Narrow Results

In this paper, we define a new algebro-geometric invariant of three-manifolds resulting from Dehn surgery along a hyperbolic knot complement in S³. We establish a Casson-type invariant for these three-manifolds. In the last section, we explicitly calculate the character variety of the figure-eight knot and discuss some applications, as well as the computation of our new invariants for some three-manifolds resulting from Dehn surgery along the figure-eight knot.

We prove that the character variety of a family of one-relator groups has only one defining polynomial and we provide the means to compute it. Consequently, we give a basis for the Kauffman bracket skein module of the exterior of the rational link L_p/q of two components modulo the (A + 1)-torsion.

We study the twisted Alexander polynomial from the viewpoint of the SL(2, ℂ)-character variety of nonabelian representations of a knot group. It is known that if a knot is fibered, then the twisted Alexander polynomials associated with nonabelian SL(2, ℂ)-representations are all monic. In this paper, we show that for a 2-bridge knot there exists a curve component in the SL(2, ℂ)-character variety such that if the knot is not fibered then there are only finitely many characters in the component for which the associated twisted Alexander polynomials are monic. We also show that for a 2-bridge knot of genus g, in the above curve component for all but finitely many characters the associated twisted Alexander polynomials have degree 4g - 2.

For a fibered knot in the 3-sphere the twisted Alexander polynomial associated to an SL(2, ℂ)-character is known to be monic. It is conjectured that for a nonfibered knot there is a curve component of the SL(2, ℂ)-character variety containing only finitely many characters whose twisted Alexander polynomials are monic, i.e. finiteness of such characters detects fiberedness of knots. In this paper, we discuss the existence of a certain curve component which relates to the conjecture when knots have nonmonic Alexander polynomials. We also discuss the similar problem of detecting the knot genus.

We determine the PSL₂(ℂ) and SL₂(ℂ) character varieties of the once-punctured torus bundles with tunnel number one, i.e. the once-punctured torus bundles that arise from filling one boundary component of the Whitehead link exterior. In particular, we determine "natural" models for these algebraic sets, identify them up to birational equivalence with smooth models, and compute the genera of the canonical components. This enables us to compare dilatations of the monodromies of these bundles with these genera. We also determine the minimal polynomials for the trace fields of these manifolds. Additionally, we study the action of the symmetries of these manifolds upon their character varieties, identify the characters of their lens space fillings, and compute the twisted Alexander polynomials for their representations to SL₂(ℂ).

We explicitly calculate the universal character ring of the (-2, 2m + 1, 2n)-pretzel link and show that it is reduced for all integers m and n.

Let 𝖥_r be a free group of rank r, 𝔽_q a finite field of order q, and let SL_n(𝔽_q) act on Hom(𝖥_r, SL_n(𝔽_q)) by conjugation. We describe a general algorithm to determine the cardinality of the set of orbits Hom(𝖥_r, SL_n(𝔽_q))/SL_n(𝔽_q). Our first main theorem is the implementation of this algorithm in the case n = 2. As an application, we determine the E-polynomial of the character variety Hom(𝖥_r, SL₂(ℂ))//SL₂(ℂ), and of its smooth and singular locus. Thus we determine the Euler characteristic of these spaces.

The author constructs the moduli of representations whose images generate the subalgebra of upper triangular matrices (up to inner automorphisms) of the full matrix ring for any groups and any monoids.

We compute the Hodge polynomials for the moduli space of representations of an elliptic curve with two marked points into SL(2, ℂ). When we fix the conjugacy classes of the representations around the marked points to be diagonal and of modulus one, the character variety is diffeomorphic to the moduli space of strongly parabolic Higgs bundles, whose Betti numbers are known. In that case we can recover some of the Hodge numbers of the character variety. We extend this result to the moduli space of doubly periodic instantons.

