Bridge Knot
An 
-bridge knot is a knot with bridge number 
. The set of 2-bridge knots is identical to the set of rational knots. If 
is a 2-bridge knot, then the BLM/Ho polynomial 
and Jones polynomial 
satisfy
where 
(Kanenobu and Sumi 1993). Kanenobu and Sumi also give a table containing the number of distinct 2-bridge knots of 
crossings for 
to 22, both not counting and counting mirror images as distinct.
 |
 |
 |
| 3 |
0 |
0 |
| 4 |
0 |
0 |
| 5 |
|
|
| 6 |
|
|
| 7 |
|
|
| 8 |
|
|
| 9 |
|
|
| 10 |
45 |
85 |
| 11 |
91 |
182 |
| 12 |
176 |
341 |
| 13 |
352 |
704 |
| 14 |
693 |
1365 |
| 15 |
1387 |
2774 |
| 16 |
2752 |
5461 |
| 17 |
5504 |
11008 |
| 18 |
10965 |
21845 |
| 19 |
21931 |
43862 |
| 20 |
43776 |
87381 |
| 21 |
87552 |
175104 |
| 22 |
174933 |
349525 |
REFERENCES:
Kanenobu, T. and Sumi, T. "Polynomial Invariants of 2-Bridge Links through 20 Crossings." Adv. Studies Pure Math. 20, 125-145, 1992.
Kanenobu, T. and Sumi, T. "Polynomial Invariants of 2-Bridge Knots through 22-Crossings." Math. Comput. 60, 771-778 and S17-S28, 1993.
Schubert, H. "Knotten mit zwei Brücken." Math. Z. 65, 133-170, 1956.
