Linking Number

The linking number, a topological invariant for a given link, represents the linking of two knots or links in three-dimensional space. Various methods are available to measure it for a particular link, each offering a distinct perspective:

1. Vector Calculus Method: Involves the use of vector calculus and line integrals to express the linking number in terms of the vector functions representing the two curves.
2. Gauss's Linking Number Formula: Utilizes Gauss's formula, expressing the linking number as a line integral along one of the curves involving the tangent vector and the position vector of the other curve.
3. Writhe and Twist Method: Breaks down the linking number into the writhe and twist components, where the writhe represents the overall coiling of one curve around the other, and the twist measures local twisting.
4. Determinant Method: Involves constructing a determinant using the components of the tangent vectors of the two curves and integrating it over the entire curve.
5. Knot Diagram Method: Converts the curves into knot diagrams, simplifying the calculation of the linking number by counting the crossings in the diagram.
6. Homotopy Method: Utilizes homotopy theory to calculate the linking number by continuously deforming one curve to the other while monitoring a certain quantity related to the linking.
7. Algebraic Topology Methods: Leverages concepts from algebraic topology, such as homology and cohomology, to define and compute linking numbers in a more abstract and general framework.