Timeline for Cancellation of 2-component links
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 12, 2015 at 19:00 | vote | accept | James Propp | ||
| Oct 11, 2015 at 3:57 | comment | added | Ryan Budney | Your definition appears to be equivalent to the Blair-Burke-Koytcheff definition via the isotopy extension theorem. | |
| Oct 9, 2015 at 19:08 | comment | added | James Propp | It is easy to show that my kind of deformation gives an equivalence relation that allows for the definition of the stacking operation (as illustrated in Figure 1 on page 3 of arxiv.org/abs/1308.1594) at the level of equivalence classes. | |
| Oct 9, 2015 at 19:02 | comment | added | James Propp | My kind of isotopy between links is given by a continuous map $f: [0,1] \times \{1,2\} \times [0,1] \rightarrow M$ such that $f(0,\cdot,\cdot)$ parametrizes one link and $f(1,\cdot,\cdot)$ parametrizes the other (where a link parametrization is a continuous injection from $\{1,2\} \times [0,1]$ to $M$; the discrete coordinate indexes the 2 components of the 2-component link and the continuous coordinate parametrizes each link from one endpoint to the other). Are you saying I should call this homotopy rather than isotopy? Anyway, it's the kind of equivalence relation I'm trying to ask about. | |
| Oct 9, 2015 at 17:12 | comment | added | Ryan Budney | @JamesPropp: I do not understand what kind of isotopy relation you are using. I had assumed you were using a standard one that allowed for the definition of the stacking operation. If your isotopy relation is too weak, then stacking isn't well-defined. | |
| Oct 9, 2015 at 16:32 | comment | added | James Propp | Definition 2.1 defines two such links to be equivalent iff there an isotopy of the slab (denoted by $M$ in the paper) that fixes the boundary and maps one link to the other. This is a priori stronger than the notion of isotopy that I had in mind, namely a way of deforming one link into the other that pays no attention to "how the rest of the slab gets out of the way" during the course of the deformation. But maybe the two notions of equivalence coincide. Do they? | |
| Oct 7, 2015 at 18:36 | history | edited | Ryan Budney | CC BY-SA 3.0 |
added 66 characters in body
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| Oct 7, 2015 at 18:30 | history | answered | Ryan Budney | CC BY-SA 3.0 |