Alexander and Jones polynomials of surgerized tst links

Abstract

This paper is a continuation on the 2012 paper on "Cutting Twisted Solid Tori (TSTs)", in which we considered twisted solid torus links (tst links). We generalize the notion of tst links to "surgerized tst links": recall that when performing $\Phi^\mu(n(\tau), d(\tau), M)$ on a tst $\langle \tau \rangle$ where $M$ is odd, we obtain the tst link, $[\Phi^\mu(n(\tau), d(\tau), M)]$ that contains a trivial knot as one of its components. We then perform another operation $\Phi^{\mu'}(n(\tau '), d(\tau '), M')$ on that trivial knot to create a new link, which we call a "surgerized tst link" (stst link). If $M'$ is odd, we can repeat the process to give more complicated stst links. We compute braid words, Alexander and Jones polynomials of such links.