On a behavior of a slice of the SL2(C)-character variety of a knot group under the connected sum

AbstractWe observe a behavior of a slice (an algebraic subset) S0(K) of the SL2(C)-character variety of a knot group under the connected sum of knots. It turns out that the number of 0-dimensional components of S0(K) is additive under the connected sum of knots

A Diagrammatic Construction of the (sl (N,C),ρ)-Weight System (Special Section for Workshop of Topology in Sendai 2002)

In this paper, we first give a diagrammatic analogue of the Young symmetrizer. By using this, the (sl?(N,C),ρ)-weight system for an arbitrary finite-dimensional irreducible representation ρ is formulated in a diagrammatic way. The formula is useful for the calculations of the (sl?(N,C),ρ)-weight system in the sense that we do not need actual constructions of the representations of sl?(N,C) essentially. Hence by using this and the modified Kontsevich integral we can get the quantum (sl?(N,C),ρ)-invariant for any finite-dimensional irreducible representation without actual constructions of the representations of sl?(N,C). The diagrammatic construction is a generalization of the formula given in “Remarks on the (sl?(N,C),ad?)-weight system"