Link number

In mathematics , the link number is an invariant that describes the link between two non-interpenetrating, closed curves in three-dimensional space. The number of links is always a whole number and can be positive or negative depending on the orientation (direction of travel) of the curves. Intuitively, the number of entanglements represents the number of turns of the curves around one another.

Calculation of the number of links

We consider two closed curves and , for each of which an orientation (direction of passage) is given. The link number of and is defined by the number of positive and negative crossings and in a link diagram as follows (see figure for the convention for positive and negative crossings): ${\displaystyle J}$ ${\displaystyle K}$ ${\displaystyle lk(J,K)\,}$ ${\displaystyle J}$ ${\displaystyle K}$ ${\displaystyle k_{+}}$ ${\displaystyle k_{-}}$

${\displaystyle lk(J,K)={\frac {k_{+}-k_{-}}{2}}}$.

literature

• Charles Livingston: Knot Theory for Beginners. Vieweg, Braunschweig et al. 1995, ISBN 3-528-06660-1 ( Vieweg Mathematics ).