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Proposition: Equivalent Knot Diagrams

Two knot diagrams $K$ and $K'$ are called equivalent and denoted by $K\sim K'$, if there is a finite sequence of diagrammatic moves (i.e. either Reidemeister moves or planar isotopy moves) $(K_n)_{n\in\mathbb N}$, such that

$$K=K_0\sim K_1\sim K_2\sim\ldots\sim K_n=K'.$$

In particular, the diagrammatic moves define an equivalence relation on knot diagrams.

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References

Bibliography

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