Timeline for How are the Conway polynomial and the Alexander polynomial different?
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11 events
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| Mar 14, 2022 at 8:45 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
added the (alexander-polynomial) tag
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| May 16, 2019 at 19:27 | comment | added | Kyle Miller | @DanielMoskovich Conway's original definition of his potential ("An enumeration of knots and links, and some of their algebraic properties", section 6) is multivariable, with each string given a label. The skein relation is between strings of the same label, though Conway also gives a skein relation for strings of different labels. He mentions how it's equivalent to the multivariable Alexander polynomial by a change of variables (and normalization). But, "we have not found a satisfactory explanation of these identities, although we have verified them by [...] the 'L-matrix' definition." | |
| Nov 13, 2010 at 20:36 | answer | added | Sergey Melikhov | timeline score: 4 | |
| Jul 9, 2010 at 19:24 | comment | added | Theo Johnson-Freyd | Now I wonder the following. Suppose we look for R-matrices where each string carries a (possibly different) continuous variable, and the crossing for strings with variables $x,y$ differs from the usual "flip" by $O(xy)$. We should be able to write down examples. One neato way I could imagine them arising in nature is as follows. Suppose that for some quantum group there is an "asymptotic freeness" that braidings get "closer and closer" to symmetric as the dimensions of the representations rise. Then an appropriate rescaling should do the trick, I would think, where the $x\approx\dim^{-1}$. | |
| Jul 9, 2010 at 19:20 | comment | added | Theo Johnson-Freyd | Oh, I see. For you, an "analogue for links" should have a variable for each component? (I see now that I missed the word "multivariable" in "There is a multivariable version for links".) That wouldn't be very skein-y.... I think in the usual quantum-group approach to tangle invariants, the variable controls the "nonsymmetry" of the crossing. You can, of course, also introduce a discrete variable for each link component, running over the finite-dimensional representations of the quantum group. But that's not, I think, what you're looking for. | |
| Jul 8, 2010 at 16:11 | comment | added | Daniel Moskovich | The "Conway polynomial" for links is a single variable polynomial (of course), and is thus not an analogue to the multivariable Alexander polynomial. You can't recover the multivariable Alexander from the "Conway", certainly not by a change of variables. That might be too much to expect, but a proper (i.e. non-stupid) generalization of the Conway polynomial to some multi-variable polynomial for links seems to be something people are after. I wish I understood this better. There's some sentence that should be in this comment about the relationship to the HOMFLYPT. | |
| Jul 8, 2010 at 2:06 | comment | added | Theo Johnson-Freyd | I also don't understand the "no analogue known for links" comment. The skein relations require that the Conway polynomial be defined for links, because changing a crossing to two parallel strands changes the number of components of the link. | |
| Jul 8, 2010 at 0:37 | comment | added | algori | Daniel -- could you explain what you mean by "there is no analogue known for links"? After a change of variable in HOMFLY one gets a polynomial invariant of oriented links that satisfies "positive crossing minus negative crossing equals $t$ times no crossing" | |
| Jul 8, 2010 at 0:04 | comment | added | algori | Qiaochu -- in a sense you are right: the coefficients of the Conway polynomial are finite type invariants on the nose; the coefficients of the Alexander polynomial are functions of finite type invariants but are not of finite type themselves. | |
| Jul 7, 2010 at 23:46 | comment | added | Qiaochu Yuan | Isn't the Conway polynomial the one which comes more directly from quantum groups? | |
| Jul 7, 2010 at 22:17 | history | asked | Daniel Moskovich | CC BY-SA 2.5 |