“Reidemeister moves” for self-intersecting loops
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:
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
asked 8 hours ago
David G. Stork
1,67811130
add a comment |
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:
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
asked 8 hours ago
David G. Stork
1,67811130
add a comment |
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:
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
asked 8 hours ago
David G. Stork
1,67811130
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:
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
asked 8 hours ago
David G. Stork
1,67811130
asked 8 hours ago
David G. Stork
1,67811130
asked 8 hours ago
David G. Stork
1,67811130
asked 8 hours ago
David G. Stork
1,67811130
asked 8 hours ago
David G. Stork
1,67811130
1,67811130
add a comment |
add a comment |
1 Answer
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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.
answered 8 hours ago
Andy Putman
31.2k5132212
1
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
8 hours ago
add a comment |
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1 Answer
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1 Answer
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active
oldest
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oldest
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votes
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.
answered 8 hours ago
Andy Putman
31.2k5132212
1
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
8 hours ago
add a comment |
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.
answered 8 hours ago
Andy Putman
31.2k5132212
1
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
8 hours ago
add a comment |
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.
answered 8 hours ago
Andy Putman
31.2k5132212
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.
answered 8 hours ago
Andy Putman
31.2k5132212
answered 8 hours ago
Andy Putman
31.2k5132212
answered 8 hours ago
Andy Putman
31.2k5132212
answered 8 hours ago
Andy Putman
31.2k5132212
31.2k5132212
1
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
8 hours ago
add a comment |
1
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
8 hours ago
1
1
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
8 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
8 hours ago
add a comment |
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