Narrow Results

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.

Twisted Alexander invariants of knots are well defined up to multiplication of units. We get rid of this multiplicative ambiguity via a combinatorial method and define normalized twisted Alexander invariants. We then show that the invariants coincide with sign-determined Reidemeister torsion in a normalized setting, and refine the duality theorem. We further obtain necessary conditions on the invariants for a knot to be fibered, and study behavior of the highest degrees of the invariants.

We introduce a new relation for high dimensional non-spherical knots, which is motivated by the codimension two surgery theory: a knot is a pull back of another knot if the former is obtained as the inverse image of the latter by a certain degree one map between the ambient spheres. We show that this relation defines a partial order for (2n-1)-dimensional simple fibered knots for n≥3. We also give some related results concerning cobordisms and isotopies of knots together with several important explicit examples.

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.

We define a new combinatorial complex computing the hat version of link Floer homology over ℤ/2ℤ, which turns out to be significantly smaller than the Manolescu–Ozsváth–Sarkar one.

A torti-rational knot, denoted by K(2α, β|r), is a knot obtained from the 2-bridge link B(2α, β) by applying Dehn twists an arbitrary number of times, r, along one component of B(2α, β). We determine the genus of K(2α, β|r) and solve a question of when K(2α, β|r) is fibered. In most cases, the Alexander polynomials determine the genus and fiberedness of these knots. We develop both algebraic and geometric techniques to describe the genus and fiberedness by means of continued fraction expansions of β/2α. Then, we explicitly construct minimal genus Seifert surfaces. As an application, we solve the same question for the satellite knots of tunnel number one.

We introduce a new technique for studying classical knots with the methods of virtual knot theory. Let K be a knot and J be a knot in the complement of K with lk(J, K) = 0. Suppose there is covering space , where V(J) is a regular neighborhood of J satisfying V(J) ∩ im(K) = ∅ and Σ is a connected compact orientable 2-manifold. Let K′ be a knot in Σ × (0, 1) such that π_J(K′) = K. Then K′ stabilizes to a virtual knot , called a virtual cover of K relative to J. We investigate what can be said about a classical knot from its virtual covers in the case that J is a fibered knot. Several examples and applications to classical knots are presented. A basic theory of virtual covers is established.

We give a new infinite family of non-fibered 2-bridge knots whose knot groups are bi-orderable.

We provide a complete set of two moves that suffice to relate any two open book decompositions of a given 3-manifold. One of the moves is the usual plumbing with a positive or negative Hopf band, while the other one is a special local version of Harer’s twisting, which is presented in two different (but stably equivalent) forms. Our approach relies on 4-dimensional Lefschetz fibrations, and on 3-dimensional contact topology, via the Giroux-Goodman stable equivalence theorem for open book decompositions representing homologous contact structures.

We show that if a fibered knot $K$ is expressed as a band-connected sum of $K_1,\dots ,K_n$, then each $K_i$ is fibered, and the genus of $K$ is greater than or equal to that of the connected sum of $K_1,\dots ,K_n$.

We show that every tunnel number 1 genus 1 knot can be changed into a genus 3 fibered knot by a crossing change.

We use knot Floer surgery exact sequences and torsion invariants to compute the knot Floer homology of certain fibered knots in the double cover of S³ branched along the closure of an alternating braid.