Author Topic: 8** is the first ordered polyhedron knot of different genus  (Read 2431 times)

Stagehand

  • Full Member
  • ***
  • Posts: 29
8** is the first ordered polyhedron knot of different genus
« on: September 29, 2014, 11:32:23 PM »
8** is the first ordered polyhedron knot of different genus.  Under the present ordering of knots and links, 8** will be found represented as a four-link knot of 12 non-alternating crossings.  Its saturated over-under patterning is not expressible on a flat page but only on a toroid.  This lack of true expression in the current tables is the case for all polyhedra knots of different genus and is the case for all knots seen as such degenerate polyhedra.  This could account for all non-alternating knots and links.  The first ordered single-line polyhedra of different genus is 11D, a permutation of joining the rings of 8** without degeneracy (twisting).
« Last Edit: September 30, 2014, 08:58:23 PM by Stagehand »

KnotMe

  • Sr. Member
  • *****
  • Posts: 570
    • The Dao of Silk
Re: 8** is the first ordered polyhedron knot of different genus
« Reply #1 on: September 30, 2014, 02:46:02 AM »
Care to diagram that in a few steps for those of us who are slow?   :o

Stagehand

  • Full Member
  • ***
  • Posts: 29
Re: 8** is the first ordered polyhedron knot of different genus
« Reply #2 on: September 30, 2014, 09:02:22 PM »
Thanks Knotme for your interest.  These objects generate a series like the antiprisms, so after the fashion of Kepler we could call them "toroiprisms."  They are unquitting generators of two lines with one line always crossing over on the top and bottom edge, and the other line always crossing over on the inside and the outside.  Simply involve the two lines anywhere, but as a personal choice I would not simply twist or drop a crossing.
« Last Edit: September 30, 2014, 10:31:46 PM by Stagehand »