On page 9 of his book "An Introduction to Knot Theory", Lickorish just mentions the fact that the determinant of the rational knot $K(p,q)$ is $|p|$.
I'm trying to find a proof of this fact (I looked into other books and papers but did not find this again). It says "see Chapter 9" but I did not really see where in chapter 9 he proves this fact (also I must say that I did not look into Goeritz Matrix yet).
Is there another way to prove this?
Thanks!
Rational tangles (and their closures, rational knots/links) can be nicely understood through double branched covers, which is related to something called the Montesinos trick.
For the following, the illustrations come from the Sketches of Topology blog.
Recall that a $2$-tangle is a $1$-manifold $T$ properly embedded in a $3$-ball $B$ such that $\partial T$ is four points in $\partial B$, and equivalence of tangles is isotopy rel $\partial B$. One characterization of a rational tangle is one that is two trivial ball-arc pairs (in the terminology of Lickorish) glued along a disk in their respective boundaries. That is, there exists a properly embedded disk $D\subset B-T$ such that $B-D$ is a disjoint union of trivial ball-arc pairs. In the following ball, $D$ is the blue disk, and the two yellow disks demonstrate triviality of the ball-arc pairs.
The double branched cover of a rational tangle $(B,T)$ has a nice characterization as being a solid torus $S^1\times D^2$ with the branch locus $T'$ being trivial:
The disk $D$ lifts to two copies in the solid torus, and the yellow disks lift to four disks that fuse into being two meridian disks.
A rational knot/link (a.k.a. a two-bridge knot/link) is a knot that is formed from a pair of rational tangles glued along their boundaries. We may assume one of these tangles is trivial, and then this is called a closure of a rational tangle. The double branched cover is from taking two copies of the solid torus and gluing them together. That is, it is a lens space. I'm not exactly sure about conventions for $K(p,q)$, but to match what you state from Lickorish, the knot $K(p,q)$ must have as its double branched cover $L(p,q)$, which is from doing $p/q$ Dehn surgery on an unknot in $S^3$.
If $X_2$ is a double branched cover, $\lvert H_1(X_2)\rvert$ is the determinant of the knot (see Chapter 9 of Lickorish). Since $H_1(L(p,q))=\mathbb{Z}/p\mathbb{Z}$, then $\lvert K(p,q)\rvert = \lvert p\rvert$.