The sources I'm looking at are giving me conflicting information. One paper gives the presentation $$\langle x,y|xyx=yxy, x^2y=yx^2\rangle,$$

while another paper asserts that Example 12 from Fox's A Quick Trip through Knot Theory is in fact the two-twist spun trefoil, which has the presentation $$\langle x,y|xy^2=yx, y^2x=xy\rangle=\langle x,y|y^3=1, xyx^{-1}=y^{-1}\rangle.$$

Finally, a third paper gives the presentation as $$\langle x,y|xyx=yxy, yxy^{-1}=xyx^{-1}\rangle$$ citing a paper by Zeeman that appears to only talk about the 5-twist spun trefoil.

Through a little bit of work, I can see that the first and third groups are the same, but the second one seems distinct from them both. Also, if I use these presentations to construct the first Alexander ideal, I get different ideals, and as far as I know, the 'right' ideal should be $\langle 3, t+1\rangle$. Is there a definitive source as to which of these are right, or where my confusion might be coming from?


1 Answer 1


All of these presentations are right and present isomorphic groups.

You will get the second presentation in the form $\left<x,t | t^3=1, xtx^{-1}=t^{-1}\right>$ from the first presentation by substitution $t=yx^{-1}$.


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