In this note, we exhibit concrete examples of characterizing slopes for the knot $12n242$, also known as the $(-2,3,7)$-pretzel knot. Although it was shown by Lackenby that every knot admits infinitely many characterizing slopes, the nonconstructive nature of the proof means that there are very few hyperbolic knots for which explicit examples of characterizing slopes are known.

Relations between the Atiyah–Patodi–Singer rho invariant and signatures of links have been known for a long time, but they were only partially investigated. In order to explore them further, we develop a versatile cut-and-paste formula for the rho invariant, which allows us to manipulate manifolds in a convenient way. With the help of this tool, we give a description of the multivariable signature of a link $L$ as the rho invariant of some closed three-manifold $Y_L$ intrinsically associated with $L$. We study then the rho invariant of the manifolds obtained by the Dehn surgery on $L$ along integer and rational framings. Inspired by the results of Casson and Gordon and Cimasoni and Florens, we give formulas expressing this value as a sum of the multivariable signature of $L$ and some easy-to-compute extra terms.

A slope r is called a left orderable slope of a knot $K \subset S^3$ if the 3-manifold obtained by r-surgery along K has left orderable fundamental group. Consider double twist knots $C(2m, \pm 2n)$ and $C(2m+1, -2n)$ in the Conway notation, where $m \ge 1$ and $n \ge 2$ are integers. By using continuous families of hyperbolic ${\mathrm {SL}}_2(\mathbb {R})$ -representations of knot groups, it was shown in [8, 16] that any slope in $(-4n, 4m)$ (resp. $ [0, \max \{4m, 4n\})$ ) is a left orderable slope of $C(2m, 2n)$ (resp. $C(2m, - 2n)$ ) and in [6] that any slope in $(-4n,0]$ is a left orderable slope of $C(2m+1,-2n)$ . However, the proofs of these results are incomplete, since the continuity of the families of representations was not proved. In this paper, we complete these proofs, and, moreover, we show that any slope in $(-4n, 4m)$ is a left orderable slope of $C(2m+1,-2n)$ detected by hyperbolic ${\mathrm {SL}}_2(\mathbb {R})$ -representations of the knot group.

We characterize the $(1,1)$ knots in the 3-sphere and lens spaces that admit non-trivial L-space surgeries. As a corollary, 1-bridge braids in these manifolds admit non-trivial L-space surgeries. We also recover a characterization of the Berge manifold among 1-bridge braid exteriors.

We give a method for constructing a Legendrian representative of a knot in $S^{3}$ which realizes its maximal Thurston–Bennequin number under a certain condition. The method utilizes Stein handle decompositions of $D^{4}$ , and the resulting Legendrian representative is often very complicated (relative to the complexity of the topological knot type). As an application, we construct infinitely many knots in $S^{3}$ each of which yields a reducible 3-manifold by a Legendrian surgery in the standard tight contact structure. This disproves a conjecture of Lidman and Sivek.

We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus $g$ must have slope $2g-1$, leading to a proof of the cabling conjecture for positive knots of genus 2. Our techniques also produce bounds on the maximum Thurston–Bennequin numbers of cables.

We show that the manifold resulting from $r$-surgery on the knot ${{5}_{2}}$, which is the two-bridge knot corresponding to the rational number $3/7$, has a left-orderable fundamental group if the slope $r$ satisfies $0\,\le \,r\,\le \,4$.

We show that any exceptional non-trivial Dehn surgery on a twist knot, except the trefoil, yields a 3-manifold whose fundamental group is left-orderable. This is a generalization of a result of Clay, Lidman, and Watson, and also gives a new supporting evidence for a conjecture of Boyer, Gordon, andWatson.

We study lens space surgeries along two different families of 2-component links, denoted by ${A}_{m, n} $ and ${B}_{p, q} $, related with the rational homology $4$-ball used in J. Park’s (generalized) rational blow down. We determine which coefficient $r$ of the knotted component of the link yields a lens space by Dehn surgery. The link ${A}_{m, n} $ yields a lens space only by the known surgery with $r= mn$ and unexpectedly with $r= 7$ for $(m, n)= (2, 3)$. On the other hand, ${B}_{p, q} $ yields a lens space by infinitely many $r$. Our main tool for the proof are the Reidemeister-Turaev torsions, that is, Reidemeister torsions with combinatorial Euler structures. Our results can be extended to the links whose Alexander polynomials are same as those of ${A}_{m, n} $ and ${B}_{p, q} $.

We consider orientable closed connected 3-manifolds obtained by performing Dehn surgery on the components of some classical links such as Borromean rings and twisted Whitehead links. We find geometric presentations of their fundamental groups and describe many of them as 2-fold branched coverings of the 3-sphere. Finally, we obtain some topological applications on the manifolds given by exceptional surgeries on hyperbolic 2-bridge knots.

We give a necessary condition for a two-bridge knot or link S(p,q) to be 2-adjacent to another two-bridge knot or link S(r,s). In particular, we show that if the trivial knot or link is 2-adjacent to S(p,q), then S(p,q) is trivial, that if S(p,q) is 2-adjacent to its mirror image, then S(p,q) is amphicheiral, and that for a prime integer p, if S(p,q) is 2-adjacent to S(r,s), then S(p,q)=S(r,s) or S(r,s)=S(1,0).

For a non-trivial knot in the 3-sphere, only integral Dehn surgery can create a closed 3-manifold containing a projective plane. If we restrict ourselves to hyperbolic knots, the corresponding claim for a Klein bottle is still true. In contrast to these, we show that non-integral surgery on a hyperbolic knot can create a closed non-orientable surface of any genus greater than two.

Let $K$ be a knot in ${{S}^{3}}$ . This paper is devoted to Dehn surgeries which create 3-manifolds containing a closed non-orientable surface $\hat{S}$ . We look at the slope $p/q$ of the surgery, the Euler characteristic $\mathcal{X}(\hat{S})$ of the surface and the intersection number $s$ between $\hat{S}$ and the core of the Dehn surgery. We prove that if $\mathcal{X}(\hat{S})\,\ge \,15\,-3q$, then $s\,=\,1$. Furthermore, if $s\,=\,1$ then $q\,\le \,4\,-\,3\,\mathcal{X}(\hat{S})$ or $K$ is cabled and $q\,\le \,8\,-5\mathcal{X}(\hat{S})$ . As consequence, if $K$ is hyperbolic and $\mathcal{X}(\hat{S})\,=\,-1$ , then $q\,\le \,7$.

In this paper we present a short definition of the Witten invariants of 3-manifolds. We also give simple proofs of invariance of those obtained for r = 3 and r = 4. Our definition is extracted from the 1993 paper of Lickorish and the Prasolov-Sossinsky book, where it is dispersed over 20 pages. We show by several examples that it is indeed convenient for calculations.