= Knot theory
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Knot theory studies <closed curves> in three-dimensional <Euclidean space> that cannot intersect themselves, and asks which can be deformed into one another without cutting or passing one strand through another. Its basic object is a <knot>: a <smooth embedding> of a <circle>, considered up to <ambient isotopy>. Several disjoint <knots> form a <link>. More generally, <knot theory> studies such <embeddings> in other <manifolds> and in higher dimensions.
The central surprise is that a flexible <circle> can carry permanent information about how it sits in space. A <trefoil knot> and an <unknot> each consist of just one <circle>, but no <ambient isotopy> turns one into the other. A <knot diagram> translates this spatial question into a finite drawing; <Reidemeister moves> describe the changes of drawing that preserve the <knot>. <Knot invariants> translate it into computations: unequal <Alexander polynomials of a knot>, <Jones polynomials>, or <knot groups> certify different <knots>. Equal values of a particular <knot invariant> generally leave the question open; even an isomorphism of <knot groups> alone is not a complete classification.
An early foundation for working with <knot diagrams> is Kurt Reidemeister's 1927 paper \i[Elementare Begründung der Knotentheorie], published in volume 5 of \i[Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg], pages 24–32.https://doi.org/10.1007/BF02952507{ref} Marc Lackenby's authored notes \i[Elementary knot theory], Section 2.4, explain the <Reidemeister move> theorem and its role in searching for a certificate that two <knot diagrams> represent the same <knot>.https://people.maths.ox.ac.uk/lackenby/ekt11214.pdf{ref}
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