Related papers: Left-orderable surgeries on the knot $6_2$ via hyp…
This paper employs knot invariants and results from hyperbolic geometry to develop a practical procedure for checking the cosmetic surgery conjecture on any given one-cusped manifold. This procedure has been used to establish the following…
In this paper, by using the regulator map of Beilinson-Deligne, we show that the quantization condition posed by Gukov is true for the SL_2(\mathbb{C}) character variety of the hyperbolic knot in S^3. Furthermore, we prove that the…
Concerning the set of exceptional surgery slopes for a hyperbolic knot, Lackenby and Meyerhoff proved that the maximal cardinality is 10 and the maximal diameter is 8. Their proof is computer-aided in part, and both bounds are achieved…
Distinct knots K, K' can sometimes share a common p/q-framed Dehn surgery. A folk conjecture held that for a fixed pair of knots, this can occur for at most one value of p/q. We disprove this conjecture by constructing pairs of distinct…
We study certain linear representations of the knot group that induce augmentations of knot contact homology. This perspective on augmentations enhances our understanding of the relationship between the augmentation polynomial and the…
This paper initiates the study of circular orderability of $3$-manifold groups, motivated by the L-space conjecture. We show that a compact, connected, $\mathbb{P}^2$-irreducible $3$-manifold has a circularly orderable fundamental group if…
In these lecture notes, the representation theory of the Heisenberg group as well as Howe's construction of the metaplectic group by means of twisted convolution operators with generalized, complex Gaussians are reviewed, and it is shown…
We show that the fundamental group of the double branched cover of an infinite family of homologically thin, non-quasi-alternating knots is not left-orderable, giving further support for a conjecture of Boyer, Gordon, and Watson that an…
Berge in [1] defined doubly primitive knots, which yield lens spaces by Dehn surgery. At the same paper he listed the knots into several types. In this paper we will prove the list is complete when $\tau>1$. The invariant $\tau$ is a…
We review the representation theory of the quantum group $U_\epsilon sl_2\mathbb{C}$ at a root of unity $\epsilon$ of odd order, focusing on geometric aspects related to the 3-dimensional quantum hyperbolic field theories (QHFT). Our…
The paper concerns two classical problems in knot theory pertaining to knot symmetry and knot exteriors. In the context of a knotted handlebody $V$ in a $3$-sphere $S^3$, the symmetry problem seeks to classify the mapping class group of the…
We consider knots and links in handlebodies that have hyperbolic complements and operations akin to composition. Cutting the complements of two such open along separating twice-punctured disks such that each of the four resulting…
We show that any parabolic generating pair of a genus-one hyperbolic 2-bridge knot group is equivalent to the upper or lower meridian pair. As an application, we obtain a complete classification of the epimorphisms from 2-bridge knot groups…
We present a series of algorithms for skein manipulation in a genus-2 handlebody, implementing a novel strand sorting method to reduce any skein to a skein in a 2-punctured disk. This reduction guarantees resolution as a linear combination…
We prove Riley's conjecture on the number of parabolic SL(2,R) representations of 2-bridge knot groups.
Given any knot k, there exists a hyperbolic knot tilde k with arbitrarily large volume such that the knot group pi k is a quotient of pi tilde k by a map that sends meridian to meridian and longitude to longitude. The knot tilde k can be…
Any profinite isomorphism between two cusped finite-volume hyperbolic 3-manifolds carries profinite isomorphisms between their Dehn fillings. With this observation, we prove that some cusped finite-volume hyperbolic 3-manifolds are…
Earlier work with Robert Gompf and Abigail Thompson classified, via a natural slope indexed by the rationals, all two-component links which contain the square knot and from which $(S^1 \times S^2) \# (S^1 \times S^2)$ can be obtained by…
We show that for a hyperbolic knot complement, all but at most 12 Dehn fillings are irreducible with infinite word-hyperbolic fundamental group.
Exceptional Dehn surgeries have been classified for 2-bridge knots and Montesinos knots of length at least 4. In this paper we classify all toroidal Dehn surgeries on Montesinos knots of length 3.