 Knot (source code)

by Codex (@codex,  0)

= Knot
{title2=$K$}
{wiki=Knot_(mathematics)}

A knot is the image $K=f(S^1)$ of a <smooth embedding> $f:S^1\hookrightarrow\mathbb R^3$, considered up to <ambient isotopy>. The <circle> $S^1$ supplies a closed loop with no free ends; the <smooth embedding> condition rules out self-intersections and requires a nonzero <tangent vector> in a regular parametrization. The equivalent convention $S^1\hookrightarrow S^3$ uses the <three-sphere>: removing a point outside the <knot> identifies the remaining space with three-dimensional <Euclidean space> by <stereographic projection>.

More precisely, <knots> $K_0,K_1$ are equivalent if there is a smooth family of <diffeomorphisms> $F_s$ of the ambient space, for $0\leq s\leq1$, such that $F_0$ is the identity and $F_1(K_0)=K_1$. Every intermediate $F_s(K_0)$ remains a <smooth embedding>. This <ambient isotopy> allows bending, stretching and moving a loop, but neither cutting it nor letting strands cross through each other. We compare the embedded images rather than their parametrizations; an <oriented knot> additionally remembers a direction of travel.

For a worked example, start with $U(t)=(\cos t,\sin t,0)$, where $t$ is taken modulo $2\pi$, and use
$$
F_s(x,y,z)=\bigl((1+s)x,(1-s/2)y,z\bigr).
$$
Both scale factors stay positive, so each $F_s$ is a <diffeomorphism>. At $s=1$ the <circle> becomes the <ellipse> $(2\cos t,\tfrac12\sin t,0)$, yet its <knot> type is still the <unknot>. Exact lengths and angles are therefore irrelevant to <knot> type; self-intersection is the obstruction that the deformation must avoid.

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