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Tame Knot Diagrams can be represented by two different discrete structures, namely, Grid Diagrams and Knot Mosaics. This report proposes two polynomial time algorithms for translations between Grid Diagrams and Knot Mosaics. It is shown that that the time complexity of both algorithms is O(\\ensuremath{n^{3}}). These results prove that Grid Diagrams and Knot Mosaics are topologically equivalent. This equivalence is efficiently computable. We also conjecture that the two Cromwell moves of Grid Diagrams, i.e. Castling and Stabilization, are equivalent to sequences of planar moves defined for Knot Mosaics. These equivalences are also conjectured to be polynomially computable.
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