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Most knot invariants in knot theory are discovered by finding a property of knot diagrams which is invariant under the three Reidemeister moves. Now in principle, any knot invariant can be described in a diagram-independent way, that is, as a property of the three-dimensional knot itself without reference to diagrams of the knot. But in practice, it can take years between the development of a knot invariant and the discovery of a diagram-independent description of it.

So my question is, for what knot invariants are a diagram-independent description not yet known?

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  • $\begingroup$ Does tricolorability have such a description? $\endgroup$ – Wojowu Sep 11 at 10:30
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    $\begingroup$ Many knot invariants such as the volume (and other hyperbolic invariants) or invariants coming from gauge theory are not described (and certainly were not discovered) in the diagrammatic way you suggest. It might be better to ask which diagrammatic invariants (do or) do not have known diagram-independent descriptions. $\endgroup$ – Danny Ruberman Sep 11 at 12:02
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    $\begingroup$ @Wojowu: tri-colorings correspond to (certain) representations of the fundamental group of the knot complement into $S_3$. More generally, $p$-colorings correspond to (certain) representations into the dihedral group with $2p$ elements. $\endgroup$ – Marco Golla Sep 11 at 12:36

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