I am trying to determine whether a knot is a Montesinos knot from its Conway notation. If a knot is Montesinos (or 2-bridge), I also wish to obtain its associated fraction decomposition $K\left(\frac{p_1}{q_1}, \dots,\frac{p_r}{q_r} \right)$ from its Conway notation. I think that I understand how to do this in most cases. For example:
- The knot $6_2$ has Conway notation [3 1 2]. We use the partial fraction decomposition to obtain that this knot is the rational knot $K\left(\frac{4}{11} \right)$.
- The knot $8_5$ has Conway notation [3; 3; 2]. We again use the partial fraction decomposition, noting that the semicolon separates the rational tangles, and we obtain that this is the Montesinos knot $K\left(\frac{1}{3},\frac{1}{3},\frac{1}{2} \right)$.
I have been trying to read Conway's paper "An enumeration of knots and links, and some of their algebraic properties", but I still do not understand some of the symbols used in the Conway notation. It seems that if either the asterisk "*" or the colon ":" appears in the Conway notation, the knot will not be Montesinos or 2-bridge. However, I do not understand what a parenthesis "( )" or a period "." does in the Conway notation. Initially, I thought that parentheses or periods implied that the knot would not be Montesinos or 2-bridge, but I was wrong, since we have the following:
- The knot 12a_338 has Conway notation [(2 2; 2 1) 1 2 2]. It turns out that this is the Montesinos knot $K\left(\frac{2}{3},\frac{2}{5},\frac{5}{7} \right)$.
- The knot 12n_404 has Conway notation [.3.2 0.-2 1.2 0]. And it turns out it is the Montesinos knot $K\left(\frac{2}{3},-\frac{1}{3},-\frac{4}{11} \right)$.
Summary of my question: What are the the parentheses "( )" and the periods "." doing in the Conway notation? It would be extremely useful to just knot how to determine the fractions from the Conway notation of the two knots above, 12a_338 and 12n_404.
Note: I am getting the Conway notation (of knots up to 12 crossings) from KnotInfo (https://knotinfo.math.indiana.edu/).