Knot Recognition as a Proof of Work
If there is an Arthur-Merlin protocol for knottedness similar to the [GMW85] and [GS86] Arthur-Merlin protocols for Graph Non Isomorphism, then I believe such a cryptocurrency proof-of-work could be designed, wherein each proof-of-work shows that two knots are not likely to be equivalent/isotopic.

In more detail, as is well known in the Graph Non Isomorphism protocol of [GMW85], Peggy the prover wishes to prove to Vicky the verifier that two (rigid) graphs $G_0$ and $G_1$ on $V$ vertices are not isomorphic.  Vicky may secretly toss a random coin $i\in\{0,1\}$, along with other coins to generate a permutation $\pi\in\ S_V$, and may present to Peggy a new graph $\pi(G_i)$.  Peggy must deduce $i$.  Clearly Peggy is only able to do this if the two graphs are not isomorphic.

Similarly, and more relevant for the purposes of a proof-of-work, as taught by [GS86] an Arthur-Merlin version of the same protocol includes Arthur agreeing with Merlin on $G_0$, $G_1$, given as for example adjacency matrices. Arthur randomly picks a hash function $H:\{0,1\}^*\rightarrow\{0,1\}^k$, along with an image $y$.  Arthur provides $H$ and $y$ to Merlin.  Merlin must find a $(i,\pi)$ such that $H(\pi(G_i))=y$.

That is, Merlin looks for a preimage of the hash $H$, the preimage being a permutation of one of the two given adjacency matrices.  As long as $k$ is chosen correctly, if the two graphs $G_0$ and $G_1$ are not isomorphic then there will be a higher chance that a preimage will be found, because the number of adjacency matrices in $G_0 \cup G_1$ may be twice as large than if $G_0\cong G_1$.

In order to convert the above [GS86] protocol to a proof-of-work, identify miners as Merlin, and identify other nodes as Arthur.  Agree on a hash $H$, which, for all purposes, may be the $\mathsf{SHA256}$ hash used in Bitcoin.  Similarly, agree that $y$ will always be $0$, similar to the Bitcoin requirement that the hash begins with a certain number of leading $0$’s. 


The network agrees to prove that two rigid graphs $G_0$ and $G_1$ are not isomorphic.  The graphs may be given by their adjacency matrices
A miner uses the link back to the previous block, along with her own Merkle root of financial transactions, call it $B$, along with her own nonce $c$, to generate a random number $Z=H(c\Vert B)$
The miner calculates $W= Z\:mod\: 2V!$ to pick $(i,\pi)$
The miner confirms that  $\pi(G_i)\neq G_{1-i}$ - that is, to confirm that the randomly chosen $\pi$ is not a proof that the graphs are isomorphic
If not, the miner calculates a hash $W=H(\pi(G_i))$
If $W$ begins with the appropriate number of $0$’s, then the miner “wins” by publishing $(c,B)$
Other nodes can verify that $Z=H(c\Vert B)$ to deduce $(i,\pi)$, and can verify that $W=H(\pi(G_i))$ begins with the appropriate difficulty of $0$’s


The above protocol is not perfect, some kinks I think would need to be worked out.  For example, it's not clear how to generate two random graphs $G_0$ and $G_1$ that satisfy good properties of rigidity, for example, nor is it clear how to adjust the difficulty other than by testing for graphs with more or less vertices.  However, I think these are probably surmountable. 

But for a similar protocol on knottedness, replace random permutations on the adjacency matrix of one of the two graphs $G_1$ and $G_2$ with some other random operations on knot diagrams or grid diagrams... or something.  I don’t think random Reidemeister moves work, because the space becomes too unwieldy too quickly.

[HTY05] proposed an Arthur-Merlin protocol for knottedness, but unfortunately there was an error and they withdrew their claim. 

[Kup11] showed that, assuming the Generalized Riemann Hypothesis, knottedness is in $\mathsf{NP}$, and mentions that this also puts knottedness in $\mathsf{AM}$, but I’ll be honest I don’t know how to translate this into the above framework; the $\mathsf{AM}$ protocol of [Kup11] I think involves finding a rare prime $p$ modulo which a system of polynomial equations is $0$.  The prime $p$ is rare in that $H(p)=0$, and the system of polynomial equations corresponds to a representation of the knot complement group.

Of note, see this answer to a similar question on a sister site, which also addresses the utility of such "useful" proofs-of-work.



References:

[GMW85] Oded Goldreich, Silvio Micali, and Avi Wigderson.  Proofs that Yield Nothing but their Validity, 1985.

[GS86] Shafi Goldwasser, Michael Sipser.  Private Coins versus Public Coins in Interactive Proof Systems, 1986.

[HTY05]  Masao Hara, Seiichi Tani, and Makoto Yamamoto.  UNKNOTTING is in $\mathsf{AM} \cap \mathsf{coAM}$, 2005.

[Kup11] Greg Kuperberg.  Knottedness is in $\mathsf{NP}$, modulo GRH,  2011.