In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

subject of knot theory, the average crossing number of a

knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...

is the result of averaging over all directions the number of crossings in a knot diagram of the knot obtained by projection onto the plane orthogonal to the direction. The average crossing number is often seen in the context of

physical knot theory Physical may refer to: *Physical examination In a physical examination, medical examination, or clinical examination, a medical practitioner examines a patient for any possible medical signs or symptoms of a medical condition. It generally co ...

Definition

More precisely, if ''K'' is a smooth knot, then for almost every unit vector ''v'' giving the direction, orthogonal projection onto the plane perpendicular to ''v'' gives a knot diagram, and we can compute the crossing number, denoted ''n''(''v''). The average crossing number is then defined as the integral over the unit sphere: :

\frac\int_ n(v) \, dA

where ''dA'' is the area form on the 2-sphere. The integral makes sense because the set of directions where projection doesn't give a knot diagram is a set of measure zero and ''n''(''v'') is locally constant when defined.

Alternative formulation

A less intuitive but computationally useful definition is an

integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...

similar to the Gauss linking integral. A derivation analogous to the derivation of the linking integral will be given. Let ''K'' be a knot, parameterized by :

f: S^1 \rightarrow \mathbb R^3.

Then define the map from the

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...

to the

2-sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...

g: S^1 \times S^1 \rightarrow  S^2

by :

g(s, t) = \frac.

(Technically, one needs to avoid the diagonal: points where ''s'' = ''t'' .) We want to count the number of times a point (direction) is covered by ''g''. This will count, for a generic direction, the number of crossings in a knot diagram given by projecting along that direction. Using the degree of the map, as in the linking integral, would count the number of crossings with ''sign'', giving the writhe. Use ''g'' to pull back the area form on ''S''² to the torus ''T''² = ''S''¹ × ''S''¹. Instead of integrating this form, integrate the absolute value of it, to avoid the sign issue. The resulting integral is :

\frac\int_ \frac \, ds\, dt.

Definition

Alternative formulation

References

Further reading