Linking number
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In mathematics, the linking number is a simple invariant for links (i.e. submanifolds of three-dimensional space homeomorphic to a number of circles). It is initially defined for two-component links, but can easily be extended to deal with one-component links, i.e. knots.
The linking number of a link is calculated as follows:
- Choose a direction on each component.
- Draw a diagram of the link.
- Look at the crossings of the components.
- For each crossing, note the directions at the crossing point.
- As in the picture, the crossings may come in either of the two forms:
| Missing image Knot-crossing-plus.png Image:knot-crossing-plus.png | Missing image Knot-crossing-minus.png Image:knot-crossing-minus.png | |
| (+) | (−) |
- (We assume here that the horizontal line goes above the vertical. You might need to tilt your diagram until you get one of these pictures).
- Let's choose a convention: a plus sign for the first form of crossing, and a negative for the second type.
- Add the plusses and minuses for all the crossings. You are guaranteed to get a whole even number. Its half is the linking number.
The linking number of a knot
A knot is a 1-component link. The aforementioned method for calculating a linking number disregards any twists and knots within single link component, so for a knot you always get zero as the linking number.
This is a true fact, though not completely trivial. (Note that there are many non-trivial knots, with irreducible crossings). To see why this is so, consider a single component T of a link (or a knot). Every crossing i where both vertical and horizontal lines are in T, is considered twice in the calculation of the linking number: Going over T, crossing i will be met with twice, once going along the vertical, the second time going over the horizontal line. These have different signs, which add to zero.
To get an interesting knot invariant, one can do the following. Every knot has a two-sided Seifert surface (not unique!). Being a two-sided surface, you can 'puff' it somewhat to get a 3-dimensional body, whose edge is the union of two (possibly intersecting) disks and a band. The band has two edges, both of them copies of the knot you started with. The linking number of the disjoint union of these two knots is the linking number for the knot you started with, and is an invariant - though the Seifert surface is in general not unique.