knot theory In the mathematical field of 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 ...

, each link and

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 ' ...

can have an assigned knot thickness. Each realization of a link or knot has a thickness assigned to it. The thickness τ of a link allows us to introduce a scale with respect to which we can then define the

ropelength In physical knot theory, each realization of a link or knot has an associated ropelength. Intuitively this is the minimal length of an ideally flexible rope that is needed to tie a given link, or knot. Knots and links that minimize ropelength are ...

of a link.

Definition

There exist several possible definitions of thickness that coincide for smooth enough curves.

Global radius of curvature

The thickness is defined using the simpler concept of the local thickness τ(''x''). The local thickness at a point ''x'' on the link is defined as :

\tau(x)=\inf r(x,y,z),\,

where ''x'', ''y'', and ''z'' are points on the link, all distinct, and ''r''(''x'', ''y'', ''z'') is the radius of the circle that passes through all three points (''x'', ''y'', ''z''). From this definition we can deduce that the local thickness is at most equal to the local radius of curvature. The thickness of a link is defined as :

\tau(L) = \inf \tau(x).

Injectivity radius

This definition ensures that a normal tube to the link with radius equal to τ(''L'') will not self intersect, and so we arrive at a "real world" knot made out of a thick string.

References

{{DEFAULTSORT:Knot Thickness Knot theory