Dihedral Linking Invariants

A Fox p-colored knot $K$ in $S^3$ gives rise to a corresponding $p$-fold dihedral branched cover $M$ of $S^3$ along $K$. The pre-image of the knot $K$ under the covering map is a $\dfrac{p+1}{2}$-component link $L$ in $M$, and the set of pairwise linking numbers of the components of $L$ is an invariant of $K$. This powerful invariant played a key role in the development of early knot tables, and appears in formulas for many other important knot and manifold invariants. We give an algorithm for computing this invariant for all odd $p$, generalizing an algorithm of Perko. We then extend this algorithm to compute linking numbers of arbitrary curves in a $p$-fold dihedral branched cover of $S^3$ along $K$. As an application, we compute Kjuchukova's ribbon obstruction $\Xi_p$ using a method of the first author and Kjuchukova. We also tabulate the dihedral linking invariant for all $p$-colorings of prime knots of crossing number less than or equal to 13, with $p\geq 3$ prime. Finally, we demonstrate the strength of the dihedral linking invariant by comparing it to several polynomial invariants. For example, the dihedral linking invariant distinguishes more than 98% of the 1183 prime non-mutant knot pairs with the same Fox coloring invariant and the same HOMFLY-PT polynomial through 13 crossings.