A 6-Letter ‘DNA’ for Baskets with Handles

James Mallos
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James Mallos: Sculptor, Washington, DC 20002, USA

Mathematics, 2019, vol. 7, issue 2, 1-14

Abstract: Fabric surfaces, made using techniques such as crochet and net-making, are typically worked in a linear order that meanders, without crossing itself, to ultimately visit and build the entire surface. For a closed basket, whose surface is a topological sphere, it is known that the construction can be described by a codeword on a 4-letter alphabet via Mullin’s encoding of plane graphs. Mullin’s code exemplifies the formal language known as the Shuffled Dyck Language with 2 Types of Parenthesis ( S D L 2 ). Besides its 4-letter alphabet, S D L 2 has some other similarities to DNA: Any word can be ‘evolved’ via a sequence of local mutations (rewriting rules), and ‘gene-splicing’ two S D L 2 words, by an insertion or concatenation, produces another S D L 2 word. However, S D L 2 comes up short when we attempt to make a basket with handles. I show that extending the language to S D L 3 , by addition of a third type of parenthesis, succeeds for orientable surfaces with handles—provided an appropriate choice of cut graph is made.

Keywords: Mullin encoding; shuffled Dyck language; chord diagrams; basket making; crochet; basket map; A-trail; non-crossing Eulerian circuit; topological surface; polygonal schema (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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