2011/08/19 by Karoline P. Null, Null, Karoline P.
Computer Science · Mathematics · #20B10 #22C05 #22F05 #22F30 #22F50 #57M05 #57M27 #57M60 #57N65 #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #General Topology (math.GN) #Geometric Topology (math.GT) #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1108.3885
openalex publication_date 2011/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We demonstrate how a 3-manifold, a Heegaard diagram, and a group presentation can each be interpreted as a pair of signed permutations in the symmetric group Sd. We demonstrate the power of permutation data in programming and discuss an algorithm we have developed that takes the permutation data as input and determines whether the data represents a closed 3-manifold. We therefore have an invariant of groups, that is given any group presentation, we can determine if that fixed presentation presents a closed 3-manifold. (The proposed techniques begin with a pair of signed permutations and builds a finite group presentation. The finite group presentation results in a finite class of associated 3-manifolds. Notice that a negative answer only implies the fixed presentation does not result in a closed 3-manifold under this construction, but says nothing about an isomorphic form of the group presentation.)