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• W3102443920 abstract "The work of Jørgensen and Thurston shows that there is a finite number <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N left-parenthesis v right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>N(v)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of orientable hyperbolic <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=3> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=application/x-tex>3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-manifolds with any given volume <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=v> <mml:semantics> <mml:mi>v</mml:mi> <mml:annotation encoding=application/x-tex>v</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this paper, we construct examples showing that the number of hyperbolic knot complements with a given volume <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=v> <mml:semantics> <mml:mi>v</mml:mi> <mml:annotation encoding=application/x-tex>v</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can grow at least factorially fast with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=v> <mml:semantics> <mml:mi>v</mml:mi> <mml:annotation encoding=application/x-tex>v</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A similar statement holds for closed hyperbolic <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=3> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=application/x-tex>3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-manifolds, obtained via Dehn surgery. Furthermore, we give explicit estimates for lower bounds of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N left-parenthesis v right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>N(v)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=v> <mml:semantics> <mml:mi>v</mml:mi> <mml:annotation encoding=application/x-tex>v</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for these examples. These results improve upon the work of Hodgson and Masai, which describes examples that grow exponentially fast with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=v> <mml:semantics> <mml:mi>v</mml:mi> <mml:annotation encoding=application/x-tex>v</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our constructions rely on performing volume preserving mutations along Conway spheres and on the classification of Montesinos knots." @default.
• W3102443920 created "2020-11-23" @default.
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• W3102443920 date "2015-01-16" @default.
• W3102443920 modified "2023-09-25" @default.
• W3102443920 title "Factorial growth rates for the number of hyperbolic 3-manifolds of a given volume" @default.
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