Properties and Examples
- Any two unlinked curves have linking number zero. However, two curves with linking number zero may still be linked (e.g. the Whitehead link).
- Reversing the orientation of either of the curves negates the linking number, while reversing the orientation of both curves leaves it unchanged.
- The linking number is chiral: taking the mirror image of link negates the linking number. Our convention for positive linking number is based on a right-hand rule.
- The winding number of an oriented curve in the x-y plane is equal to its linking number with the z-axis (thinking of the z-axis as a closed curve in the 3-sphere).
- More generally, if either of the curves is simple, then the first homology group of its complement is isomorphic to Z. In this case, the linking number is determined by the homology class of the other curve.
- In physics, the linking number is an example of a topological quantum number. It is related to quantum entanglement.
Read more about this topic: Linking Number
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