits a nontrivial decomposition along a separating vertical annulus that intersects K in two points. Here we prove that every knot K ⊂ × [0, 1] is concordant to a prime satellite knot and a prime hyperbolic knot. For homologically trivial knots in × [0, 1], we prove this can be done so that the Alexander polynomial is preserved. This generalizes the corresponding results for classical knot concordance, due to Bleiler, Kirby–Lickorish, Livingston, Myers, Nakanishi, and Soma. The new challenge for virtual knots lies in proving primeness. Contrary to the classical case, not every hyperbolic knot in × [0, 1] is prime and not every composite knot is a satellite. Our results are obtained using a generalization of tangles in 3-balls we call complementary tangles. Properties of complementary tangles are studied in detail. Keywords Virtual knot · Concordance · Prime knot · Satellite knot · Hyperbolic knot · Almost classical knot · Complementary tangle Mathematics Subject Classification (2010) Primary: 57M25; Secondary: 57M27
1 Introduction 1.1 Motivation Let I = [0, 1]. Two oriented knots K 0 , K 1 in the 3-sphere are said to be concordant if there is a smoothly and properly embedded annulus A ⊂ S 3 × I such that K i ⊂ S 3 × {i} for i = 0, 1 and ∂ A = K 1 −K 0 . Here −K is K with the opposite orientation. A knot in S 3 is said to be prime (or locally trivial) if every 2-sphere that intersects K transversely in two points bounds a 3-ball that intersects K in an unknotted arc. Kirby–Lickorish [30] showed that every
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Micah Chrisman [email protected] Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA
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Geometriae Dedicata
knot is concordant to a prime knot. Livingston [34] proved that every knot is concordant to a prime satellite knot. One may even choose the prime knot so that the Alexander polynomial is preserved (Bleiler [4], Nakanishi [42]). Myers [40] further showed that every knot in S 3 is concordant to a knot that is not only prime but hyperbolic. The same result holds true for links in S 3 (see also Soma [46]). In [49], Turaev introduced a new notion of concordance for knots in thickened surfaces × I , where is closed and oriented. Two knots K 0 ⊂ 0 × I , K 1 ⊂ 1 × I are said to be (virtually) concordant if there is a compact oriented 3-manifold W and a smoothly and properly embedded annulus A ⊂ W × I such that
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