The 6-ring is a tubular surface of genus 13, obtained by gluing together twenty-four 12-gons which follows the regularity of the map R13.2′{12,3}. It is constructed from six rings of two Borromean rings, has the twenty-four elements of an oriented cube and matches nicely with the 6-coloring of R13.2′{12,3}. The 6-ring has minimal twists and geometrically equivalent sets of four 12-gons. We explicitly construct this highly symmetric surface from two Borromean rings together with a detailed mapping of the 12-gons.
http://page.mi.fu-berlin.de/faniry/files/the6Ring.pdf
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