A note on HOMFLY polynomial of positive braid links

Ito, Tetsuya

doi:10.1142/s0129167x22500318

Cited by 3 publications

(4 citation statements)

References 21 publications

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“…Similarly, many classical invariants cannot distinguish positive braids amongst positive knots: for example, all positive knots have negative signature [Prz89] and positive Conway polynomials [Cro89]. Some properties of the coefficients and degrees of the HOMFLY and Jones polynomials are able to differentiate positive braid knots amongst positive knots on an ad-hoc basis; see [Sto03,Section 8] and [Ito22]. However, no general formula computing either polynomial for cables of knots in terms of the polynomials of T (p, q) and K can exist [ST09], making it difficult to distinguish braid positivity from knot positivity for infinite families.…”

Section: Resultsmentioning

confidence: 99%

Taut foliations, positive 3‐braids, and the L‐space conjecture

Krishna

2020

Journal of Topology

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We construct taut foliations in every closed 3‐manifold obtained by r‐framed Dehn surgery along a positive 3‐braid knot K in S3, where r<2g(K)−1 and g(K) denotes the Seifert genus of K. This confirms a prediction of the L‐space Conjecture. For instance, we produce taut foliations in every non‐L‐space obtained by surgery along the pretzel knot P(−2,3,7), and indeed along every pretzel knot P(−2,3,q), for q a positive odd integer. This is the first construction of taut foliations for every non‐L‐space obtained by surgery along an infinite family of hyperbolic L‐space knots. Additionally, we construct taut foliations in every closed 3‐manifold obtained by r‐framed Dehn surgery along a positive 1‐bridge braid in S3, where r

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Section: Resultsmentioning

confidence: 99%

Taut foliations, positive 3‐braids, and the L‐space conjecture

Krishna

2020

Journal of Topology

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“…The results of Corollary 1.5 combined with those of [24] provide many examples of L-space knots (that are not lens space knots) that admit a fillable positive surgery. It is natural to ask whether every L-space knot admits a fillable positive surgery; as a concrete instance we do not currently know whether the cable knot T (2, 3) 2,3 , which is an L-space knot but neither a lens space knot [20] nor a positive braid closure [31,Remark 4], [1, Example 1], admits a fillable positive surgery.…”

Section: Constructionsmentioning

confidence: 99%

Fillable contact structures from positive surgery

Mark¹,

Tosun²

2023

Preprint

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We consider the question of when the operation of contact surgery with positive surgery coefficient, along a knot K in a contact 3-manifold Y , gives rise to a weakly fillable contact structure. We show that this happens if and only if Y itself is weakly fillable, and K is isotopic to the boundary of a properly embedded symplectic disk inside a filling of Y . Moreover, if Y is a contact manifold arising from positive contact surgery along K, then any filling of Y is symplectomorphic to the complement of a suitable neighborhood of such a disk in a filling of Y .Using this result we deduce several necessary conditions for a knot in the standard 3-sphere to admit a fillable positive surgery, such as quasipositivity and equality between the slice genus and the 4-dimensional clasp number, and we give a characterization of such knots in terms of a quasipositive braid expression. We show that knots arising as the closure of a positive braid always admit a fillable positive surgery, as do knots that have lens space surgeries, and suitable satellites of such knots. In fact the majority of quasipositive knots with up to 10 crossings admit a fillable positive surgery. On the other hand, in general (strong) quasipositivity, positivity, or Lagrangian fillability need not imply a knot admits a fillable positive contact surgery.

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“…Here the list of nonzero integers represents a braid word by letting the integer k stand for the standard generator k or its inverse 1 k , depending on whether k is positive or negative. Ito gives new constraints on a suitably normalized version of the HOMFLY polynomial for positive braids [20]. The Ito-normalized HOMFLY polynomial z…”

Section: Introductionmentioning

confidence: 99%

“…According to [20,Theorem 2], if a link K is braid positive then the Ito-normalized HOMFLY polynomial should only have nonnegative coefficients. As one observes, the coefficients h 30 and h 31 are negative.…”

Section: Introductionmentioning

confidence: 99%

Census L–space knots are braid positive, except for one that is not

Baker,

Kegel

2024

Algebr. Geom. Topol.

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We exhibit braid positive presentations for all L-space knots in the SnapPy census except one, which is not braid positive. The normalized HOMFLY polynomial of o9_30634, when suitably normalized, is not positive, failing a condition of Ito for braid positive knots.We generalize this knot to a 1-parameter family of hyperbolic L-space knots that may not be braid positive. Nevertheless, as pointed out by Teragaito, this family yields the first examples of hyperbolic L-space knots whose formal semigroups are actual semigroups, answering a question of Wang. Further, the roots of the Alexander polynomials of these knots are all roots of unity, disproving a conjecture of Li and Ni.

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A note on HOMFLY polynomial of positive braid links

Cited by 3 publications

References 21 publications

Taut foliations, positive 3‐braids, and the L‐space conjecture

Taut foliations, positive 3‐braids, and the L‐space conjecture

Fillable contact structures from positive surgery

Census L–space knots are braid positive, except for one that is not

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