Cubical complexes

2002 ◽  
pp. 49-55
Author(s):  
Victor Buchstaber ◽  
Taras Panov
Keyword(s):  
Cubical Complexes

scholarly journals Topology of random -dimensional cubical complexes

2021 ◽  
Vol 9 ◽  
Author(s):  
Matthew Kahle ◽  
Elliot Paquette ◽  
Érika Roldán
Keyword(s):  
Fundamental Group ◽  
Random Graphs ◽  
High Probability ◽  
Cubical Complex ◽  
Sharp Threshold ◽  
Cubical Complexes ◽  
Simple Connectivity ◽  
Dimensional Cube ◽  
Dimensional Face ◽  
Connectivity Threshold

Abstract We study a natural model of a random $2$ -dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$ -face is included independently with probability p. Our main result exhibits a sharp threshold $p=1/2$ for homology vanishing as $n \to \infty $ . This is a $2$ -dimensional analogue of the Burtin and Erdoős–Spencer theorems characterising the connectivity threshold for random graphs on the $1$ -skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial–Meshulam theorem for random $2$ -dimensional simplicial complexes. However, the models exhibit strikingly different behaviours. We show that if $p> 1 - \sqrt {1/2} \approx 0.2929$ , then with high probability the fundamental group is a free group with one generator for every maximal $1$ -dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold, even in the strong ‘hitting time’ sense. This is in contrast with the simplicial case, where the thresholds are far apart. The proof depends on an iterative algorithm for contracting cycles – we show that with high probability, the algorithm rapidly and dramatically simplifies the fundamental group, converging after only a few steps.

scholarly journals The Tits Alternative for Cat(0) Cubical Complexes

2005 ◽  
Vol 37 (5) ◽  
pp. 706-710 ◽  
Author(s):  
Michah Sageev ◽  
Daniel T. Wise
Keyword(s):  
Cubical Complexes ◽  
Tits Alternative

scholarly journals Geodesics in CAT(0) cubical complexes

2012 ◽  
Vol 48 (1) ◽  
pp. 142-163 ◽  
Author(s):  
Federico Ardila ◽  
Megan Owen ◽  
Seth Sullivant
Keyword(s):  
Cubical Complexes

scholarly journals The optimal number of faces in cubical complexes

1971 ◽  
Vol 8 (3) ◽  
pp. 245-257 ◽  
Author(s):  
Bernt Lindström
Keyword(s):  
Optimal Number ◽  
Cubical Complexes ◽  
Number Of Faces

Multiplexer logic circuit design using cubical complexes

1980 ◽  
Vol 16 (13) ◽  
pp. 495
Author(s):  
A.B. Ektare ◽  
D.P. Mital
Keyword(s):  
Circuit Design ◽  
Logic Circuit ◽  
Cubical Complexes
Sign in / Sign up

Export Citation Format

Share Document