Asymptotic Rasmussen invariant

Baader, Sebastian

doi:10.1016/j.crma.2007.06.020

Cited by 12 publications

(19 citation statements)

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“…These properties seem to have not been established in the literature so far, although some have been claimed without proof in [15] and so we provide proofs for completeness. Our proofs are in spirit close to the constructions of Baader [3] and Brandenbursky [8]. More concretely, the proofs of Lemma A.1 and Proposition A.4 are based on the following fundamental observation: Given two n-braids α and β, the closure of αβ and the connected sum of the closures of α and β are related by a connected cobordism of Euler characteristic n − 1.…”

Section: Appendix Homogenization Of Concordance Homomorphismsmentioning

confidence: 69%

“…The concordance C induces a bijection between the connected components of the links β and α: Connected components of the links are related if they are contained in the same subannulus of C. We pick i m − 1 such that under this bijection the connected component of β that contains the strand that ends left-most on the top of β gets map to the connected component of α that contains the strand of α that ends ith on the top of α. For example, let β be the 3-braid a 3 2 and α be the 3-braid a 3 1 . The closure of both of these are an unknot disjoint union a T 2,3 (that is, a trefoil).…”

Section: Proof Of Lemmamentioning

confidence: 99%

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Braids with as many full twists as strands realize the braid index

Feller

Hubbard

2019

Journal of Topology

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We characterize the fractional Dehn twist coefficient of a braid in terms of a slope of the homogenization of the Upsilon function, where Upsilon is the function‐valued concordance homomorphism defined by Ozsváth, Stipsicz, and Szabó. We use this characterization to prove that n‐braids with fractional Dehn twist coefficient larger than n−1 realize the braid index of their closure. As a consequence, we are able to prove a conjecture of Malyutin and Netsvetaev stating that n‐times twisted braids realize the braid index of their closure. We provide examples that address the optimality of our results. The paper ends with an appendix about the homogenization of knot concordance homomorphisms.

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Section: Appendix Homogenization Of Concordance Homomorphismsmentioning

confidence: 69%

Section: Proof Of Lemmamentioning

confidence: 99%

Braids with as many full twists as strands realize the braid index

Feller

Hubbard

2019

Journal of Topology

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“…A very special case of Theorem 2 was proved in [3]: the asymptotic Rasmussen invariant equals twice the asymptotic signature invariant.…”

Section: Theorem 1 Let Be a Linear Saddle Invariant With Values In Rmentioning

confidence: 99%

“…They proved the existence of an asymptotic signature invariant for orbits of a smooth volume-preserving vector field on a compact domain of R 3 and related it to the asymptotic linking number. Recently we could prove the existence of an asymptotic Rasmussen invariant [3]. Both the signature and Rasmussen's invariant are so-called concordance invariants.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic concordance invariants for ergodic vector fields

Baader¹

2010

Comment. Math. Helv.

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Abstract. We study the asymptotics of a family of link invariants on the orbits of a smooth volume-preserving ergodic vector field on a compact domain of the 3-space. These invariants, called linear saddle invariants, include many concordance invariants and generate an infinitedimensional vector space of link invariants. In contrast, the vector space of asymptotic linear saddle invariants is 1-dimensional, generated by the asymptotic signature. We also determine the asymptotic slice genus and relate it to the asymptotic signature.Mathematics Subject Classification (2010). 57M27, 37A05.

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“…(See a brief description of this representation in §1 below.) For example, it was discovered that certain link invariants are quasimorphisms of braid groups (see [12,2]). …”

Section: Introductionmentioning

confidence: 99%

Pseudocharacters of braid groups and prime links

Malyutin

2010

St. Petersburg Math. J.

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Abstract. Pseudocharacters of groups have recently found an application in the theory of classical knots and links in R 3 . More precisely, there is a relationship between pseudocharacters of Artin's braid groups and the properties of links represented by braids. In the paper, this relationship is investigated and the notion of kernel pseudocharacters of braid groups is introduced. It is proved that if a kernel pseudocharacter φ and a braid β satisfy |φ(β)| > C φ , where C φ is the defect of φ, then β represents a prime link (i.e., a link that is noncomposite, nonsplit, and nontrivial). Furthermore, the space of braid group pseudocharacters is studied and a way is described to obtain nontrivial kernel pseudocharacters from an arbitrary braid group pseudocharacter that is not a homomorphism. This makes it possible to employ an arbitrary nontrivial braid group pseudocharacter for the recognition of prime knots and links.

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Asymptotic Rasmussen invariant

Cited by 12 publications

References 13 publications

Braids with as many full twists as strands realize the braid index

Braids with as many full twists as strands realize the braid index

Asymptotic concordance invariants for ergodic vector fields

Pseudocharacters of braid groups and prime links

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