The Knight Move Conjecture is false

Manolescu, Ciprian; Marengon, Marco

doi:10.1090/proc/14694

Cited by 12 publications

(15 citation statements)

References 9 publications

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“…Khovanov thin knots have torsion order at most 1. Prior to the work of Manolescu and Marengon [19], the largest known torsion order was 2. Their work exhibits a knot with torsion order at least 3.…”

Section: Introductionmentioning

confidence: 99%

Knot cobordisms, bridge index, and torsion in Floer homology

Juhász

Miller

Zemke

2020

Journal of Topology

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Given a connected cobordism between two knots in the 3-sphere, our main result is an inequality involving torsion orders of the knot Floer homology of the knots, and the number of local maxima and the genus of the cobordism. This has several topological applications: The torsion order gives lower bounds on the bridge index and the band-unlinking number of a knot, the fusion number of a ribbon knot, and the number of minima appearing in a slice disk of a knot. It also gives a lower bound on the number of bands appearing in a ribbon concordance between two knots. Our bounds on the bridge index and fusion number are sharp for Tp,q and Tp,q#T p,q , respectively. We also show that the bridge index of Tp,q is minimal within its concordance class. The torsion order bounds a refinement of the cobordism distance on knots, which is a metric. As a special case, we can bound the number of band moves required to get from one knot to the other. We show that knot Floer homology also gives a lower bound on Sarkar's ribbon distance, and exhibit examples of ribbon knots with arbitrarily large ribbon distance from the unknot.

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Section: Introductionmentioning

confidence: 99%

Knot cobordisms, bridge index, and torsion in Floer homology

Juhász

Miller

Zemke

2020

Journal of Topology

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“…The success of the Khovanov polynomial in predicting the s-invariant is somewhat less surprising. The knight move conjecture [27], which is false [42], is satisfied by all the knots in our data set. It states Kh(K; q, −q −4 ) = q s (q + q −1 ) .…”

Section: Learning From Polynomial Invariantsmentioning

confidence: 69%

(K)not machine learning

Craven¹,

Hughes²,

Jejjala³

et al. 2022

Preprint

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“…It follows that the ranks of the cohomology groups are completely determined by the chromatic polynomial. We remark that the original knight move conjecture is false for Khovanov homology [13]. The reader is referred to [6,10] for some investigations on the torsion in chromatic cohomology.…”

Section: Introductionmentioning

confidence: 93%

A categorification for the signed chromatic polynomial

Cheng¹,

Lei²,

Wang³

et al. 2021

Preprint

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By coloring a signed graph by signed colors, one obtains the signed chromatic polynomial of the signed graph. For each signed graph we construct graded cohomology groups whose graded Euler characteristic yields the signed chromatic polynomial of the signed graph. We show that the cohomology groups satisfy a long exact sequence which corresponds to signed deletion-contraction rule. This work is motivated by Helme-Guizon and Rong's construction of the categorification for the chromatic polynomial of unsigned graphs.

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The Knight Move Conjecture is false

Cited by 12 publications

References 9 publications

Knot cobordisms, bridge index, and torsion in Floer homology

Knot cobordisms, bridge index, and torsion in Floer homology

(K)not machine learning

A categorification for the signed chromatic polynomial

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