Histogram of polynomial frequencies.
We take the set of knots, with equivalence up to equality in a chosen invariant. The plot below shows a histogram of sizes of equivalence classes. For example, over all prime knots (up to mirror) with crossing numbers 3-16, 495,163 have a unique Jones invariant and 179,862 form isolated pairs sharing the same Jones invariant.
Invariant
Interactive plot: zoom, pan and toggle your desired invariants!
| stat | A2 | Alexander | B1 | Alexander+BV | BV | HFK2 | HFK2T1 | HOMFLYPT | HOMFLYPT Homology 3-11 | HOMFLYPT Homology Partial 3-13 | Jones | Khovanov | KhovanovT1 | KhovanovOdd | KR3 3-15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| %single (all) | 76.7959 | 44.5147 | 85.5137 | 97.9227 | 97.9428 | 60.7208 | 44.4434 | 76.8210 | 96.3684 | 91.5873 | 58.8677 | 62.9613 | 61.9049 | 61.9325 | 79.9157 |
| %pair (all) | 17.2328 | 18.0847 | 12.1518 | 1.8859 | 1.8743 | 18.8077 | 18.2282 | 17.2151 | 3.3722 | 7.0262 | 21.3830 | 20.8199 | 21.1333 | 20.9766 | 15.7068 |
| %triple (all) | 2.5711 | 8.9164 | 0.4013 | 0.0574 | 0.0527 | 6.3858 | 8.8478 | 2.5657 | 0.2594 | 1.0122 | 7.2286 | 6.2950 | 6.5339 | 6.5180 | 2.1830 |
| max-dupes (all) | 24 | 923 | 16 | 11 | 1919 | 410 | 510 | 24 | 3 | 7 | 61 | 52 | 67 | 52 | 15 |
| %single (a) | 82.2569 | 54.3418 | 87.1409 | 97.9217 | 97.9632 | 60.2190 | 54.3418 | 82.2597 | 96.4945 | 88.2174 | 68.1661 | 68.4017 | 68.4017 | 68.4017 | 84.0453 |
| %pair (a) | 14.0273 | 19.1845 | 10.7860 | 1.9414 | 1.9116 | 19.1823 | 19.1845 | 14.0251 | 3.1365 | 10.0728 | 19.2855 | 19.1567 | 19.1567 | 19.1567 | 12.8042 |
| %triple (a) | 1.5072 | 8.6973 | 0.3619 | 0.0362 | 0.0292 | 7.6017 | 8.6973 | 1.5067 | 0.3690 | 1.3543 | 5.6784 | 5.6232 | 5.6232 | 5.6232 | 1.7013 |
| max-dupes (a) | 20 | 143 | 16 | 8 | 1354 | 85 | 143 | 20 | 3 | 6 | 28 | 28 | 28 | 28 | 10 |
| %single (n) | 74.7671 | 39.0687 | 84.8347 | 97.9258 | 97.9353 | 62.5091 | 38.6449 | 74.7973 | 96.0699 | 97.2598 | 54.5497 | 60.8344 | 59.2272 | 59.3720 | 78.1057 |
| %pair (n) | 18.4738 | 17.7209 | 12.7217 | 1.8664 | 1.8613 | 18.8224 | 18.0133 | 18.4535 | 3.9301 | 2.5586 | 22.4403 | 21.5686 | 22.0579 | 21.8037 | 17.0825 |
| %triple (n) | 2.9437 | 9.1390 | 0.4177 | 0.0638 | 0.0615 | 5.7821 | 8.9628 | 2.9365 | 0.0000 | 0.1651 | 7.9474 | 6.5316 | 6.8856 | 6.8560 | 2.3273 |
| max-dupes (n) | 24 | 922 | 16 | 10 | 565 | 390 | 494 | 24 | 2 | 4 | 61 | 52 | 64 | 52 | 15 |
Brief descriptions of rows. All the types {all, alternating, non-alternating} are considered separately.
| Name | Description |
|---|---|
| %single | Percentage of equivalence classes of size 1, in all equivalence classes |
| %double | Percentage of equivalence classes of size 2, in all equivalence classes |
| %triple | Percentage of equivalence classes of size 3, in all equivalence classes |
| max-dupes | Maximum size of an equivalence class |