Khovanov-Rozansky homology and the braid index of a knot

Kawamuro, Keiko

doi:10.1090/s0002-9939-09-09743-3

Cited by 7 publications

(8 citation statements)

References 17 publications

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“…It is known that P (5 1 ) = P (m(10 132 )) but we obviously see that H(5 1 ) = H(m(10 132 )). This implies that the reduced HOM-FLY homology is strictly stronger than the HOMFLY polynomial (as verified in several contexts, for example, see [Kaw09]). This pair appears in [Bar02] as an example that the two knots has the same Jones polynomial but distinct Khovanov homology.…”

Section: Introductionmentioning

confidence: 61%

Computations of HOMFLY homology

Nakagane¹,

Sano²

2021

Preprint

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Section: Introductionmentioning

confidence: 61%

Computations of HOMFLY homology

Nakagane¹,

Sano²

2021

Preprint

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“…We finish this section by showing that Theorem determines the braid index in infinitely many cases where the Morton–Franks–Williams inequality is not sharp. Elrifai in (see also ) proved that for all knots and links of braid index three, the Morton–Franks–Williams inequality is sharp except for the families of knots which are closures of

K_{k} = {(a_{1} a_{2} a_{2} a_{1})}^{2 k} a_{1} a_{2}^{- 2 k - 1}

and

L_{k} = {(a_{1} a_{2} a_{2} a_{1})}^{2 k + 1} a_{1} a_{2}^{- 2 k + 1}

for

k

a positive integer, and their mirror images. Example The families of 3‐braids

K_{k}

and

L_{k}

( for

k ⩾ 2

and

k ⩾ 1

, respectively ) have fractional Dehn twist coefficients strictly larger than two ( and hence have braid index three by Theorem ) .…”

Section: Examples and Optimalitymentioning

confidence: 99%

“…We finish this section by showing that Theorem 1.1 determines the braid index in infinitely many cases where the Morton-Franks-Williams inequality [17,37,38] is not sharp. Elrifai in [12] (see also [29]) proved that for all knots and links of braid index three, the Morton-Franks-Williams inequality is sharp except for the families of knots which are closures of K k = (a 1 a 2 a 2 a 1 ) 2k a 1 a −2k−1 2 and L k = (a 1 a 2 a 2 a 1 ) 2k+1 a 1 a −2k+1 2 for k a positive integer, and their mirror images.…”

Section: −S ω(α) Rmentioning

confidence: 99%

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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“…Suppose a knot K has a braid diagram of b strands with writhe w. Denote by K m,k the (m, k)-cable of K. Then the braid diagram of K leads to an obvious braid diagram of K m,k of mb strands with writhe mw+k(m−1). [3,9,12,17,18,25] for related results. 1.5.…”

Section: 4mentioning

confidence: 99%

Colored Morton–Franks–Williams Inequalities

2012

International Mathematics Research Notices

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We generalize the Morton-Franks-Williams inequality [3,14] to the colored sl(N ) link homology defined in [30], which gives infinitely many new bounds for the braid index and the self linking number. A key ingredient of our proof is a composition product for the general MOY graph polynomial, which generalizes that of Wagner [27].2000 Mathematics Subject Classification. Primary 57M25. Key words and phrases. braid index, self linking number, Morton-Franks-Williams inequality, Khovanov-Rozansky homology, Reshetikhin-Turaev invariant, colored sl(N ) link homology.

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Khovanov-Rozansky homology and the braid index of a knot

Cited by 7 publications

References 17 publications

Computations of HOMFLY homology

Computations of HOMFLY homology

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

Colored Morton–Franks–Williams Inequalities

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