Kashaev's volume conjecture

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introduction

  • The hyperbolic volume of a knot complement can be calculated using the Jones polynimials of the ca
  • \(SU(2)\) connections on \(S^3-K\) should be sensitive to the flat \(SL_2(C)\) connection defining its hyperbolic structure
  • hyperbolic volume is closely related to the Cherm-Simons invariant
  • volume conjecture has its complexified version


Kashaev invariant

  • invariant of a link using the R-matrix
  • calculate the limit of the Kashaev invariant
  • related with the colored Jones polynomial

optimistic limit

  • volume conjecture
  • idea of the optimistic limit


examples

  • \(4_1\) figure eight knot
  • \(5_2\)
  • \(6_1\)


known examples

  • figure eight knot
  • Borromean ring
  • torus knots
  • whitehead chains
  • all links of zero volume
  • twist knows is (almost) done


history

  • 1995 Kashaev constructed knot invariants \(\langle K \rangle_N\)
  • 1997 Kashaev proposed that the asymptotic behaviour of the 1995 invariant involves the volume of the hyperbolic 3-manifold
  • 2001 [MM01] Murakami-Murakami found that \(\langle K \rangle_N\) can be obtained from evaluating the colored Jones polynomial at the \(N\)-th root of unity

related items


computational resource


encyclopedia


expositions


articles

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Spacy 패턴 목록

  • [{'LOWER': 'volume'}, {'LEMMA': 'conjecture'}]