The fact that every smooth knot type admits a Legendrian representative is a classical result in contact topology. However, the analogous surjectivity question was open at the parametric level. In this work we address the $n>1$ case. We prove that for every $n\geq 3$, every knot type $\mathcal K$, every Legendrian representative $\mathcal L$ and every formal Legendrian representative $\mathcal{FL}$, the associated group homomorphisms $\pi_n(\mathcal{L})\to\pi_n(\mathcal{K})$ and $\pi_n(\mathcal{FL})\to\pi_n(\mathcal{K})$ are never surjective. We then show that surjectivity at the $\pi_2$-level depends on the knot type. This work thus proves the presence of rigidity for parametric families at every higher homotopy level beyond $\pi_1$.
It is shown that every locally flatly embedded genus $g \in \{1,2\}$ surface in the $4$-sphere with knot group $\mathbb{Z}$ is unknotted. The same proof establishes that any two genus $g \in \{1,2\}$ surfaces in $D^4$ with knot group $\mathbb{Z}$ and common boundary an Alexander polynomial one knot are isotopic rel. bo...
A classical theorem of Maillet asserts that every nonconstant rational function over $\mathbb{Q}$ maps Liouville numbers to Liouville numbers. In 1984, Mahler asked whether a transcendental entire function can have the same property. We prove a strong smooth counterpart: writing $\mathscr{L}$ for the set of Liouville n...
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 $...
Let $K\subset S^3$ be a non-trivial knot and let $\mathrm{IS}(K)$ be the simplicial complex whose vertices are the ambient isotopy classes of incompressible Seifert surfaces in the exterior $E(K)$, a finite set of distinct vertices spanning a simplex exactly when its classes admit simultaneously pairwise disjoint repre...
Guan-Cheng Pan, Cheng-Song You, Jun-Wei Zhou et al.· 0 citations
Let $\Omega\subsetneq\mathbb C$ be a non-empty simply connected domain and $0<p<2$. Bouchala and Koskela [Proc. Amer. Math. Soc. (2024)] proved that there exists a quasiconformal map $f:\mathbb D\to\Omega$ such that $\int_0^1\ell(f(r S^1))^p\,dr<\infty$, and asked whether this conclusion holds at $p=2$, or even for eve...
In this paper we consider area-preserving twist maps $f:\mathbf{T}^2\to\mathbf{T}^2$ with vertical rotation sets reduced to a single irrational number $\alpha$. Our first result states that under these hypotheses, there exists a unique $f$-invariant foliation $\mathcal{F}_G=\{\operatorname{Graph}(\phi_t)\}_{t\in\mathbf...
S. A. Zanata, Julian Lazaro Aguirre· 1 citation
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