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$\frac{1}{4\pi} \oint_{\gamma_1}\oint_{\gamma_2} \frac{\mathbf{r}_1 - \mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|^3} \cdot (d\mathbf{r}_1 \times d\mathbf{r}_2)$

is an integer when $\gamma_1, \gamma_2: S^1 \to \mathbb{R}^2$ mathbb{R}^3$ are non-intersecting differentiable curves.

Seriously?

This number tells you how many times $\gamma_1$ winds around $\gamma_2$ (The linking number). My wife was a math and biochem major as an undergraduate interested in applying knot theory to genomics, and she and I spent countless hours trying to make sense of this without any knowledge of cohomology. Builds character I guess.
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$\frac{1}{4\pi} \oint_{\gamma_1}\oint_{\gamma_2} \frac{\mathbf{r}_1 - \mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|^3} \cdot (d\mathbf{r}_1 \times d\mathbf{r}_2)$

is an integer when $\gamma_1, \gamma_2: S^1 \to \mathbb{R}^2$ are non-intersecting differentiable curves.

Seriously?

This number tells you how many times $\gamma_1$ winds around $\gamma_2$ (The linking number). My wife was a math and biochem major as an undergraduate interested in applying know knot theory to genomics, and she and I spent countless hours trying to make sense of this without any knowledge of cohomology. Builds character I guess.
show/hide this revision's text 1

$\frac{1}{4\pi} \oint_{\gamma_1}\oint_{\gamma_2} \frac{\mathbf{r}_1 - \mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|^3} \cdot (d\mathbf{r}_1 \times d\mathbf{r}_2)$

is an integer when $\gamma_1, \gamma_2: S^1 \to \mathbb{R}^2$ are non-intersecting differentiable curves.

Seriously?

This number tells you how many times $\gamma_1$ winds around $\gamma_2$ (The linking number). My wife was a math and biochem major as an undergraduate interested in applying know theory to genomics, and she and I spent countless hours trying to make sense of this without any knowledge of cohomology. Builds character I guess.