Timeline for Reference for a theorem on crossing changes of links
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Jun 16, 2013 at 20:39 | comment | added | Scott Taylor | (continued) Assume that $k_1$ and $k_2$ are composite. By Theorem 1 from the Eudave-Munoz paper the tangle $(B,t)$ cannot be prime. By the definition of prime tangle either this means that a strand of $(B,t)$ has a local knot (in which case, you can show the crossing change occurs in a factor of $k_1$ and $k_2$) or the tangle $(B,t)$ is rational (in which case, $k_1$ and $k_2$ are 2-bridge and so can't be composite). | |
| Jun 16, 2013 at 20:34 | comment | added | Scott Taylor | A crossing change is rational tangle replacement of distance 2. If you remove a regular neighborhood of the crossing you get a tangle $(B,t)$. Putting the crossing in one way produces a knot $k_1$ and putting it in the other way (with the crossing reversed) produces a knot $k_2$. The rational tangles are the intersections of $k_1$ and $k_2$ with the complement of $B$. Each of those rational tangles has a disc with boundary in the four-punctured sphere $\partial B - t$. The discs can be isotoped to intersect in two arcs, and so the distance between the rational tangles is 2 | |
| Jun 14, 2013 at 10:09 | comment | added | Springfield | The results in the cited paper are about rational tangles, while the statement in Scharlemann's paper is about summands of a link. I can't really figure out how to proof Theorem 2.2 starting from the paper of Eudave-Mun ̃oz. | |
| Jun 13, 2013 at 17:39 | history | edited | Scott Taylor | CC BY-SA 3.0 |
Corrected answer to answer the question that was asked.
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| Jun 13, 2013 at 17:33 | history | answered | Scott Taylor | CC BY-SA 3.0 |