Braids, Fibered Knots, and Concordance Questions

Hubbard, Diana; Kawamuro, Keiko; Kose, Feride Ceren; Martin, Gage; Plamenevskaya, Olga; Raoux, Katherine; Truong, Linh; Turner, H. H.

doi:10.1007/978-3-030-80979-9_7

Association for Women in Mathematics Series

2021

DOI: 10.1007/978-3-030-80979-9_7

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Braids, Fibered Knots, and Concordance Questions

Diana Hubbard

Keiko Kawamuro

Feride Ceren Kose

et al.

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“…Indeed, let K be a knot that is the closure of a positive braid of the form in (C) or (D) with ` 0. Then we have(29) g T .K/ D alt.K/ D dalt.K/ D r C ` 1;…”

mentioning

confidence: 99%

The upsilon invariant at 1 of 3–braid knots

Truöl

2023

Algebr. Geom. Topol.

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We provide explicit formulas for the integer-valued smooth concordance invariant .K/ D ‡ K .1/ for every 3-braid knot K. We determine this invariant, which was defined by Ozsváth, , 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/ C .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. 57K10; 20F36, 57K18Ã 2`: By Murasugi's classification of the conjugacy classes of 3-braids [45, Proposition 2.1], indeed all 3-braid knots -except for the torus knots that are closures of 3-braidsare covered by Theorem 1.1. However, for torus knots the invariant can be calculated explicitly by a combinatorial, inductive formula in terms of their Alexander polynomial [46, Theorem 1.15]; see (12) below. Hence, we have indeed determined .K/ for all 3-braid knots K.As an application of Theorem 1.1, we show that the following invariants coincide for positive 3-braid knots -knots that are the closure of positive 3-braids.

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“…Indeed, let K be a knot that is the closure of a positive braid of the form in (C) or (D) with ` 0. Then we have(29) g T .K/ D alt.K/ D dalt.K/ D r C ` 1;…”

mentioning

confidence: 99%

The upsilon invariant at 1 of 3–braid knots

Truöl

2023

Algebr. Geom. Topol.

View full text Add to dashboard Cite

show abstract

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Braids, Fibered Knots, and Concordance Questions

Cited by 1 publication

References 49 publications

The upsilon invariant at 1 of 3–braid knots

The upsilon invariant at 1 of 3–braid knots

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