I recently came upon a recursive formula for the (ordinary) signatures of torus knots. The formula, which I found in Murasugi's book "Knot Theory and Applications" (Springer, 2007), originally appeared in a paper by Gordon, Litherland and Murasugi, "Signatures of Covering Links" (<http://www.maths.ed.ac.uk/~aar/papers/glm.pdf>). It is apparent from the formula that torus knots have even signatures. I wondered if this is a property specific to torus knots, so I went and poked around the Knot Atlas for a while. I did not see any knot of odd signature. (I did not check every knot, but checked enough to get discouraged.) 


My question is: does there exist a knot of odd signature? 


Secondly, why do all torus knots (and, quite possibly, all knots) have even signature? 



Parity of knot signatures
This is exercise 3.4 in [Livingston's book (Knot Theory, Carus math. monographs, vol 24, page 123).](http://books.google.com/books?id=KXAS3KRZGRMC&pg=PA123&lpg=PA123&dq=knot%20signature%20is%20even&source=bl&ots=OxeFQFpqv_&sig=r9zshQSSogYUAPIFEfrFdy8A8Vw&hl=en&sa=X&ei=bapyU_SxI8mRyAT81oDwBg&ved=0CHgQ6AEwCA#v=onepage&q=knot%20signature%20is%20even&f=false)


