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May
18
revised Question about the Weeks Manifold
Added reference to paper
May
18
comment Question about the Weeks Manifold
I should add that Regina is able to do this calculation for any triangulation of a closed or cusped manifold. The algorithm is at least exponential in the number of tetrahedra but is still practical even when this number becomes quite large. For example in 2009 Burton, Rubinstein and Tillmann used it to show that the Seifert-Weber dodecahedral space is not Haken using a triangulation with 23 tetrahedra. See arxiv.org/abs/0909.4625
May
17
answered Question about the Weeks Manifold
May
3
comment Computing minimal polynomials using LLL
@JoeSilverman Yes but shouldn't the bounds on the degree and height of $\alpha$ mean that $a$ determines a unique algebraic number, and so a unique polynomial? Perhaps a more basic question would be: is it true that $|a_i| \leq N$ (or some similar bound)?
May
1
comment Computing minimal polynomials using LLL
@JoeSilverman: I should also add that the PARI documentation for algdep says that "the polynomial which is obtained is not necessarily the "correct" one". However its code requests a 0.99---LLL Reduced Basis using ZM_lll. With $\delta=0.99$ being so close to 1, we get "better guarantees for the basis (in principle smaller basis vectors) but longer running times".
May
1
comment Computing minimal polynomials using LLL
@JoeSilverman: Thanks for the suggestion. I've only had a brief chance to look at PARI but the relevant code appears to be "lindep2" in ./basemath/bibli1.c (the precision to work to (equiv. the choice of $N$) is chosen here not in algdep0). It appears that if the binary expansion of $a$ has $i$ bits before the decimal and $j$ bits afterwards then PARI uses $N = 10^{32+i}$ if $j = 0$ and $N = 10^{0.8 j}$ otherwise (or a user given precision).
May
1
comment Computing minimal polynomials using LLL
@Igor: I guess I have two questions: 1) What exactly does Cohen mean by "subtle"? 2) Have there been any further developments in how to choose $N$ in the 20 years since Cohen's book?
May
1
asked Computing minimal polynomials using LLL
Mar
23
awarded  Enlightened
Mar
23
awarded  Nice Answer
Mar
13
answered 2-bridge knots in the Rolfsen's table
Mar
9
awarded  Nice Answer
Jan
10
awarded  Yearling
Dec
19
awarded  Necromancer
Nov
5
comment Fantastic properties of Z/2Z
So I guess that the correct reference should be Gromov's "Asymptotic invariants of infinite groups" but it is ~300 pages and I haven't read it. Danny gave a talk at Cornell and in the first ~15 mins he covers the density model and gives a sketch of this result. It's available here: cornell.edu/video/danny-calegari-random-groups-diamonds-glass
Nov
4
answered Fantastic properties of Z/2Z
Oct
13
awarded  Notable Question
Sep
24
revised Is there a table of (fibred knot) monodromies?
Added link to more data.
Jul
2
awarded  Curious
May
9
comment Does small Perron-Frobenius eigenvalue imply small entries for integral matrices?
Thanks, Do you know of any reference for this result?