Let $K_{p,q}$ be a $(p,q)$-cable of the non-trivial knot $K$ in $S^3$. Let $V_{p,q}$ and $V$ denote the Seifert matrices of $K_{p,q}$ and $K$, respectively.
Is it possible to obtain a closed formula for the matrix $V_{p,q}$ in terms of $V$?
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2 Answers
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It also appeared in the article of H. Seifert as Theorem II:
Seifert, H. (1950). On the homology invariants of knots. The Quarterly Journal of Mathematics, 1(1), 23-32.
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$\begingroup$ Wow. It is surprising to me. $\endgroup$– user160180Commented Aug 12, 2020 at 20:03
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2$\begingroup$ You can even draw the whole Seifert surface: sketchesoftopology.wordpress.com/2009/11/18/… $\endgroup$ Commented Aug 13, 2020 at 8:07
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A formula for the Seifert matrix of an arbitrary satellite can be found in the proof of Theorem 6.15 of Lickorish's "An introduction to knot theory".
answered Aug 12, 2020 at 14:40