Short version: **Does anyone study equivalence classes generated by a given set of "moves" (in the sense of, but not limited to, Reidemeister moves) on the set of knot diagrams?**


Yes, I understand that the concept of knot has a natural geometrical significance, that one usually views knot diagrams as a tool to study underlying knots, that the value of Reidemeister moves lies in how they preserve and generate the equivalence relation of isotopy. So, yes, I see why a knot theorist might reasonably have little interest, say, in looking at local moves not preserving isotopy.


So I'm asking this question in the spirit of abstraction for its own sake. But there is a precedent. A "symmetry theorist" studies groups because they capture the set of symmetries of important objects. But combinatorial group theory studies equivalence classes of strings under moves...and symmetry, if it enters the story at all, does so as a tool.


That said, it would be interesting if one could add an extra move to the Reidemeister moves that produced a coarser but computationally tractable classification.


No need to retread the ground here -- <https://en.wikipedia.org/wiki/Reidemeister_move> -- so, for example, I already understand that knot theorists know what happens with only Reidemeister type II and III moves. I am interested in stories like this, where one gets a finer equivalence relation that isotopy, but equally interested in sets of local moves that don't preserve isotopy and thus generate equivalence relations either coarser to, or simply incomparable with, isotopy.



Knot diagrams, sets of moves and equivalence relations
Very much so. There are a number of small industries centred around studying equivalence classes of knot diagrams generated by a set of moves.


1. The study of [claspers](https://en.wikipedia.org/wiki/Clasper_(mathematics)). For example, $C\_k$-moves are a special type of clasper surgeries. MathSciNet indicates [123 citations](https://mathscinet.ams.org/mathscinet-getitem?mr=1735632 "Geom. Topol. 4, 1-83 (2000), doi:10.2140/gt.2000.4.1, EuDML:120433, available at https://www.emis.de/journals/UW/gt/GTVol4/paper1.abs.html. zbMATH review at https://zbmath.org/0941.57015") for Habiro's fundamental paper *Claspers and finite type invariants of links*, providing some coarse measure of the vitality of the topic.
2. Replacing one rational tangle in a knot diagram by another generates an equivalence relation which has been deeply studied using [quandles](https://en.wikipedia.org/wiki/Racks_and_quandles). See *e.g.* [J. Przytycki's introductory lectures](https://arxiv.org/abs/math/0405248 "Przytycki, Józef H. From 3-moves to Lagrangian tangles and cubic skein modules. Bryden, John M. (ed.), Advances in topological quantum field theory. Proceedings of the NATO Advanced Research Workshop on new techniques in topological quantum field theory, Kananaskis Village, Canada, August 22–26, 2001. Dordrecht: Kluwer Academic Publishers. NATO Science Series II: Mathematics, Physics and Chemistry 179, 71-125 (2004). zbMATH review at https://zbmath.org/1114.57008").
3. Dehn surgery, where the surgery curve is required to belong to some specified part of a knot group or link group (in the kernel of its representation to some fixed group, for instance) generates equivalence relations on knot diagrams modulo combinatorial "twisting" moves, which have been studied by Cochran-Orr-Gerges, and (excuse the self promotion) by myself and Andrew Kricker, and by Litherland and Wallace. The techniques for studying these equivalence relations have been topological rather than combinatorial.
4. There are a number of setting in which one allows Reidemeister moves plus some crossing changes, but not others. In the theory of [finite type invariant](https://en.wikipedia.org/wiki/Finite_type_invariant), one fixes a some crossings (considers them in resolutions of "double points"), and allows crossing changes away from them. The equivalence classes are detected by the finite-type invariant of type the number of "fixed" crossings. In a similar-sounding vein, a free virtual knot is a virtual knot where we allow crossing changes away from virtual crossings. They have a rich theory- see *e.g.* [this Manturov paper](https://doi.org/10.1090/S0077-1554-2012-00188-5 "Manturov, V. O. Parity, free knots, groups, and invariants of finite type. Trans. Mosc. Math. Soc. 2011, 13 p. (2011); translation from Tr. Mosk. Mat. O.-va 2011, No. 1, 157-169 (2011). zbMATH review at https://zbmath.org/1246.57019").


