Let $K$ be a hyperbolic knot and let $\rho_n$ be the $n$-dimensional irreducible representation induced from a lift of its holonomy representation. Motivated by Goda's asymptotic volume formula and the complexified Volume Conjecture, we study whether the higher-dimensional twisted Alexander invariants associated with $\rho_n$ detect the complex volume of the knot complement. We compute $$ \frac{\pi}{2} \log \left( \frac{A_{K,n-2}(1)A_{K,n+2}(1)} {A_{K,n}(1)^2} \right) $$ for all hyperbolic knots with at most six crossings. Our numerical experiments indicate that these values approach $$ \operatorname{Vol}(S^3\setminus K) +i\,2\pi^2\operatorname{CS}(S^3\setminus K) $$ modulo $i\pi^2\mathbb{Z}$. Based on these computations, we propose a complexified analogue of Goda's asymptotic volume formula.
Let $C_{p, 2}(\mathbb{P}^{n})$ denote the Chow variety parametrizing effective algebraic $p$-cycles of degree two on $\mathbb{P}^{n}$. We show that its rational cohomology is pure Tate and concentrated in even degrees. An anti-invariant Chern class identity yields a presentation of the cohomology ring as a fiber produc...
Let $\Sigma$ be a closed oriented surface of genus $>1$ and $M$ a complete hyperbolic 3-manifold with a marking $i:\Sigma\longrightarrow M$. We consider the case that $M$ has no parabolic cusps and at least one of the two ends is simply degenerate. For $\varGamma=\pi_1(\Sigma)$, let $\rho_M:\varGamma\longrightarrow \ma...
Let $X$ be a compact Riemann surface of genus $g\geq2$, and let $\M$ be the moduli space of rank-two trace-free Higgs bundles with fixed determinant of odd degree. For the normalization of the Hitchin metric used in this paper, we prove that the volume of geodesic ball in the Hitchin moduli space is given by $$ \Vol_{g...
For a finite abelian group $G$ acting linearly on the plane let $\mathcal{X}$ be the associated quotient stack. We analyze a fine moduli space which parameterizes zero dimensional substacks of $\mathcal{X}$ whose corresponding subschemes of the plane have a given representation type, denoted by $\mathcal{X}^{[\rho]}$....
For immersions of a closed $2h$-dimensional manifold $\Sigma$ into $\mathbb R^m$, we consider curvature energies that are invariant under ambient isometries and dilations, with the principal term $\int_{\Sigma} \big|\nabla^{(h-1)} \vec{\mathrm{I\!I}}\big|_g^2\,d\text{vol}_g$ and lower-order submanifold invariants. Foll...
Let $\Gamma<\operatorname{PSL}_2(\mathbb R)$ be a torsion-free, non-elementary, finitely generated Fuchsian group with parabolic elements, and let $\rho:\Gamma\to G$ be a type-preserving, Zariski dense, positive representation into a connected split semisimple real algebraic group. We prove quantitative local mixing fo...
Dongryul M. Kim, Hee Oh, Wen-Yu Pan· 0 citations
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