On the Question of Genericity of Hyperbolic Knots

Malyutin, A. V.

doi:10.1093/imrn/rny220

Cited by 13 publications

(26 citation statements)

References 114 publications

(91 reference statements)

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“…Now, we deduce Theorem 1 from these propositions. The argument is similar to that of Proposition 3.6 in [Mal18].…”

Section: The Idea Of the Proof Of Theoremsupporting

confidence: 66%

“…Definition 2 (Regular knots; see [Mal18]). If P is a knot and x is a real number, we say that P is x-regular if we have x • cr(P ) ≤ cr(K) whenever P is a factor of a knot K. If there exists a knot K such that P is a factor of K and cr(K) < x•cr(P ), we say that P is non-x-regular.…”

Section: The Idea Of the Proof Of Theoremmentioning

confidence: 99%

“…Wellknown examples here are hyperbolic surfaces and pseudo-Anosov automorphisms in mapping class groups of surfaces. See [Mal18] for more details and references.…”

Section: Introductionmentioning

confidence: 99%

“…The present paper deals with the case of links. Statistics show that the overwhelming majority of prime knots and links of small complexity is of hyperbolic type (see [HTW98,Bur18,Mal18]), and it was widely believed for some time that generic prime knots and links are hyperbolic (see, e. g., [Ad94,Rat06]). However, in the cases of knots, links, and 3-manifolds there is a counter-argument to the conjecture that hyperbolic elements are generic.…”

Section: Introductionmentioning

confidence: 99%

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Hyperbolic links are not generic

Malyutin

2019

Preprint

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We show that if K is a nontrivial knot then the proportion of satellites of K among all of the prime non-split links of n or fewer crossings does not converge to 0 as n approaches infinity. This implies in particular that the proportion of hyperbolic links among all of the prime non-split links of n or fewer crossings does not converge to 1 as n approaches infinity. We consider unoriented link types.

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“…Now, we deduce Theorem 1 from these propositions. The argument is similar to that of Proposition 3.6 in [Mal18].…”

Section: The Idea Of the Proof Of Theoremsupporting

confidence: 66%

Section: The Idea Of the Proof Of Theoremmentioning

confidence: 99%

“…Wellknown examples here are hyperbolic surfaces and pseudo-Anosov automorphisms in mapping class groups of surfaces. See [Mal18] for more details and references.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Hyperbolic links are not generic

Malyutin

2019

Preprint

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“…That is, given two knots K 1 and K 2 of minimal crossing numbers c(K 1 ) and c(K 2 ) respectively, is it true that c(K 1 #K 2 ) = c(K 1 + K 2 )? A positive answer to this question would not only help the understanding of this most fundamental knot invariant, but also contradict other conjectures, for example that the percentage of hyperbolic knots among all prime knots of minimal crossing number at most n approaches 100 as n goes to infinity [7].…”

Section: Introductionmentioning

confidence: 93%

Crossing numbers of composite knots and spatial graphs

Bode¹

2018

Topology and its Applications

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We study the minimal crossing number c(K 1 #K 2 ) of composite knots K 1 #K 2 , where K 1 and K 2 are prime, by relating it to the minimal crossing number of spatial graphs, in particular the 2ntheta-curve θ n K 1 ,K 2 that results from tying n of the edges of the planar embedding of the 2n-theta-graph into K 1 and the remaining n edges into K 2 . We prove that for large enough n we have c(θ n K 1 ,K 2 ) = n(c(K 1 )+c(K 2 )). We also formulate additional relations between the crossing numbers of certain spatial graphs that, if satisfied, imply the additivity of the crossing number or at least give a lower bound for c(K 1 #K 2 ). arXiv:1709.05118v2 [math.GT]

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A table of n-component handlebody links of genus n+1 up to six crossings

Bellettini¹,

Paolini²,

Paolini³

et al. 2022

Math. Proc. Camb. Phil. Soc.

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A handlebody link is a union of handlebodies of positive genus embedded in 3-space, which generalises the notion of links in classical knot theory. In this paper, we consider handlebody links with a genus two handlebody and $n-1$ solid tori, $n>1$ . Our main result is the classification of such handlebody links with six crossings or less, up to ambient isotopy.

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On the Question of Genericity of Hyperbolic Knots

Cited by 13 publications

References 114 publications

Hyperbolic links are not generic

Hyperbolic links are not generic

Crossing numbers of composite knots and spatial graphs

A table of n-component handlebody links of genus n+1 up to six crossings

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