= Knot diagram
{wiki}
A knot diagram is a generic projection of a <knot> to a <Euclidean plane> together with overcrossing and undercrossing information at every double point. Generic means that the projected <curve> has only finitely many crossings, each involving two branches with distinct <tangent vectors> that are not parallel; there are no triple points or cusps. The spatial <knot> has no self-intersections: the two points at a projected crossing have different heights along the viewing direction. A gap in the lower strand records which one passes underneath.
Forgetting the over/under information leaves a <knot shadow>. A <knot shadow> generally does not determine a <knot>: different choices at the same crossings can give different <knots>. Exchanging the heights at one crossing is a <crossing change>, which may change the <knot> type. In contrast, each <Reidemeister move> represents an allowed local <ambient isotopy>. Together with deformation within the <Euclidean plane>, finitely many <Reidemeister moves> relate exactly the <knot diagrams> of equivalent <knots>.
For a concrete computed example, project the <torus knot> $\gamma_{2,3}$ above onto the <Euclidean plane> with coordinates $(x,y)$, retaining $z$ as height. Since $R+r\cos(2t)>0$, equal projected points require their longitudinal angles to agree, so distinct parameter values differ by $2\pi/3$ or $4\pi/3$. Equating the radii then gives exactly four unordered pairs, with all parameters in units of $\pi$:
$$
(1/6,5/6),\quad(1/3,5/3),\quad(2/3,4/3),\quad(7/6,11/6).
$$
At each pair, the two heights are opposite and have absolute value $r\sqrt3/2$; the projected <tangent vectors> are not parallel. Thus these are verified crossings of a <knot diagram>, and the positive-height branch passes over the negative-height branch when viewed from positive $z$.
\Image[/knot-crossing-diagram.png]
{title=One trefoil projection: a shadow and its four crossings with verified over/under gaps}
If $c(D)$ denotes the number of crossings in a <knot diagram> $D$ of $K$, then $c(K)=\min_D c(D)$ is the <crossing number of a knot>. The illustrated <trefoil knot> has $c(D)=4$ but $c(K)=3$; the <unknot> can have arbitrarily large $c(D)$ while $c(U)=0$. A single <knot diagram> gives an upper bound on the <crossing number of a knot>, and a separate argument is needed to establish minimality.
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