Linking numbers of Montesinos links

Abstract

The linking number of an oriented two-component link is an invariant indicating how intertwined the two components are. Tuler proved that the linking number of a two-component rational $\frac{p}{q}$-link is $$\sum^{\frac{|p|}{2}}_{k=1} (-1)^{\big\lfloor (2k-1) \frac{q}{p} \big\rfloor }.$$ In this paper, we provide a simple proof the above result, and introduce the numerical algorithm to find linking numbers of rational links. Using this result, we find linking numbers between any two components in a Montesinos link.

Figures (9)

• Figure 1: Assigning the signs of crossings in a link.
• Figure 2: Operations for rational tangles.
• Figure 3: Pillowcase forms of rational tangles with $\frac{2}{1}$, $-\frac{4}{3}$ and $\frac{5}{3}$.
• Figure 4: The structure of a Montesinos link and the example of $M \left( \left. \frac{3}{2}, -\frac{3}{1}, \frac{5}{3} \right| 2 \right)$.
• Figure 5: The construction of the branched covering of $\partial P$.

Theorems & Definitions (11)

• Proposition 1
• proof : Alternative proof of Proposition \ref{['prop:link']}
• Proposition 2
• proof
• Example
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5