Related papers: Limiting characters at ideal points detecting twic…
We provide a partial classification of the 3-strand pretzel knots $K = P(p,q,r)$ with unknotting number one. Following the classification by Kobayashi and Scharlemann-Thompson for all parameters odd, we treat the remaining families with $r$…
Let G be a simple complex algebraic group and g its Lie algebra. We show that the g-Witten-Reshetikhin-Turaev quantum invariants determine a deformation-quantization, C_q[X_G(torus)], of the coordinate ring of the G-character variety of the…
In this note, we attempt to find counterexamples to the conjecture that the ideal form of a knot, that which minimizes its contour length while respecting a no-overlap constraint, also minimizes the volume of the knot, as determined by its…
We show that under a suitable transversality condition, the intersection of two rational subtori in an algebraic torus $(\C^*)^n$ is a finite group which can be determined using the torsion part of some associated lattice. Applications are…
We give a criterion for distinguishing a prime knot $K$ in $S^3$ from every other knot in $S^3$ using the finite quotients of $\pi_1(S^3\setminus K)$. Using recent work of Baldwin-Sivek, we apply this criterion to the hyperbolic knots…
We study when the Thurston norm is detected by twisted Alexander polynomials associated to representations of the 3-manifold group to SL(2, C). Specifically, we show that the hyperbolic torsion polynomial determines the genus for a large…
We extend the recent classification of Hilbert schemes with two Borel-fixed points to arbitrary characteristic. We accomplish this by synthesizing Reeves' algorithm for generating strongly stable ideals with the basic properties of…
We prove that 0 is a characterizing slope for infinitely many knots, namely the genus-1 knots whose knot Floer homology is 2-dimensional in the top Alexander grading, which we classified in recent work and which include all $(-3,3,2n+1)$…
By a result of Klyachko the Euler characteristic of moduli spaces of stable bundles of rank two on the projective plane is determined. Using similar methods we extend this result to bundles of rank three. The fixed point components…
We assign to a finite $CW$-complex and an element in its first cohomology group a twisted version of the $L^2$-Euler characteristic and study its main properties. In the case of an irreducible orientable $3$-manifold with empty or toroidal…
A conjecture, known as the Shokurov-Koll\'ar connectedness principle, predicts the following. Let $(X,B)$ be a pair, and let $f \colon X \rightarrow S$ be a contraction with $-(K_X + B)$ nef over $S$; then, for any point $s \in S$, the…
Motivated by recent work with Mazorchuk, we characterize the conditions under which the intersection of two principal order ideals in the Bruhat order is boolean. That characterization is presented in three versions: in terms of reduced…
We show that all twist knots, certain double twist knots and some other 2-bridge knots are minimal elements for the partial ordering on the set of prime knots. The key to these results are presentations of their character varieties using…
In this paper we study welded knots and their invariants. We focus on generating examples of non-trivial knotted ribbon tori as the tube of welded knots that are obtained from classical knot diagrams by welding some of the crossings.…
A 1-bridge torus knot in a 3-manifold of genus $\le 1$ is a knot drawn on a Heegaard torus with one bridge. We give two types of normal forms to parameterize the family of 1-bridge torus knots that are similar to the Schubert's normal form…
The cosmetic crossing conjecture posits that switching a non-trivial crossing in a knot diagram always changes the knot type. Generalizing work of Balm, Friedl, Kalfagianni and Powell, and of Lidman and Moore, we give an Alexander…
We investigate limit linear series on chains of elliptic curves, giving a simple proof of a conjecture of Farkas stating the existence of curves with a theta-characteristic with a given number of sections for the expected range of genera.…
In this paper, we characterize closed incompressible surfaces of genus two in the complements of 3-bridge knots and links. This characterization includes that of essential 2-string tangle decompositions for 3-bridge knots and links.
We classify closed curves on a once-punctured torus with a single self-intersection from a combinatorial perspective. We determine the number of closed curves with given word-length and with zero, one, and arbitrary self-intersections.
In the present paper, we will show that for any integer n>0 there are infinitely many twisted torus knots with n-string essential tangle decompositions.