Critical point counts in knot cobordisms: abelian and metacyclic invariants

Livingston, Charles

doi:10.1090/btran/139

Cited by 1 publication

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“…Further, each knot K satisfies 2g(K) = |σ(K)|, g the genus of K, implying the existence of a definite Seifert surface. These examples were found with the help of KnotInfo [14].…”

Section: Proof Of Main Resultsmentioning

confidence: 99%

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2023

Proc. Amer. Math. Soc. Ser. B

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We prove that a family of links, which includes all special alternating knots, does not admit non-nugatory crossing changes which preserve the isotopy type of the link. Our proof incorporates a result of Lidman and Moore [Trans. Amer. Math. Soc. 369 (2017), pp. 3639-3654] on crossing changes to knots with L-space branched double-covers, as well as tools from Scharlemann and Thompson's [Comment. Math. Helv. 64 (1989), pp. 527-535] proof of the cosmetic crossing conjecture for the unknot.Conjecture 1.1 (Cosmetic crossing conjecture). For any knot L ⊂ S 3 , only a nugatory crossing admits a cosmetic crossing change.Conjecture 1.1 has been affirmed for two-bridge knots [19] and fibered knots [11], and significant partial results exist for genus one knots and satellite knots [1,2,9,10]. Further, Lidman and Moore have verified the conjecture for all knots L ⊂ S 3 such that the branched double-cover Σ(L) is an L-space, and L has squarefree determinant [13]; their work has been extended by Ito [8].In this note, we prove the cosmetic crossing conjecture for all special alternating knots in S 3 . (The case of special alternating knots with square-free determinant is included in [13].) Theorem 1.2. Let L ⊂ S 3 be a special alternating knot. Then L admits no cosmetic, non-nugatory crossing change.Actually, we prove Conjecture 1.1 for a family of oriented links which includes all non-split special alternating links with certain orientations, and some nonalternating links-see Theorem 3.2.

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“…Further, each knot K satisfies 2g(K) = |σ(K)|, g the genus of K, implying the existence of a definite Seifert surface. These examples were found with the help of KnotInfo [14].…”

Section: Proof Of Main Resultsmentioning

confidence: 99%

Untitled

2023

Proc. Amer. Math. Soc. Ser. B

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Critical point counts in knot cobordisms: abelian and metacyclic invariants

Abstract: For a pair of knots K 1 K_1 and K 0 K_0 , we consider the set of four-tuples of integers ( g , c 0 , c 1 , c 2 ) (g, c_0,c_1, c_2) for which there is a cobordism from K 1 K_1 to K 0 K_0 of genus g g havin… Show more

Cited by 1 publication

References 24 publications

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