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In this paper, we find a more straightforward problem that is equivalent to one of the major challenges in knot theory: the classification of links in the 3-sphere. More precisely, we provide a simpler braid description for all links in the 3-sphere in terms of generalised T-links. With this, we translate the problem of classifying links in the 3-sphere into a problem of counting the number of generalised T-links that represent the same link. Generalised T-links are a natural generalisation of twisted torus links and Lorenz links, two families of links that have been extensively studied by many people. Moreover, we generalise the bunch algorithm to construct links embedded in universal Lorenz-like templates and provide an upper volume bound that is quadratic in the trip number. We use the upper bound obtained from the generalised bunch algorithm for generalised T-links to establish an upper bound for the sum of the volumes of the hyperbolic pieces of all closed, orientable, connected 3-manifolds., Comment: 24 pages, 6 figures, V2 change in abstract

Every finite collection of oriented closed geodesics in the modular surface has a canonically associated link in its unit tangent bundle coming from the periodic orbits of the geodesic flow. We study the volume of the associated link complement with respect to its unique complete hyperbolic metric. We provide the first lower volume bound that is linear in terms of the number of distinct exponents in the code words corresponding to the collection of closed geodesics., Comment: 20 pages, 8 figures

In the 1970s, Williams developed an algorithm that has been used to construct modular links. We introduce the notion of bunches to provide a more efficient algorithm for constructing modular links in the Lorenz template. Using the bunch perspective, we construct parent manifolds for modular link complements and provide the first upper volume bound that is independent of word exponents and quadratic in the braid index. We find families of modular knot complements with upper volume bounds that are linear in the braid index. A classification of modular link complements based on the relative magnitudes of word exponents is also presented., Comment: 29 pages, 10 figures

We construct a sequence of geodesics on the modular surface such that the complement of the canonical lifts to the unit tangent bundle are arithmetic 3-manifolds., Comment: version accepted to PJM

Every oriented closed geodesic on the modular surface has a canonically associated knot in its unit tangent bundle coming from the periodic orbit of the geodesic flow. We study the volume of the associated knot complement with respect to its unique complete hyperbolic metric. We show that there exist sequences of closed geodesics for which this volume is bounded linearly in terms of the period of the geodesic's continued fraction expansion. Consequently, we give a volume's upper bound for some sequences of Lorenz knots complements, linearly in terms of the corresponding braid index. Also, for any punctured hyperbolic surface we give volume's bounds for the canonical lift complement relative to some sequences of sets of closed geodesics in terms of the geodesics length., Comment: 36 pages, 22 figures. New result on volume's upper bound of some Lorenz knots. A generalization for volume's bounds in terms of the geodesic length

Let $S$ be a surface of negative Euler characteristic and consider a finite filling collection $\Gamma$ of closed curves on $S$ in minimal position. An observation of Foulon and Hasselblatt shows that $PT(S) \setminus \hat{\Gamma}$ is a finite-volume hyperbolic 3-manifold, where $PT(S)$ is the projectivized tangent bundle and $\hat\Gamma$ is the set of tangent lines to $\Gamma$. In particular, $vol(PT(S) \setminus \hat{\Gamma})$ is a mapping class group invariant of the collection $\Gamma$. When $\Gamma$ is a filling pair of simple closed curves, we show that this volume is coarsely comparable to Weil-Petersson distance between strata in Teichm\"uller space. Our main tool is the study of stratified hyperbolic links $\bar\Gamma$ in a Seifert-fibered space $N$ over $S$. For such links, the volume of $N\setminus\bar\Gamma$ is coarsely comparable to expressions involving distances in the pants graph., Comment: paper has been accepted for publication by the Journal of Topology

Let $\bar\gamma$ be a link in a Seifert fibered space $M$ over a hyperbolic $2$-orbifolds $\mathcal O$ that projects injectively to a filling multicurve of closed geodesics $\gamma$ in $\mathcal O.$ We prove that the complement $M_{\bar\gamma}$ of $\bar\gamma$ in $M$ admits a hyperbolic structure of finite volume and give combinatorial bounds of its volume., Comment: V3: fixed many typos, clarified some arguments and improved expositions

Every closed geodesic $\gamma$ on a surface has a canonically associated knot $\widehat\gamma$ in the projective unit tangent bundle. We study, for $\gamma$ filling, the volume of the associated knot complement with respect to its unique complete hyperbolic metric. We provide a lower bound for the volume relative to the number of homotopy classes of $\gamma$-arcs in each pair of pants of a pants decomposition of the surface., Comment: 28 pages, 12 figures