Unoriented Prime Alternating Links

Below is a table listing the number of prime alternating links for each component/crossing number combination. The crossings numbers are in the left column and the components are in the top row.

123456789101112Sum
211
311
4112
5213
63328
776114
818146139
9414212196
101231214391297
11367384146171915
121,2881,4085001001113,308
134,8785,1002,07434123112,417
1419,53621,8548,2061,55618113151,347
1585,26392,23437,2227,193653291222,595
16379,799427,079172,67833,2163,8853011611,016,975
171,769,9792,005,800829,904173,54919,1221,1293614,799,520
188,400,2859,716,8484,194,015876,173105,5398,42847119123,301,779
1940,619,38548,184,01821,207,6954,749,914599,43343,5131,813431115,405,815
20199,631,989241,210,386110,915,68425,644,8023,368,608282,89816,613708221581,071,711
21990,623,8571,228,973,463581,200,584141,228,38719,967,9111,707,14789,2252,7705112,963,793,396
224,976,016,4856,301,831,9443,091,592,835786,648,328115,822,29010,710,211673,34430,6711,01625115,283,327,150
2325,182,878,92132,663,182,52116,547,260,9934,388,853,201689,913,11767,959,9624,265,763169,4804,05459179,544,488,072
24128,564,665,125170,407,462,90089,132,658,20924,737,359,7874,078,666,332425,398,30229,712,6521,469,67553,6171,429291417,377,448,058
Sum159,965,097,353210,903,116,128109,490,080,80930,085,576,5754,908,467,107506,111,93534,759,9351,673,36858,7621,515311515,894,943,519

The most recent run of the enumeration program from 2 to 23 crossings was completed on Mon April 23rd, 2007. The runtime was 1712634 seconds (2 weeks 5 days 19 hours 43 minutes 54 seconds) on a single computer. The enumeration of the 24 crossing links took approximately 80 hours on cluster of 40 computers and was completed on Sept 30th 2007. Our first algorithm generated numbers close to these at the end of summer 2005. These numbers agreed with the existing tables of links due to Thistlethwaite et al. and with Rankin and Flints previous work on enumerating the prime alternating knots up to 23 crossings. Early 2007 we developed a simpler and faster enumeration scheme whose results did not agree with those generated by the algorithm used in 2005/2006. This led us to find and fix a bug in the code for the original algorithm and now both produce the same results.

Oriented Prime Alternating Links

The table below is formatted the same as the one above, but it list the number of oriented prime alternating links.

1234567891011Sum
2022
311
4123
5213
635311
779218
8182415461
94377302152
10144239131414559
114908865528342,015
121,9263,8032,1116739388,614
138,13915,39710,9052,683212537,341
1434,77073,15048,07414,8222,5122229173,559
15159,486329,254244,49481,01210,6885439825,486
16730,0991,595,0791,216,136415,16779,3538,799523184,045,174
173,462,8437,704,3826,137,5792,384,565447,19038,3871,3251220,176,283
1816,593,42137,956,85332,006,16812,693,5492,747,732360,62628,6291,24023102,388,241
1980,689,811190,156,365164,961,23971,601,15916,848,2402,091,185128,3993,25223526,479,673
20397,782,507957,467,036872,933,793395,677,87799,225,49415,127,8021,474,13989,6242,889442,739,781,205
211,977,282,4824,894,915,4964,605,897,8712,211,591,299608,018,77897,277,1118,769,351406,7667,7493414,404,166,937
229,941,282,45925,147,320,99424,599,611,55612,428,621,8253,595,309,394637,664,64473,343,7215,600,130268,3056,7326376,429,029,823
2350,336,761,633130,481,594,997131,974,739,09369,708,724,73221,685,188,1684,162,525,590490,793,68933,976,6101,235,30518,40263408,875,558,282
Sum62,754,790,286161,719,134,053162,257,809,75584,831,809,49726,007,877,8674,915,094,928574,539,80140,077,6601,514,30325,222137503,102,673,443


Since we generate the symmetry group of a link when computing its master array, during the enumeration of the unoriented prime alternating links, we are able to separate the those links with non-trivial symmetry groups. For a asymmetric link, each orientation is unique, otherwise there would be a symmetry of the link mapping one to the other. Thus we only need to compute the orientations of the symmetric links. Since this set is much smaller, the computation only took 27048 seconds (7 hours 30 minutes 48 seconds) on a single computer.

Future Enumeration Work

Enumerate the number of chiral/achiral links, minimal diagrams etc...

Other Interesting Link properties

T-Fixed Knots

During our enumeration work we discovered some asymmetric knots that are invariant under our T operator. For instance, this knot, is the smallest asymmetric knots with this property. If you turn group 21 or group 22, the knot is unchanged. Below 19 crossings there are only 3 knots that exhibit this property, 2 symmetric and 1 asymmetric: 
1,2,3,4,5,6,7,8,9,10,11,3,4,9,8,5,2,1,6,7,10,11
1,2,3,4,5,6,7,8,6,9,10,11,9,12,2,1,13,14,15,13,4,5,11,10,14,15,12,7,8,3
1,2,3,4,5,6,7,8,9,10,11,12,4,13,8,9,2,14,15,3,13,7,10,1,14,15,12,5,6,11