The work of J{\o}rgensen and Thurston shows that there is a finite number
N(v) of orientable hyperbolic 3-manifolds with any given volume v. In this
paper, we construct examples showing that the number of hyperbolic knot
complements with a given volume v can grow at least factorially fast with v. A
similar statement holds for closed hyperbolic 3-manifolds, obtained via Dehn
surgery. Furthermore, we give explicit estimates for lower bounds of N(v) in
terms of v for these examples. These results improve upon the work of Hodgson
and Masai, which describes examples that grow exponentially fast with v. Our
constructions rely on performing volume preserving mutations along Conway
spheres and on the classification of Montesinos knots.Comment: 13 pages, 6 figure

Millichap, Christian

English

arXiv

The work of Jørgensen and Thurston shows that there is a finite number N(v) of orientable hyperbolic 3-manifolds with any given volume v. In this paper, we construct examples showing that the number of hyperbolic knot complements with a given volume v can grow at least factorially fast with v. A similar statement holds for closed hyperbolic 3-manifolds, obtained via Dehn surgery. Furthermore, we give explicit estimates for lower bounds of N(v) in terms of v for these examples. These results improve upon the work of Hodgson and Masai, which describes examples that grow exponentially fast with v. Our constructions rely on performing volume preserving mutations along Conway spheres and on the classification of Montesinos knots

DigitalCommons@Linfield

Factorial Growth Rates for the Number of Hyperbolic 3-Manifolds of a Given Volume

The work of Jørgensen and Thurston shows that there is a finite number 

  
    
      N
      (
      v
      )
    
    N(v)
  

 of orientable hyperbolic 

  
    3
    3
  

-manifolds with any given volume 

  
    v
    v
  

. In this paper, we construct examples showing that the number of hyperbolic knot complements with a given volume 

  
    v
    v
  

 can grow at least factorially fast with 

  
    v
    v
  

. A similar statement holds for closed hyperbolic 

  
    3
    3
  

-manifolds, obtained via Dehn surgery. Furthermore, we give explicit estimates for lower bounds of 

  
    
      N
      (
      v
      )
    
    N(v)
  

 in terms of 

  
    v
    v
  

 for these examples. These results improve upon the work of Hodgson and Masai, which describes examples that grow exponentially fast with 

  
    v
    v
  

. Our constructions rely on performing volume preserving mutations along Conway spheres and on the classification of Montesinos knots.</p

Factorial growth rates for the number of hyperbolic 3-manifolds of a given volume

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