Representations of the quantum groups at roots of unity and its application to knot theory

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We did research rerated to the ‘Volume Conjecture’ which gives a relation between the quantum invariant related to the quantum group Uq(sl2), and, by noticing the non-semisimple representation of it, such representations and the related knot invariants were studied. By applying the volume conjecture, the geometric structure of a knot complement and the volumes of polyhedrons in a three space with a constant curvature were studied. Moreover, new invariants of knots in a three manifolds were constructed from the non-semisimple representations.