# Signature/nullity function for a link obtained by parallel pushoffs of a knot?

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.

• 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. – Marco Golla Mar 6 '19 at 1:07
• 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. – Anthony Conway Mar 7 '19 at 13:51

## 1 Answer

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.

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