Trace-free characters and abelian knot contact homology I
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| Format: | Preprint |
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2017
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| _version_ | 1866918344101199872 |
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| author | Nagasato, Fumikazu |
| author_facet | Nagasato, Fumikazu |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1708_00851 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Trace-free characters and abelian knot contact homology I Nagasato, Fumikazu Geometric Topology 57K31 (Primary), 57K18 (Secondary) 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. |
| title | Trace-free characters and abelian knot contact homology I |
| topic | Geometric Topology 57K31 (Primary), 57K18 (Secondary) |
| url | https://arxiv.org/abs/1708.00851 |