“Reidemeister moves” for self-intersecting loops












3














Reidemeister showed that just three basic moves (which manipulated crossings) were needed to test whether two knots were equivalent or "ambient isotopic." Suppose we expand the class of topological objects to permit self-intersection, as here:



self-intersecting



The notion of ambient isotopy applies to such objects. What is the smallest class of moves that guarantees one can transform such an object into any other ambient isotopic object?



I suspect the Reidemeister moves need be augmented by fairly trivial moves passing segments above or beneath the intersected crossing, but how would one prove that?






knot-theory







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    3














    Reidemeister showed that just three basic moves (which manipulated crossings) were needed to test whether two knots were equivalent or "ambient isotopic." Suppose we expand the class of topological objects to permit self-intersection, as here:



    self-intersecting



    The notion of ambient isotopy applies to such objects. What is the smallest class of moves that guarantees one can transform such an object into any other ambient isotopic object?



    I suspect the Reidemeister moves need be augmented by fairly trivial moves passing segments above or beneath the intersected crossing, but how would one prove that?






    knot-theory







    share|cite|improve this question

























      3












      3








      3


      1





      Reidemeister showed that just three basic moves (which manipulated crossings) were needed to test whether two knots were equivalent or "ambient isotopic." Suppose we expand the class of topological objects to permit self-intersection, as here:



      self-intersecting



      The notion of ambient isotopy applies to such objects. What is the smallest class of moves that guarantees one can transform such an object into any other ambient isotopic object?



      I suspect the Reidemeister moves need be augmented by fairly trivial moves passing segments above or beneath the intersected crossing, but how would one prove that?






      knot-theory







      share|cite|improve this question













      Reidemeister showed that just three basic moves (which manipulated crossings) were needed to test whether two knots were equivalent or "ambient isotopic." Suppose we expand the class of topological objects to permit self-intersection, as here:



      self-intersecting



      The notion of ambient isotopy applies to such objects. What is the smallest class of moves that guarantees one can transform such an object into any other ambient isotopic object?



      I suspect the Reidemeister moves need be augmented by fairly trivial moves passing segments above or beneath the intersected crossing, but how would one prove that?






      knot-theory




      knot-theory






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 6 hours ago









      David G. Stork

      1,67811130




      1,67811130






















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          These objects are called spatial graphs. A set of Reidemeister moves for them was found by Kauffman; see



          L H Kauffman, Invariants of graphs in three-space, Trans. Amer. Math. Soc. 311 (1989) 697–710.






          share|cite|improve this answer





















          • Perfect. Thanks so much for the reference. (accept) . Indeed, the additional moves are fairly straightforward (including the "twist" of segments at an intersection).
            – David G. Stork
            5 hours ago













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          active

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          active

          oldest

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          3














          These objects are called spatial graphs. A set of Reidemeister moves for them was found by Kauffman; see



          L H Kauffman, Invariants of graphs in three-space, Trans. Amer. Math. Soc. 311 (1989) 697–710.






          share|cite|improve this answer





















          • Perfect. Thanks so much for the reference. (accept) . Indeed, the additional moves are fairly straightforward (including the "twist" of segments at an intersection).
            – David G. Stork
            5 hours ago


















          3














          These objects are called spatial graphs. A set of Reidemeister moves for them was found by Kauffman; see



          L H Kauffman, Invariants of graphs in three-space, Trans. Amer. Math. Soc. 311 (1989) 697–710.






          share|cite|improve this answer





















          • Perfect. Thanks so much for the reference. (accept) . Indeed, the additional moves are fairly straightforward (including the "twist" of segments at an intersection).
            – David G. Stork
            5 hours ago
















          3












          3








          3






          These objects are called spatial graphs. A set of Reidemeister moves for them was found by Kauffman; see



          L H Kauffman, Invariants of graphs in three-space, Trans. Amer. Math. Soc. 311 (1989) 697–710.






          share|cite|improve this answer












          These objects are called spatial graphs. A set of Reidemeister moves for them was found by Kauffman; see



          L H Kauffman, Invariants of graphs in three-space, Trans. Amer. Math. Soc. 311 (1989) 697–710.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 5 hours ago









          Andy Putman

          31.2k5132212




          31.2k5132212












          • Perfect. Thanks so much for the reference. (accept) . Indeed, the additional moves are fairly straightforward (including the "twist" of segments at an intersection).
            – David G. Stork
            5 hours ago




















          • Perfect. Thanks so much for the reference. (accept) . Indeed, the additional moves are fairly straightforward (including the "twist" of segments at an intersection).
            – David G. Stork
            5 hours ago


















          Perfect. Thanks so much for the reference. (accept) . Indeed, the additional moves are fairly straightforward (including the "twist" of segments at an intersection).
          – David G. Stork
          5 hours ago






          Perfect. Thanks so much for the reference. (accept) . Indeed, the additional moves are fairly straightforward (including the "twist" of segments at an intersection).
          – David G. Stork
          5 hours ago




















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