Narrow Results

Let D be a clasp disk in S³ with n singularities and let U be a regular neighborhood of D. Say D bounds the knot Γ. We show that the 3-complex X=(U\Γ)∪ c * Bd(U) 3-deforms to . In particular, π₁(X)=ℤ for all 3-complexes X constructed in this manner. We observe that each X in this class corresponds to an unknotting scheme for the participating knot Γ.

We propose a method to compute complex volume of 2-bridge link complements. Our construction sheds light on a relationship between cluster variables with coefficients and canonical decompositions of link complements.

Bing, Bothe and Shilepsky studied knots that were wild at every point, and Bothe developed an invariant for a certain subclass of these knots. This paper develops invariants which distinguish ambient isotopy equivalence classes of these knots (which include the "Bing sling") and shows that there is an uncountable number of inequivalent of Bing sling type knots. As an aside, it is also shown that it is possible for inequivalent wild knots to have homeomorphic complements.

A solenoid is an inverse limit of circles. When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse limits) have different fundamental groups. Embeddings of the same solenoid can give different groups; in particular, the nicest embeddings are unknotted at each level, and give an Abelian fundamental group, while other embeddings have non-Abelian groups. We show using geometry that every solenoid has uncountably many embeddings with nonhomeomorphic complements.

The Gukov–Manolescu series, denoted by $F_K$, is a conjectural invariant of knot complements that, in a sense, analytically continues the colored Jones polynomials. In this paper we use the large color $R$-matrix to study $F_K$ for some simple links. Specifically, we give a definition of $F_K$ for positive braid knots, and compute $F_K$ for various knots and links. As a corollary, we present a class of “strange identities” for positive braid knots.

Tied links in $S^3$ were introduced by Aicardi and Juyumaya as standard links in $S^3$ equipped with some non-embedded arcs, called ties, joining some components of the link. Tied links in the Solid Torus were then naturally generalized by Flores. In this paper, we study this new class of links in other topological settings. More precisely, we study tied links in the lens spaces $L(p,1)$, in handlebodies of genus $g$, and in the complement of the $g$-component unlink. We introduce the tied braid monoids $T{M}_{g,n}$ by combining the algebraic mixed braid groups defined by Lambropoulou and the tied braid monoid, and we formulate and prove analogues of the Alexander and the Markov theorems for tied links in the 3-manifolds mentioned above. We also present an $L$-move braid equivalence for tied braids and we discuss further research related to tied links in knot complements and c.c.o. 3-manifolds. The theory of tied links has potential use in some aspects of molecular biology.

An interesting class of knots have complement with a remarkably simple topological description. This class includes all the arborescent knots with only even weights hence, in particular, the two bridge knots and many knots of ten or fewer crossings. For these knots, there are choices of minimal genus Seifert surfaces S such that all taut, depth one foliations of the knot complement, having S as sole compact leaf, can be classified up to isotopy. These foliations correspond exactly to the lattice points over the open faces of the unit ball in a Thurston-like norm on the relative homology of the complement of S.

We construct infinitely many examples of pairs of isospectral but non-isometric $1$-cusped hyperbolic $3$-manifolds. These examples have infinite discrete spectrum and the same Eisenstein series. Our constructions are based on an application of Sunada’s method in the cusped setting, and so in addition our pairs are finite covers of the same degree of a 1-cusped hyperbolic 3-orbifold (indeed manifold) and also have the same complex length spectra. Finally we prove that any finite volume hyperbolic 3-manifold isospectral to the figure-eight knot complement is homeomorphic to the figure-eight knot complement.