arXiv subjects

Fumikazu Nagasato

Publications and source records attributed to Fumikazu Nagasato.

7 recordsLinked to original sources

arXiv2026

Trace-free characters and abelian knot contact homology I

We study the structure underlying Ng's conjecture, which relates the degree $0$ abelian knot contact homology of a knot $K$ to the coordinate ring of the $SL_2(\mathbf{C})$-character variety $X(Σ_2 K)$ of the $2$-fold branched cover of the $3$-sphere branched along $K$. Our approach is based on the study of (meridionally) trace-free characters of knot groups. For each knot $K$, they form a closed algebraic subset $S_0(K)$ of the $SL_2(\mathbf{C})$-character variety of $K$, defined by the trace-free condition on meridians. The subset $S_0(K)$, called the trace-free slice of $K$, has a natural connection to $X(Σ_2K)$. We show that the trace-free slice admits the structure of a $2$-fold branched cover of a closed algebraic set, called the fundamental variety, whose coordinate ring coincides with the nilradical quotient of the complexification of degree $0$ abelian knot contact homology. Using this framework, we introduce the notion of \emph{ghost characters} and prove that Ng's conjecture holds for a knot $K$ if and only if $K$ admits no ghost characters. This criterion establishes Ng's conjecture for all 2-bridge and 3-bridge knots.

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arXiv2026

Trace-free characters and abelian knot contact homology II

We show that the $(4,5)$- and $(5,6)$-torus knots admit ghost characters. Consequently, these knots provide counterexamples to Ng's conjecture, which proposes an isomorphism between the complexification of degree $0$ abelian knot contact homology and the coordinate ring of the character variety of the $2$-fold branched cover of the $3$-sphere branched along a knot. While Ng's conjecture has been verified for all $2$-bridge and $3$-bridge knots, we demonstrate, via ghost characters, how this isomorphism fails for these torus knots.

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arXiv2025

The ghost character of the (4,5)-torus knot and its applications

We show that the (4,5)-torus knot $T_{4,5}$ admits exactly one ghost character. We then show that this ghost character provides the following two important results. (1) It is known that for any knot $K$ every (meridionally) trace-free $\SL_2(\C)$-representation of the knot group $G(K)$ yields an $\SL_2(\C)$-representation of the fundamental group $π_1(Σ_2K)$ of the 2-fold branched cover $Σ_2K$ of the 3-sphere along $K$. This correspondence often but not always provides all $\SL_2(\C)$-representations of $π_1(Σ_2K)$. We show by using the ghost character that $T_{4,5}$ is the simplest torus knot such that $π_1(Σ_2T_{4,5})$ admits an $\SL_2(\C)$-representation which cannot be realized by any trace-free $\SL_2(\C)$-representations. (2) We show that $T_{4,5}$ is the simplest torus knot that provides a counterexample to Ng's conjecture, concerned with a polynomial map $h^*$ between the character variety $X(Σ_2K)$ of $π_1(Σ_2K)$ and the fundamental variety $F_2(K)$. More precisely, the map $h^*$ is surjective but not injective, and hence not an isomorphism for $T_{4,5}$.

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arXiv2016

On minimality of two-bridge knots

A knot is called minimal if its knot group admits epimorphisms onto the knot groups of only the trivial knot and itself. In this paper, we determine which two-bridge knot $\mathfrak{b}(p,q)$ is minimal where $q \leq 6$ or $p \leq 100$.

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arXiv2014

Some families of minimal elements for the partial ordering on prime knots

We show that all twist knots, certain double twist knots and some other 2-bridge knots are minimal elements for the partial ordering on the set of prime knots. The key to these results are presentations of their character varieties using Chebyshev polynomials and a criterion for irreducibility of a polynomial of two variables. These give us an elementary method to discuss the number of irreducible components of the character varieties, which concludes the result essentially.

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arXiv2011

On the geometry of the slice of trace--free SL(2,C)-characters of a knot group

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.

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arXiv2006

Finiteness of a section of the $SL(2,\mathbb{C})$-character variety of knot groups

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 $l$.

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