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Ravels arising from Montesinos tangles. (English) Zbl 1393.57009
The authors do a study at the intersection of graph theory and knot theory and characterise the conditions for a Montesinos tangle to become a ravel, a spatial graph that is non-planar but contains no nontrivial knots or links. They use results on vertex closure by replacing crossings by vertices or without doing this. The main results of the paper are Theorems 3.3. and 4.7. which are proven nicely in detail:
Theorem 3.3. Let $$T=T_1+\cdots +T_n$$ be a Montesinos tangle such that $$n$$ is minimal and not both $$T_1$$ and $$T_n$$ are trivial vertical tangles. If $$T$$ is rational, then the vertex closure $$V(T)$$ is planar. If $$T$$ is not rational and some rational subtangle $$T_i$$ has $$\infty$$-parity, then $$V(T)$$ contains a nontrivial knot or link. Otherwise, $$V(T)$$ is a ravel.
Theorem 4.7. Let $$T=T_1+\cdots +T_n$$ be a projection of a Montesinos tangle in standard form, and let $$T'$$ be obtained from $$T$$ by replacing at least one crossing by a vertex. Then the vertex closure $$V(T')$$ is a ravel if and only if $$T'$$ is an exceptional vertex insertion.
This is a very good and informative paper which can be applied to many problems in either side of these two areas.
##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57M15 Relations of low-dimensional topology with graph theory 05C10 Planar graphs; geometric and topological aspects of graph theory
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