Let $K$ be an oriented knot in $S^3$ together with a framing $n$. Let $K(a,b)$ be the oriented link obtained by taking $a$ copies of the $n$-pushoff of $K$ with the same same orientation as $K$ and $b$ copies of the $n$-pushoff of $K$ with the opposite orientation as $K$. In particular, $K(1,0) = K$ and $K(0,1)$ is the reverse of $K$.

What can be said about the signature function $\sigma_{K(a,b)}(\lambda)$ in terms of $\sigma_K(\lambda)$?

Edit: The answer is yes - in fact there is a nice formula for more general satellite constructions. In "Signatures of iterated torus knots" by Litherland, it is show that $\sigma_{K(a,b)}(\lambda) = \sigma_K(\lambda^{a-b})$ (See Theorem 2 and Remark 2 following it).

Can we say anything about the nullity $n_{K(a,b)}(\lambda)$ in terms of some info about $K$? In light of the above Edit, I should probably just ask about how the nullity function behaves under satellite operations.

  • 1
    $\begingroup$ There is an "obvious" Seifert surface for $K_n(a,b)$. Suppose $a>b$; then you can pair $b$ copies of oppositely oriented push-offs of $K$ with annuli (each of which twists $n$ times), and add a copy of the Seifert surface for the $(a-b,(a-b)n)$-cable of $K$, $J$. (Then you probably need to add tubes to make it connected.) But the signature should be related to that of $J$ and that of $b$ negative $n$-Hopf links. $\endgroup$ – Marco Golla Mar 6 '19 at 1:07
  • $\begingroup$ I might be wrong, but I suspect you are trying to apply Gilmer's surgery formula for Casson-Gordon invariants. If this is the case, you might want to take a look at Theorem 6.7 of this paper: unige.ch/math/folks/cimasoni/signature.pdf . It gives a surgery formula which uses the multivariable signature instead of Levine-Tristram signatures of cables. $\endgroup$ – Anthony Conway Mar 7 '19 at 13:51

As Marco mentions, it appears you are studying cables. These are particular examples of satellite operations. As you already noticed, Litherland proves that if $S$ is a winding number $n$ satellite with pattern $P$ and companion $C$, then $$ \sigma_S(\omega)=\sigma_P(\omega)+\sigma_C(\omega^n). $$ For the nullity, I think the formula is similar: $$ \eta_S(\omega)=\eta_P(\omega)+\eta_C(\omega^n). $$ I think a proof can be found in here in a more general setting, namely "splicing": https://arxiv.org/pdf/1802.01836.pdf (Theorem 5.2). The way to recover satellites from splices is for instance explained here: https://arxiv.org/pdf/1409.5873.pdf (Section 2.4).

For the nullity, you can also prove this "by hand". I think that the most painless way is to view $\eta_K(\omega)$ as the dimension of a twisted homology vector space: $ \eta_K(\omega)=\operatorname{dim}_{\mathbb{C}} H_1(S^3 \setminus K;\mathbb{C}^\omega)$. You can then run a Mayer-Vietoris argument for the satellite knot exterior and the result will follow.

  • $\begingroup$ Thanks that is perfect and I found your review article very helpful :-) $\endgroup$ – user101010 Mar 15 '19 at 2:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.