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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.

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    $\begingroup$ 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
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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.

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

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