We present two practical and widely applicable methods, including some criteria and a general procedure, for detecting Brunnian property of a link, if each component is known to be unknot. The methods are based on observation and handwork. They are used successfully for all Brunnian links known so far. Typical examples and extensive experiments illustrate their efficiency. As an application, infinite families of Brunnian links are created and we establish a general way to construct new ones in bulk.
Suppose an orientation preserving action of a finite group G on the closed surface Σg of genus g > 1 extends over the 3-torus T 3 for some embedding Σg ⊂ T 3 . Then |G| ≤ 12(g − 1), and this upper bound 12(g − 1) can be achieved for g = n 2 + 1, 3n 2 + 1, 2n 3 + 1, 4n 3 + 1, 8n 3 + 1, n ∈ Z+. Those surfaces in T 3 realizing the maximum symmetries can be either unknotted or knotted. Similar problems in non-orientable category is also discussed.Connection with minimal surfaces in T 3 is addressed and when the maximum symmetric surfaces above can be realized by minimal surfaces is identified.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.