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Many 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 is a diagram-independent description not yet known?

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    $\begingroup$ Does tricolorability have such a description? $\endgroup$
    – Wojowu
    Commented Sep 11, 2019 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
    Commented Sep 11, 2019 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
    Commented Sep 11, 2019 at 12:36
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    $\begingroup$ I would maybe disagree with your central assertion. How are you counting "knot invariants"? Many knot invariants have no known (reasonable) diagrammatic formulation. $\endgroup$
    – Ryan Budney
    Commented Oct 15, 2019 at 6:24
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    $\begingroup$ Is there a way to understand the minimal crossing number of a knot without using diagrams? $\endgroup$
    – Marc Kegel
    Commented Oct 15, 2019 at 14:10

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I'd be happy to be proven wrong, but I would argue that this is still the case of the Jones polynomial and its generalizations, and my, maybe naive, understanding is that it's one of the many reasons it was considered fairly mysterious when it was discovered.

In fact, finding such a description was one of the motivations for Witten's Jones paper (he says as much in the introduction). As beautiful as this is, and even if like many others I'm more than happy to think about this as an actual definition, this is strictly speaking not mathematically rigorous and recovers only the values of the Jones polynomial at roots of unity.

Those can, I believe, now be given a diagrammatic-free definition in the framework of TFT's but this relies on fairly recent result. I also believe the case of generic $q$ is within reach but hasn't been done yet, and is closely related to exciting recent development in low-dimensional topology, e.g. finding a rigorous construction of analytic continuation of Chern-Simons theory, factorization algebras, the AJ conjecture etc..

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