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Bridge Knot

An $n$-bridge knot is a knot with Bridge Number $n$. The set of 2-bridge knots is identical to the set of rational knots. If $L$ is a 2-Bridge Knot, then the BLM/Ho Polynomial $Q$ and Jones Polynomial $V$ satisfy

\begin{displaymath}
Q_L(z)=2z^{-1}V_L(t)V_L(t^{-1}+1-2z^{-1}),
\end{displaymath}

where $z\equiv -t-t^{-1}$ (Kanenobu and Sumi 1993). Kanenobu and Sumi also give a table containing the number of distinct 2-bridge knots of $n$ crossings for $n=10$ to 22, both not counting and counting Mirror Images as distinct.

$n$ $K_n$ $K_n+K_n^*$
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 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.




© 1996-9 Eric W. Weisstein
1999-05-26