Normal-inverse-gamma distribution

Template:More footnotes In the mathematical subject of knot theory, the average crossing number of a knot 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.

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:

$\frac{1}{4\pi }\underset{S^2}{\int }n(v)dA$

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 similar to the Gauss linking integral.

We will give a derivation analogous to the derivation of the linking integral. Let K be a knot, parametrized by

$f:S^1\to {\mathbb{ℝ}}^3.$

Then define the map from the torus to the 2-sphere

$g:S^1\times S^1\to S^2$

by

$g(s,t)=\frac{f(s)-f(t)}{|f(s)-f(t)|}.$

(Technically, we need 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 pullback the area form on S² to the torus T² = S¹ × S¹. Instead of integrating this form, we integrate the absolute value of it, to avoid the sign issue. The resulting integral is

$\frac{1}{4\pi }\underset{T^2}{\int }\frac{|(f^{\prime }(s)\times f^{\prime }(t))\cdot (f(s)-f(t))|}{|(f(s)-f(t)){|}^3}dsdt.$

Further reading

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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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