Practical Software for Triangulating and Simplifying 4-Manifolds

Rhuaidi Antonio Burke

Home/Archive/arXiv

arXiv · 2402.15087

Practical Software for Triangulating and Simplifying 4-Manifolds

Rhuaidi Antonio Burke

View original source ↗Read source document ↗Export metadata ↓

Abstract

Dimension 4 is the first dimension in which exotic smooth manifold pairs appear -- manifolds which are topologically the same but for which there is no smooth deformation of one into the other. Whilst smooth and triangulated 4-manifolds do coincide, comparatively little work has been done towards gaining an understanding of smooth 4-manifolds from the discrete and algorithmic perspective. In this paper we introduce new software tools to make this possible, including a software implementation of an algorithm which enables us to build triangulations of 4-manifolds from Kirby diagrams, as well as a new heuristic for simplifying 4-manifold triangulations. Using these tools, we present new triangulations of several bounded exotic pairs, corks and plugs (objects responsible for "exoticity"), as well as the smallest known triangulation of the fundamental K3 surface. The small size of these triangulations benefit us by revealing fine structural features in 4-manifold triangulations.

Explore related subjects

math.GT cs.CG

Keep this discovery

BibTeX RIS

Rhuaidi Antonio Burke. 2024-02-23. Practical Software for Triangulating and Simplifying 4-Manifolds. https://arxiv.org/abs/2402.15087

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Bar cohomology of links: beyond Milnor invariants

We develop bar cohomology of link complements as an invariant of links in homology spheres. In this setting, bar cohomology is a Hopf algebra which is calculable using surfaces and their intersection curves in a link complement. In this first in a sequence of works, we introduce the invariant and show that it defines a canonical subspace of the tensor Hopf algebra, which already encodes information about Milnor's link invariants and provides geometrically significant information beyond them.

math.GT↗

arXiv2026

A counterexample to the wrapping number conjecture

We exhibit an annular knot with wrapping number four whose Kauffman bracket has annular degree at most two. This disproves the wrapping number conjecture.

math.GT↗

arXiv2026

Homological lifts of Arnold invariants $J^-$ and $J^+$

Viro's Euler-integral polynomial $P_C(q)$ and the Lanzat--Polyak quantized-curvature polynomial $I_q(C)$ refine Arnold's invariants $J^-$ and $J^+$ for generic immersed one-component plane curves. We construct homological lifts of both. The bigraded region homology retains the singular homology of every connected Alexander-index region; its graded Euler characteristic is $P_C(q)$. The triply graded smoothing-circle homology is generated by the oriented circles of the orientation-preserving smoothing and decategorifies to the smoothing term in $I_q(C)$. Keeping the actual region summands and the boundary regions of every smoothing circle gives a homological refinement of the oriented smoothing configuration, or Seifert state. An infinite family proves strictness: both polynomial data and the ordinary homological lifts agree, while the component-graded region homology and the branch-decomposed circle homology distinguish every pair. Further constructions recover the full $I_q(C)$ by a vertex complex, realize the local change of its curvature integral by edge homology, and give a canonical two-state homology for unoriented curves. Viro described his Euler-integral formula as an analogue of face state-sum formulas for quantum knot polynomials. Through the categorifications developed here, we obtain one concrete homological face-state-sum model realizing that analogy.

math.GT↗