# Zero exponent sum w.r.t group words in knot group's presentation

I am reading, "Plane Curves Associated to Character Varieties of 3-Manifolds" by Cooper, Culler, Gillet, Long, and Shalen and on page 28 of this PDF the authors talk about the tractability of 2-bridge knots and the presentation of the knot complement's fundamental group, $$\langle x,y|x\,w = w\,y\rangle.$$

They then show the generating elements, $$\mathfrak{M}=x$$, and $$\mathfrak{L}=x^n\,w\,w^*$$, where $$w^*$$ is the word $$w$$ written in reverse, and $$n$$ is chosen to make the exponent sum of $$\mathfrak{L}$$ to be zero.

So. The reason why $$n=0$$ (chosen to be so?) is to undo the winding number, as in following Wirtinger's presentation method for identifying the longitude?

This seems to be the case but then in Burde and Zieschang's book, "Knots", it seems that the longitude of any two-bridge knot is in the 2nd commutator subgroup of $$\pi_1(S^3-K)$$, and so it should have zero exponent sum - still working on looking at this with the trefoil knot...

I can also see that the exponent sum should (?) be zero since $$w$$ and $$w^*$$ have the same exponents (Burde also shows this as having symmetric powers on the word $$w$$) and another proof that is easy to follow shows that the exponents start with 1 and end with -1.

Summarily, I googled around the phrase "zero exponent sum" and have come across such relatively esoteric things like Baumslag-Solitar groups, HNN extensions of finite groups, two-generator one-relator groups, and word problems. I guess I haven't had enough group theory to know where to go and would therefore hope someone could shed some light on why this exponent is zero as well as the proper path to follow to learn more about this.

P.S. Please let me know if my question needs clarification.

• I have a full answer to this, ready to be posted, but your strange tactics on this infamous previous question are making me reluctant to post it. What should I do, oh, what should I do? Please enlighten me... – Did Sep 2 '16 at 7:13
• Hi @Did, notwithstanding the previous behavior of OP, I would appreciate an answer to this question if you're still able to post it. Thanks! – William Stagner Jun 1 '18 at 14:12
• @WilliamStagner Sorry but I would find quite inconsistent to post anything on this page. Too bad... – Did Jun 1 '18 at 15:45