The minimal number of Seifert circles equals the braid index of a link

Yamada, Shǔji

doi:10.1142/9789812798329_0023

Cited by 54 publications

(64 citation statements)

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“…P r o o f of Theorem 2.1. The following proof is based on Yamada's proof of Alexander's theorem in [6], see also [3]. Let G be a disjoint union of a θ m -curve and an oriented link, and D be a diagram of G in R 2 whose vertices are placed on the top and bottom as Figure 4.…”

Section: Example 23mentioning

confidence: 99%

On the braid index of θ_m‐curve in 3‐space

Shinnoki

Takamuki

2003

Mathematische Nachrichten

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Section: Example 23mentioning

confidence: 99%

On the braid index of θ_m‐curve in 3‐space

Shinnoki

Takamuki

2003

Mathematische Nachrichten

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“…It follows from the result obtained by Yamada in [40] that the conjecture presented above is equivalent to the following one: Conjecture 5.1. For every diagram D of the link L that has the minimum number of Seifert cycles among all possible diagrams for L, the torsion w(D) of the diagram D is uniquely defined.…”

Section: Diagrams Links Seifert Graphs and Braid Indexmentioning

confidence: 84%

Invariants of knots, surfaces in R3, and foliations

Plakhta

2007

Ukr Math J

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We give a survey of some known results related to combinatorial and geometric properties of finite-order invariants of knots in a three-dimensional space. We study the relationship between Vassiliev invariants and some classical numerical invariants of knots and point out the role of surfaces in the investigation of these invariants. We also consider combinatorial and geometric properties of essential tori in standard position in closed braid complements by using the braid foliation technique developed by Birman, Menasco, and other authors. We study the reductions of link diagrams in the context of finding the braid index of links.In recent 10-15 years, finite-order Vassiliev invariants of knots and links in a three-dimensional space, together with known classical combinatorial and geometric invariants, have been extensively studied in knot theory. The problem of topological and geometric information about knots that is contained in Vassiliev invariants (also called finite-order invariants) is one of the most interesting and important problems in contemporary knot theory (see, e.g., [1][2][3][4][5]). Vassiliev's conjecture that finite-order invariants distinguish knots in a three-dimensional space still remains open. Knots that are not distinguished by invariants of finite order ≤ n are called n-equivalent. Habiro gave a geometric interpretation of the n-equivalence of knots in terms of local motions on knots; a combinatorial interpretation in terms of schemes was proposed by Gusarov. A geometric interpretation of Vassiliev invariants was given in [1]. Furthermore, several interesting relations between Vassiliev invariants and classical combinatorial and geometric invariants of knots were obtained in [6-9, 2]. However, there is still a lack of complete topological and geometric understanding of finite-order invariants.In the present paper, we describe the relationship between finite-order invariants and some classical invariants of knots, namely, genus, canonical genus, and the Alexander polynomial, and consider some combinatorial and geometric aspects of finite-order invariants. In particular, in Sec. 1, we show that local simple Habiro C n -motions on knots are equivalent, in the geometric sense, to the corresponding commutators of groups of pure braids.In the theory of finite-order invariants of knots and links in a three-dimensional space, an important role is played by the graded space (over the field Q) of trivalent diagrams associated with filtration of Vassiliev-Gusarov knots (links) and the graded algebra of linear functionals on the space of trivalent diagrams (see [10]). These functionals are called weight systems. Numerous problems arising here were first solved in terms of trivalent diagrams and then interpreted in terms of invariants themselves [10,2]. Combinatorial aspects of trivalent diagrams considered as elements of a group (vector space) A, i.e., modulo AS-and STU-relations, were investigated by many authors (see, e.g., [10,11]). In particular, their geometric interpretation was given in [10,1...

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“…(3) A different weakening of Corollary 1-allowing D to be arbitrary, but concluding only ( * ′ [D])-follows easily by combining sBi and the (proof of the) main result of [20]. In fact, after posting the first version of the present article to the xxx Mathematics Archives, I received e-mail from Takuji Nakamura informing me that Nakamura's February 1998 master's thesis at Keio University (Japan) gives a proof of the Theorem using the techniques of [20]; and another reader has since kindly shown me how to use Vogel's algorithm [19] to give yet another proof.…”

Section: Introduction; Statement Of Resultsmentioning

confidence: 99%

Positive links are strongly quasipositive

Rudolph¹

1999

Geometry &Amp; Topology Monographs

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Let S(D) be the surface produced by applying Seifert's algorithm to the oriented link diagram D. I prove that if D has no negative crossings then S(D) is a quasipositive Seifert surface, that is, S(D) embeds incompressibly on a fiber surface plumbed from positive Hopf annuli. This result, combined with the truth of the "local Thom Conjecture", has various interesting consequences; for instance, it yields an easily-computed estimate for the slice euler characteristic of the link L(D) (where D is arbitrary) that extends and often improves the "slice-Bennequin inequality" for closed-braid diagrams; and it leads to yet another proof of the chirality of positive and almost positive knots.

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The minimal number of Seifert circles equals the braid index of a link

Cited by 54 publications

References 2 publications

On the braid index of θ_m‐curve in 3‐space

On the braid index of θ_m‐curve in 3‐space

Invariants of knots, surfaces in R3, and foliations

Positive links are strongly quasipositive

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The minimal number of Seifert circles equals the braid index of a link

Cited by 54 publications

References 2 publications

On the braid index of θm‐curve in 3‐space

On the braid index of θm‐curve in 3‐space

Invariants of knots, surfaces in R3, and foliations

Positive links are strongly quasipositive

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Product

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On the braid index of θ_m‐curve in 3‐space

On the braid index of θ_m‐curve in 3‐space