[Paper Review] Flip-connectivity of triangulations of the product of a tetrahedron and simplex

Gaku Liu|arXiv (Cornell University)|Jan 22, 2016

graph theory and CDMA systems4 references4 citations

TL;DR

This paper proves that the set of triangulations of the product of a 3-simplex (tetrahedron) and an n-simplex is flip-connected, extending Santos' earlier result for the 2-simplex case. Using a novel algorithm based on quasiorders on simplices and sequential flips on specific circuits, the authors show any triangulation can be transformed into a fixed reference triangulation via a finite sequence of flips, establishing flip-connectivity for $Δ^3 \times \Delta^n$. This resolves a key case in the broader open problem of flip-connectivity for products of simplices.

ABSTRACT

A flip is a minimal move between two triangulations of a polytope. An open question is whether any two triangulations of the product of two simplices can be connected through a series of flips. This was proven in the case where one of the simplices is a triangle by Santos in 2005. In this paper we extend this to when one of the simplices is a tetrahedron.

Motivation & Objective

• To resolve the flip-connectivity problem for triangulations of $Δ^3 \times \Delta^n$, extending Santos' result for $Δ^2 \times \Delta^n$.
• To establish that any two triangulations of $Δ^3 \times \Delta^n$ can be connected via a finite sequence of flips.
• To develop a constructive algorithm that transforms any given triangulation into a fixed reference triangulation through systematic flip operations.

Proposed method

• Define a quasiorder on simplices in a triangulation to identify a special circuit in $Δ^3 \times \Delta^{n-1}$ that enables flip operations.
• Identify a set $τ$ of simplices containing negative elements of a circuit $X$, and isolate those simplices affected by potential flips.
• Apply a sequence of flips that only affect simplices in $τ$, ensuring the circuit $X$ becomes flippable.
• Use link invariance and triangulation properties to verify that each flip operation preserves the triangulation structure.
• Leverage order relations $<_{ii'}$ on labels to track changes in the quasiorder and detect contradictions when a flip is not supported.
• Prove by contradiction that each candidate circuit admits a flip, ultimately showing that a full sequence of flips can transform any triangulation into a fixed one.

Experimental results

Research questions

• RQ1 Can any two triangulations of $Δ^3 \times \Delta^n$ be connected via a sequence of flips, as is known for $Δ^2 \times \Delta^n$?
• RQ2 Is there a constructive algorithm to transform an arbitrary triangulation of $Δ^3 \times \Delta^n$ into a canonical triangulation using only flips?
• RQ3 What structural properties of the product polytope $Δ^3 \times \Delta^n$ ensure flip-connectivity despite the complexity of its triangulations?

Key findings

• The set of triangulations of $Δ^3 \times \Delta^n$ is flip-connected, meaning any two triangulations can be connected by a finite sequence of flips.
• A constructive algorithm is developed that transforms any triangulation into a fixed reference triangulation through a sequence of circuit-based flips.
• The proof relies on identifying a special circuit in $Δ^3 \times \Delta^{n-1}$ and using a quasiorder on simplices to guide the flip sequence.
• Each flip is supported on a circuit $(X^+, X^-)$ only when all maximal simplices of $τ_{X^+}$ share the same link in the triangulation.
• The method ensures that after each flip, the triangulation remains valid and the process eventually reaches the target triangulation.
• The result confirms flip-connectivity for the 3-simplex case, completing a key step toward resolving the broader open problem for products of simplices.