Publications (4)0 Total impact
-
[show abstract]
[hide abstract]
ABSTRACT: In this paper, we use Chebyshev polynomials to give presentations of the
character varieties of certain types of 2-bridge knots. This gives us an
elementary method using basic calculations to discuss the number of irreducible
components of the character varieties and thus to recover the results of Burde
on the irreducibility of non-abelian SU(2)-representation spaces in [2]. These
results can be applied to determine some minimal elements of a partial ordering
of prime knots.
01/2013;
-
[show abstract]
[hide abstract]
ABSTRACT: Let K be a knot in an integral homology 3-sphere and let B denote the 2-fold
branched cover of the integral homology sphere branched along K. We construct a
map from the slice of characters with trace free along meridians in the SL(2,
C)-character variety of the knot exterior to the SL(2, C)-character variety of
2-fold branched cover B. When this map is surjective, it describes the slice as
the 2-fold branched cover over the SL(2, C)-character variety of B with
branched locus given by the abelian characters, whose preimage is precisely the
set of metabelian characters. We show that each of metabelian character can be
represented as the character of a binary dihedral representation of the knot
group. This map is shown to be surjective for all 2-bridge knots and all
pretzel knots of type (p, q, r). An extension of this framework to n-fold
branched covers is also described.
07/2008;
-
Fumikazu Nagasato
[show abstract]
[hide abstract]
ABSTRACT: We show that for any knot there exist only finitely many irreducible metabelian characters in the $SL(2,\mathbb{C})$-character variety of the knot group, and the number is given explicitly by using the determinant of the knot. Then it turns out that for any 2-bridge knot a section of the $SL(2,\mathbb{C})$-character variety consists entirely of all the metabelian characters, i.e., the irreducible metabelian characters and the single reducible (abelian) character. Moreover we find that the number of irreducible metabelian characters gives an upper bound of the maximal degree of the A-polynomial in terms of the variable $l$.
11/2006;
-
Fumikazu Nagasato
[show abstract]
[hide abstract]
ABSTRACT: 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”.
