Crystallizations of compact 4-manifolds minimizing combinatorially defined PL-invariants

Abstract

The present paper is devoted to present a unifying survey about some special classes of crystallizations of compact PL $4$-manifolds with empty or connected boundary, called {\it semi-simple} and {\it weak semi-simple crystallizations}, with a particular attention to their properties of minimizing combinatorially defined PL-invariants, such as the {\it regular genus}, the {\it Gurau degree}, the {\it gem-complexity} and the {\it (gem-induced) trisection genus}. The main theorem, yielding a summarizing result on the topic, is an original contribution. Moreover, in the present paper the additivity of regular genus with respect to connected sum is proved to hold for all compact $4$-manifolds with empty or connected boundary which admit weak semi-simple crystallizations.

arXiv comment21 pages, 11 figures, 2 tables. arXiv admin note: text overlap with arXiv:1910.08777

References in corpus (8)

• Topology in colored tensor models via crystallization theory
• G-degree for singular manifolds
• Simple crystallizations of 4-manifolds
• 4-colored graphs and knot/link complements
• PL 4-manifolds admitting simple crystallizations: framed links and regular genus
• Computing trisections of 4-manifolds
• Combinatorial properties of the G-degree
• Reflections on trisection genus