# Alexander and Jones polynomials of surgerized tst links

## TL;DR

This paper extends the concept of twisted solid tori links to surgerized tst links, and computes their braid words, Alexander, and Jones polynomials, revealing new properties of these complex links.

## Contribution

It introduces surgerized tst links and provides explicit calculations of their braid words and polynomial invariants, expanding the understanding of tst link generalizations.

## Key findings

- Derived braid words for surgerized tst links
- Computed Alexander polynomials of these links
- Computed Jones polynomials of these 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.