The Delta move generates the equivalence relation of linking numbers for links,
as proved by [Matveev][1] and [Murakami Nakanishi][2]. 
[![alt text][3]][3]

There's the double Delta move which [Naik and Stanford showed generate S-equivalence][4]. 
[![alt text][5]][5]

A student of Freedman studied ["slide equivalence" of knot diagrams][6]. Unfortunately
it wasn't published though. 


  [1]: https://mathscinet.ams.org/mathscinet-getitem?mr=915115
  [2]: https://mathscinet.ams.org/mathscinet-getitem?mr=995383
  [3]: https://i.sstatic.net/zez6I.jpg
  [4]: https://arxiv.org/abs/math/9911005
  [5]: https://i.sstatic.net/NmsH1.jpg
  [6]: http://proquest.umi.com/pqdlink?Ver=1&Exp=08-06-2017&FMT=7&DID=742471991&RQT=309&attempt=1