The algebraic crossing number and the braid index of knots and links

Kawamuro, Keiko

doi:10.2140/agt.2006.6.2313

Cited by 15 publications

(21 citation statements)

References 20 publications

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“…There are some indications to this effect, at least for what concerns the first entry of the complexity function of [4], that is, the braid index. In fact, it is shown in [56] (see also [38]) that the braid index of the n-cabling of a link L is n-times the braid index of L, so the n-cabling of a minimal braid for L is a minimal braid for the n-cabling of L. Also, by [38], if the Jones conjecture holds for L, it also holds for its cabled versions. The question of whether this property of the braid index extends to the rest of the complexity function of [4] depends on the Seifert surface of the cabled link and its braid foliations.…”

Section: Homology 3-spheresmentioning

confidence: 97%

Cyclotomy and endomotives

Marcolli

2009

P-Adic Num Ultrametr Anal Appl

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Abstract. We compare two different models of noncommutative geometry of the cyclotomic tower, both based on an arithmetic algebra of functions of roots of unity and an action by endomorphisms, the first based on the Bost-Connes (BC) quantum statistical mechanical system and the second on the Habiro ring, where the Habiro functions have, in addition to evaluations at roots of unity, also full Taylor expansions. Both have compatible endomorphisms actions of the multiplicative semigroup of positive integers. As a higher dimensional generalization, we consider a crossed product ring obtained using Manin's multivariable generalizations of the Habiro functions and an action by endomorphisms of the semigroup of integer matrices with positive determinant. We then construct a corresponding class of multivariable BC endomotives, which are obtained geometrically from self maps of higher dimensional algebraic tori, and we discuss some of their quantum statistical mechanical properties. These multivariable BC endomotives are universal for (torsion free) Λ-rings, compatibly with the Frobenius action. Finally, we discuss briefly how Habiro's universal Witten-Reshetikhin-Turaev invariant of integral homology 3-spheres may relate invariants of 3-manifolds to gadgets over F1 and semigroup actions on homology 3-spheres to endomotives.

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Section: Homology 3-spheresmentioning

confidence: 97%

Cyclotomy and endomotives

Marcolli

2009

P-Adic Num Ultrametr Anal Appl

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“…It is well know, see [1], that the closure of a braid B gives a transverse link in the standard tight contact structure on S 3 and moreover its self-linking number is a(B) − n(b) where a(B) is the algebraic length of B and n(B) is the braid index of B. Thus the Bennequin inequality can be written To state our result for minimal braid index representatives of (strongly) quasi-positive knots we recall a conjecture of K. Kawamuro, [21]. Given a link type L with braid index b > 0 is there an integer w such that any braid B whose closure is in the link type L satisfies…”

Section: 3mentioning

confidence: 98%

“…Sharpness is known for closures of positive braids that contain a full right handed twist [13] and for alternating fibered links [23]. It is known that the MFW-inequality is not always sharp, but it is still unknown if the Braid Geography Conjecture is true or not; however, in [21] it was shown that the class of links for which the Braid Geography Conjecture is true is closed under cabling and connected sums. Theorem 1.14.…”

Section: 3mentioning

confidence: 99%

Fibered Transverse Knots and the Bennequin Bound

Etnyre

Horn-Morris

2010

International Mathematics Research Notices

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Abstract. We prove that a nicely fibered link (by which we mean the binding of an open book) in a tight contact manifold (M, ξ) with zero Giroux torsion has a transverse representative realizing the Bennequin bound if and only if the contact structure it supports (since it is also the binding of an open book) is ξ. This gives a geometric reason for the nonsharpness of the Bennequin bound for fibered links. We also give the classification, up to contactomorphism, of maximal self-linking number links in these knot types. Moreover, in the standard tight contact structure on S 3 we classify, up to transverse isotopy, transverse knots with maximal self-linking number in the knots types given as closures of positive braids and given as fibered strongly quasi-positive knots. We derive several braid theoretic corollaries from this. In particular, we give conditions under which quasi-positive braids are related by positive Markov stabilizations and when a minimal braid index representative of a knot is quasi-positive. Finally, we observe that our main result can be used to show, and make rigorous the statement, that contact structures on a given manifold are in a strong sense classified by the transverse knot theory they support.

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“…Furthermore, it contains infinitely many four tuples (x, y, z, w) where the MFW inequality is not sharp [8].…”

Section: On K and Its Mirror Image K The Mfw-inequality Is Not Sharmentioning

confidence: 99%