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Two-Bridge Links

04.04.2026

The connection between rational numbers, continued fraction expansions, and two-bridge links didn't hit me until I had been playing around with them for a while. I'm sure this way of exposition is well-known, but the sources that I was reading had other motives (usually triangulating their complements), so avoided this. In any case, when I was playing around with these links, I had a big aha moment followed quickly by a "why didn't anyone explain it to me like this?" moment. I couldn't find this explanation anywhere else, although I know it must exist, so I'm writing it here.

We'll go by example, since the general case is similar. Start with a rational number; say, \(17/5\). You can compute its continued fraction expansion via the Euclidean algorithm: \begin{align*} 17 &= 5 \cdot \color{magenta} 3 \color{black} + 2\\ 5 &= 2 \cdot \color{blue} 2 \color{black} + 1\\ 2 &= 1 \cdot \color{green} 2 \color{black} + 0. \end{align*}

We'll write \(17/5 = [\color{magenta} 3 \color{black}, \color{blue} 2 \color{black}, \color{green} 2 \color{black}] \) to mean that $$ \frac{17}{5} = \color{magenta} 3 \color{black} + \frac{1}{\color{blue} 2 \color{black}+\frac{1}{\color{green} 2 \color{black}}}. $$

All in one go, we'll give a geometric interpretation of the continued fraction expansion and define the two-bridge link associated to it. Let's call the following a strand block:

A square with two strands inside. The positive slope crosses over the negative slope

We'll read the continued fraction expansion \([\color{magenta} 3 \color{black}, \color{blue} 2 \color{black}, \color{green} 2 \color{black}]\) backwards. So to begin, stack \(\color{green} 2 \color{black}\) strand blocks on top of each other:

two strand blocks stacked on top of each other

Then, lay \(\color{blue} 2 \color{black}\) strand blocks to the right. The side lengths of the new strand blocks should be the height of the result of the previous step:

two more strand blocks to the right of step1

Then, stack \(\color{magenta} 3 \color{black}\) strand blocks on the top, ensuring the side length of each new block is the entire width of the result of the previous step:

three more strand blocks on top of the previous step

Now, if the first strand block we placed has unit length, then our original fraction \(17/5\) is the ratio of the side lengths of our final stack of blocks! To get the associated link, just remove the squares and close up the strands by dropping two new strands down parallel to the left and right edges of the stack of blocks:

the two bridge link associated to 17/5

For the general case, note that if the length of the continued fraction is even (in the example above, the length is three), then the closing of the strands is horizontal as opposed to vertical. If a minus sign appears in the continued fraction, this corresponds to using the mirror image of the strand block. Other than that, one can export the above example to the general case, alternating stacking blocks above and stacking blocks to the right.