The volume conjecture for augmented links

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The volume conjecture for augmented links

• Roland van der Veen
• University of Amsterdam

Hanoi, 9-8-2007
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The volume conjecture (Kashaev, Murakami,
Murakami)

• Let L be a link and JN(L)(q) its normalized
• colored Jones polynomial.

For split links it is false, for some non-split
links it only holds on a subsequence (say odd N)
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Theorem (RvdV)

• Let L be a link. One can add u mutually unlinked
  unknots to L
• such that the resulting augmented link A
  satisfies
• The torus decomposition of the complement of A
  consists of
• Seifert fibered pieces and a hyperbolic link
  complement B that is
• made up from 2c regular ideal octahedra.
• Corollary The volume conjecture holds for
  augmented links A.

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Example
Computation of the unnormalized colored Jones
polynomial of the link L
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For odd N we find
So the volume conjecture suggests Vol(A)
4Vol(Oct)
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The complement
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Total 2 x 2 octahedra so the volume conjecture
holds
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Thank you for your attention

• This presentation will be available at
• http//www.science.uva.nl/riveen