Notes for AKT-090917-1/0:15:48: Difference between revisions
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where the sum is over the <math>2^m</math> resolutions of <math>K</math> and <math>u(K')</math> is the number of under resolutions made in obtaining <math>K'</math>. |
where the sum is over the <math>2^m</math> resolutions of <math>K</math> and <math>u(K')</math> is the number of under resolutions made in obtaining <math>K'</math>. |
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Parallel to result in calculus that partial derivatives commute, we have that <math>V^{(m)}(K)</math> is independent of the order in which the double points are resolved. |
Parallel to the result in calculus that partial derivatives commute, we have that <math>V^{(m)}(K)</math> is independent of the order in which the double points are resolved. |
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Latest revision as of 21:23, 4 September 2011
Likewise, for an [math]\displaystyle{ m }[/math]-singular knot [math]\displaystyle{ K }[/math], we can define:
- [math]\displaystyle{ V^{(m)}(K)=\sum_{K'}{(-1)^{u(K')}V(K')} }[/math]
where the sum is over the [math]\displaystyle{ 2^m }[/math] resolutions of [math]\displaystyle{ K }[/math] and [math]\displaystyle{ u(K') }[/math] is the number of under resolutions made in obtaining [math]\displaystyle{ K' }[/math].
Parallel to the result in calculus that partial derivatives commute, we have that [math]\displaystyle{ V^{(m)}(K) }[/math] is independent of the order in which the double points are resolved.
