(related to Proposition: Equivalent Knot Diagrams)

We want to show that the existence of a finite sequence of diagrammatic moves defines an equivalence relation on knot diagrams. * Reflexivity: A knot diagram \(K\) is related to itself by a sequence of length one: \(K=K_0=K\Longleftrightarrow K\sim K.\) * Symmetry: If \(K\sim K'\), then there is a finite sequence \(K=K_0\sim K_1\sim K_2\sim\ldots\sim K_n=K'\). By reversing the order of the sequence, we get \(K'\sim K\). * Transitivity: if \(K\sim K'\) and \(K'\sim K^*\), then we can form a finite sequence of moves transforming \(K\) to \(K^*\) so that \(K\sim K^*\).

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  1. Dye, Heather: "An Invitation to Knot Theory", CRC Press, 2016