In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

subject of

knot theory In topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be und ...

, the average crossing number of a

knot A knot is an intentional complication in Rope, cordage which may be practical or decorative, or both. Practical knots are classified by function, including List of hitch knots, hitches, List of bend knots, bends, List of loop knots, loop knots, ...

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 knot theory is the study of mathematical models of knotting phenomena, often motivated by considerations from biology, chemistry, and physics (Kauffman 1991). Physical knot theory is used to study how geometric and topological characterist ...

Definition

More precisely, if ''K'' is a smooth knot, then for almost every

unit vector In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...

''v'' giving the direction, orthogonal projection onto the plane perpendicular to ''v'' gives a

knot diagram In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, the simplest k ...

, and we can compute the crossing number, denoted ''n''(''v''). The average crossing number is then defined as the

integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...

over the

unit sphere In mathematics, a unit sphere is a sphere of unit radius: the locus (mathematics), set of points at Euclidean distance 1 from some center (geometry), center point in three-dimensional space. More generally, the ''unit -sphere'' is an n-sphere, -s ...

: :

\frac\int_ n(v) \, dA

where ''dA'' is the area form on the

2-sphere A sphere (from Greek , ) is a surface analogous to the circle, a curve. In solid geometry, 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 ''center' ...

. The integral makes sense because the set of directions where projection doesn't give a knot diagram is a set of

measure zero In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has Lebesgue measure, measure zero. This can be characterized as a set that can be Cover (topology), covered by a countable union of Interval (mathematics), ...

and ''n''(''v'') is locally constant when defined.

Alternative formulation

A less intuitive but computationally useful definition is an integral similar to the

Gauss linking integral In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. In E ...

. 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 (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...

to the 2-sphere :

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 In knot theory, there are several competing notions of the quantity writhe, or \operatorname. In one sense, it is purely a property of an oriented link (knot theory), link diagram and assumes integer values. In another sense, it is a quantity that ...

. 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