A163566 - OEIS

%I #7 Dec 23 2013 13:33:03

%S 10,972,85022,8077072

%N Total number of grid diagrams of nontrivial knots on grid size n.

%C From page 3 of Baldridge/McCarthy reference.

%C Note that creating grid diagrams with sizes above 8 is time intensive and the computation to rule out unknot grid diagrams grows quickly beyond the capabilities of most computers. Baldridge and McCarty highlight some of the differences between cube diagrams and grid diagrams, list examples of small cube diagrams for all knots up to 7 crossings, and give some examples of links.

%H Scott Baldridge, Ben McCarty, <a href="http://arxiv.org/abs/0907.5401">Small examples of cube diagrams of knots</a>, July 30, 2009.

%e The table on p.3 may be shown, where b(n) = Number that lift to cube diagrams:

%e ==========================================

%e .n.|..a(n)...|...b(n)..|.b(n) as % of a(n)

%e ...|.........|.........|.................

%e .5.|......10.|.......3.|.30%.............

%e .6.|.....972.|.....261.|.27%.............

%e .7.|...85022.|...19722.|.23%.............

%e .8.|.8077072.|.1589447.|.19.7%...........

%e ==========================================

%K nonn

%O 5,1

%A _Jonathan Vos Post_, Jul 30 2009