Three-Dimensional Small Covers and Links

TL;DR

The paper investigates orientation-preserving involutions $g$ in $\mathbb{Z}_2^3$ acting on orientable 3D small covers $X(P,\lambda)$. It proves the orbit space $X/g$ is homeomorphic to $\#_{k-1} S^2 \times S^1$, with $k$ counting edges not labeled by $g$, and shows that $X/g \cong S^3$ precisely when the characteristic function $\lambda$ is Hamiltonian, making $X$ a 2-fold branched covering of $S^3$ along a link $L$ described by a Hamiltonian cycle on $P$; this link is encoded combinatorially by a bipartite chord diagram $D_C$ and can be realized as the preimage of non-cycle edges under the branch cover. The work establishes a tight link between combinatorial data $(P,\lambda)$, topological branched coverings, and the resulting links in $S^3$, and discusses rigidity, invariants, and open questions about decompositions and equivalences of these structures. It also connects Hamiltonian colorings to hyperelliptic (or 2-fold branched) structures and provides explicit examples illustrating the branching sets (e.g., Hopf link for $\mathbb{R}P^3$, $L8n8$ on the cube).

Abstract

We study certain orientation-preserving involutions on three-dimensional small covers. We prove that the quotient space of an orientable three-dimensional small cover by such an involution in $\mathbb{Z}_2^3$ is homeomorphic to a connected sum of copies of $S^2 \times S^1$. If this quotient space is a 3-sphere, then the corresponding small cover is a two-fold branched covering of the 3-sphere along a link. We provide a description of this link in terms of the polytope and the characteristic function.

Figures (19)

• Figure 1: The Hamiltonian cycle in $\Delta^3$.
• Figure 2: Two cycles in $I^3$.
• Figure 3: The Hamiltonian cycle in $I^3$.
• Figure : The Hamiltonian cycle in $P$.
• Figure : The link $L_1$

Theorems & Definitions (60)

• Theorem 1
• Theorem 2
• Example 1.1
• Example 1.2
• Definition 2.1
• Theorem 2.2: DJ
• Theorem 2.3: SmallCovers
• Proposition 2.4
• proof
• Remark 2.5