📝 SummarXiv

Abstract

Let $K$ be the connected sum in $\mathbb{R}^3$ of an arbitrary knot and the $(2,q)$-torus knot for $q\neq \pm 1$, and consider its conormal bundle $L_K$ in $T^*\mathbb{R}^3$. We prove that there is no compactly supported Hamiltonian diffeomorphism $\varphi$ on $T^*\mathbb{R}^3$ such that $\varphi(L_K)$ and the zero section $\mathbb{R}^3$ intersect cleanly along the unknot in $\mathbb{R}^3$. For the proof, we examine the knot DGA, whose homology was proved to be isomorphic to the knot contact homology by Ekholm, Etnyre, Ng and Sullivan, with coefficients in $\mathbb{Z}[\lambda^{\pm},\mu^{\pm},U^{\pm}]$. We find an algebraic constraint imposed on every $\mathbb{Q}$-valued augmentation of the knot DGA of $K$ when $q\geq 3$. This constraint involves rational roots of polynomials in $\mathbb{Q}[T]$, and we apply Hilbert's irreducibility theorem to complete the proof. Using a similar idea, we also prove an analogous result replacing the $(2,q)$-torus knot with the figure-eight knot.