# Wrong equation in a paper about cosmic topology

+ 4 like - 0 dislike
229 views

I am studying the paper  " Exact Polynomial Eigenmodes for Homogeneous Spherical 3-Manifolds"  (https://arxiv.org/pdf/math/0502566v3.pdf)

I am trying to obtain the polynomials given by the equation (43) on page 19.  I am noting that there is an error in the first and the third polynomials in equation (43).   According with the mentioned paper: but the correct expressions are Could you confirm the error and the correct expressions?  Many thanks

+ 4 like - 0 dislike

$T_8 = T'_{4a}T'_{4b} = ({{{\alpha}^{4}}+{2 i {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}) ({{{\alpha}^{4}}-{2 i {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}) =\\ {{{\alpha}^{8}}+{6 {\alpha}^{4} {\beta}^{4}}}+{{\beta}^{8}}$

and

$T_{12} = \frac {T_{4a}^{'3} + T_{4b}^{'3}}{2} = \frac{{{{{({\alpha}^{4}}+{2 i {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}})^{3}}+{{{{({\alpha}^{4}}-{2 i {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}} )^ {3}}}{2} = \\ {{{{\alpha}^{12}}-{9 {\alpha}^{8} {\beta}^{4}}}-{9 {\alpha}^{4} {\beta}^{8}}}+{{\beta}^{12}}$

while with the missing $\sqrt(3)$ , we get correctly (44) :

$T_8 = T'_{4a}T'_{4b} = ({{{\alpha}^{4}}+{2 i \sqrt{3} {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}) ({{{\alpha}^{4}}-{2 i \sqrt{3} {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}) =\\ {{{\alpha}^{8}}+{14 {\alpha}^{4} {\beta}^{4}}}+{{\beta}^{8}}$

and

$T_{12} = \frac {T_{4\alpha}^{'3} + T_{4\beta}^{'3}}{2} = \frac{{{{{({\alpha}^{4}}+{2 i \sqrt{3} {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}})^{3}}+ {{{{({\alpha}^{4}}-{2 i \sqrt{3} {\alpha}^{2} {\beta}^{2}}}+{{\beta}^{4}}} )^ {3}}}{2} = \\ {{{{\alpha}^{12}}-{33 {\alpha}^{8} {\beta}^{4}}}-{33 {\alpha}^{4} {\beta}^{8}}}+{{\beta}^{12}}$

Idem for the following. I mean the author didn't used the equation with the typo error for the next calculus.

Then it is probably just a well found typo error :) You must write to the author to release a new version.

answered Oct 13, 2016 by (360 points)
edited Oct 13, 2016 by igael

@igael, you are very right.  Many thanks.  All the best.

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverflo$\varnothing$Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.