The upsilon invariant at 1 of 3-braid knots
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2023
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Journal Article
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Abstract
We provide explicit formulas for the integer-valued smooth concordance invariant v(K) = TK (1) for every 3-braid knot K. We determine this invariant, which was defined by Ozsvath, Stipsicz and Szabo (2017), by constructing cobordisms between 3-braid knots and (connected sums of) torus knots. As an application, we show that for positive 3-braid knots K several alternating distances all equal the sum g(K) + v(K), where g(K) denotes the 3-genus of K. In particular, we compute the alternation number, the dealternating number and the Turaev genus for all positive 3-braid knots. We also provide upper and lower bounds on the alternation number and dealternating number of every 3-braid knot which differ by 1.
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23 (8)
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3763 - 3804
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Mathematical Sciences Publishers
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181199 - Low-dimensional topology with a view toward the fourth dimension and complex geometry (SNF)