# 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
• 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