Via counting over finite fields, we derive explicit formulas for the $E$-polynomials and Euler characteristics of ${\mathrm{\text{GL}}}_d$- and ${\mathrm{\text{PGL}}}_d$-character varieties of free groups. We prove a positivity property for these polynomials and relate them to the number of subgroups of finite index.

We investigate representations of Kähler groups $\mathrm{\Gamma }={\pi }_1(X)$ to a semisimple non-compact Hermitian Lie group $G$ that are deformable to a representation admitting an (anti)-holomorphic equivariant map. Such representations obey a Milnor–Wood inequality similar to those found by Burger–Iozzi and Koziarz–Maubon. Thanks to the study of the case of equality in Royden’s version of the Ahlfors–Schwarz lemma, we can completely describe the case of maximal holomorphic representations. If ${dim}_{ℂ}X\ge 2$, these appear if and only if $X$ is a ball quotient, and essentially reduce to the diagonal embedding $\mathrm{\Gamma }<\text{SU}(n,1)\to \text{SU}(nq,q)\hookrightarrow \text{SU}(p,q)$. If $X$ is a Riemann surface, most representations are deformable to a holomorphic one. In that case, we give a complete classification of the maximal holomorphic representations, which thus appear as preferred elements of the respective maximal connected components.

We determine the $\mathrm{\text{SL}}(2,ℂ)$-character variety for each odd classical pretzel knot $P(2k_1+1,2k_2+1,2k_3+1)$, and present a method for computing its A-polynomial.

We establish a gluing construction for Higgs bundles over a connected sum of Riemann surfaces in terms of solutions to the $\mathrm{\text{Sp}}(4,ℝ)$-Hitchin equations using the linearization of a relevant elliptic operator. The construction can be used to provide model Higgs bundles in all the $2g-3$ exceptional components of the maximal $\mathrm{\text{Sp}}(4,ℝ)$-Higgs bundle moduli space, which correspond to components solely consisting of Zariski dense representations. This also allows a comparison between the invariants for maximal Higgs bundles and the topological invariants for Anosov representations constructed by Guichard and Wienhard.

The skein algebra of an oriented three-manifold is a classical limit of the Kauffman bracket skein module and gives the coordinate ring of the ${SL}_2(ℂ)$-character variety. In this paper, we determine the quotient of a polynomial ring which is isomorphic to the skein algebra of a group with three generators and two relators. As an application, we give an explicit formula for the skein algebra of the Borromean rings complement in $S^3$.

We show that any tunnel number one knot group has a two generator one relator presentation in which the relator is a palindrome in the generators. We use this fact to compute the character variety for this knot groups and we show that it is an affine algebraic set .

We classify Dehn surgeries on (p,q,r) pretzel knots resulting in a manifold M(α) having cyclic fundamental group and analyze those leading to a finite fundamental group. The proof uses the theory of cyclic and finite surgeries developed by Culler, Shalen, Boyer, and Zhang. In particular, Culler-Shalen seminorms play a central role.

We show that the -character variety of the (-2, 3, n) pretzel knot consists of two (respectively three) algebraic curves when 3 ∤ n (respectively 3 | n) and given an explicit calculation of the Culler-Shalem seminorms of these curves. Using this calculation, we describe the fundamental polygon and Newton polygon for these knots and give a list of Dehn surgerise yielding a manifold with finite or cyclic fundamental group. This constitutes a new proof of property P for these knots.

In this paper we prove that if M_K is the complement of a non-fibered twist knot K in , then M_K is not commensurable to a fibered knot complement in a ℤ/2ℤ-homology sphere. To prove this result we derive a recursive description of the character variety of twist knots and then prove that a commensurability criterion developed by Calegari and Dunfield is satisfied for these varieties. In addition, we partially extend our results to a second infinite family of 2-bridge knots.

In this paper we give a formula for the A-polynomial of the (2, 2p + 1)-torus knot, for any integer p, by using noncommutative methods (the Kauffman bracket skein modules).

We define A-polynomial n-tuple for a link of n-components and apply them to construct hyperbolic link manifolds with non-integral traces.