Research on representation theoretic structures of unknot detectors

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This study is concerned with researches on representation theoretic structures of unknot detectors. In particular, we focus on the geometric structures on the algebraic subset S0(K) of the character variety X(K) defined by trace-free condition. In the researches, we found a family of polynomials whose zero-locus gives the set S0(K). Applying the polynomials, we found a filtration of S0(K) each of whose filters gives a knot invariant. This fact reveals a relationship between S0(K) and the abelian knot contact homology. The methods used in these researches can be also applied to determine some minimal elements for a partial ordering of prime knots.