Non-fibered Knots Research Articles

Heegaard presentations and Milnor pairings of some knots

We develop a procedure for computing Heegaard presentations of some knot groups, and give many examples. We introduce pseudo-Pretzel knots, and give their Heegaard presentations. As an application, we give matrix presentations of Milnor pairings of all non-fibered knots with crossing number [Formula: see text].

On a theorem of Friedl and Vidussi

A theorem of Friedl and Vidussi says that any 3-manifold [Formula: see text] and any non-fibered class in [Formula: see text] there exists a representation such that the corresponding twisted Alexander polynomial is zero. However, it seems that no concrete example of such a representation is known so far. In this paper, we provide several explicit examples of non-fibered knots and their representations with zero twisted Alexander polynomial.

TWISTED ALEXANDER POLYNOMIALS ON CURVES IN CHARACTER VARIETIES OF KNOT GROUPS

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.

Knot adjacency and fibering

It is known that the Alexander polynomial detects fibered knots and 3-manifolds that fiber over the circle. In this note, we show that when the Alexander polynomial becomes inconclusive, the notion of knot adjacency can be used to obtain obstructions to the fibering of knots and of 3-manifolds. As an application, given a fibered knot $K’$, we construct infinitely many non-fibered knots that share the same Alexander module with $K’$. Our construction also provides, for every $n\in N$, examples of irreducible 3-manifolds that cannot be distinguished by the Cochran-Melvin finite type invariants of order $\leq n$.

Kneser-Haken finiteness for bounded 3-manifolds, locally free groups, and cyclic covers

Associated to every compact 3-manifold M and positive integer b, there is a constant c( M, b) = c. Any collection F i of incompressible surfaces with Betti numbers b 1 F i < b for all i, none of which is a boundary parallel annulus or a boundary parallel disk, and no two of which are parallel, must have fewer than c members. Our estimate for c is exponential in b. This theorem is used to detect closed incompressible surfaces in the infinite cyclic covers of all non-fibered knot complements. In other terms, if the commutator subgroup of a knot group is locally free, then it is actually a finitely generated free group.

Seifert surfaces of knots inS4

This paper uses some ideas from 3-dimensional topology to study knots in S4 . We show that the Poincare conjecture implies the existence of a non-fibered knot whose complement fibers homotopically. In a different direction, we show that is an obstruction to a knot having a Seifert surface made out of Seifert fibered spaces, and hence to being ribbon. We also prove that any 3-manifold is invertibly homology cobordant to a hyperbolic 3-manifold, so that every knot in S4 has a hyperbolic Seifert surface. One of the reasons that the study of knots in the 4-sphere has a special character is that the Seifert surfaces that such knots bound are 3-dimensional. Hence the peculiar nature of the topology of 3manifolds can lead to interesting behavior of 2-knots. In this paper we give several examples of this principle. The first example is to show that the 3-dimensional Poincare conjecture implies the existence of non-fibered (topological) knots in *S 4 whose exteriors are homotopy equivalent to the exterior of a fibered knot. (Similar phenomena have been noticed by J. Hillman and C. B. Thomas [12, 13] and S. Weinberger [32].) The second instance is to see how the existence of a structure on a Seifert surface influences topological properties of the knot. Restrictions on the possible geometric structures are obtained via Gromov's norm of a 2-knot, defined below. We show that a knot with non-zero cannot have a Seifert surface which is a connected sum of Seifert-fiber ed 3-manifolds. In particular, the is seen to be an obstruction to a knot in S4 being ribbon. A similar obstruction has been found by Bruce Trace [30]. In contrast, we will show that any knot has a Seifert surface which is a hyperbolic manifold. This follows from Theorem 2.6, which states that any 3-manifold has an invertible homology cobordism to a hyperbolic manifold. A cobordism W from M to N is called invertible if there is another cobordism W, so that W U N W = M x I. Without the requirement that the homology cobordism be invertible, this theorem is due to R. Myers [23].

Companionship of knots and the Smith conjecture

This paper studies the Smith Conjecture in terms of H. Schubert’s theory of companionship of knots. Suppose J is a counterexample to the Smith Conjecture, i.e. is the fixed point set of an action of Z p {{\textbf {Z}}_p} on S 3 {S^3} . Theorem. Every essential torus in an invariant knot space C ( J ) C(J) of J is either invariant or disjoint from its translates. Since the companions of J correspond to the essential tori in C ( J ) C(J) , this often allows one to split the action among the companions and satellites of J. In particular: Theorem. If J is composite, then each prime factor of J is a counterexample, and conversely. Theorem. The Smith Conjecture is true for all cabled knots. Theorem. The Smith Conjecture is true for all doubled knots. Theorem. The Smith Conjecture is true for all cable braids. Theorem. The Smith Conjecture is true for all nonsimple knots with bridge number less than five. In addition we show: Theorem. If the Smith Conjecture is true for all simple fibered knots, then it is true for all fibered knots. Theorem. The Smith Conjecture is true for all nonfibered knots having a unique isotopy type of incompressible spanning surface.

A characterization of non-fibered knots.

A characterization of non-fibered knots.

Knots with infinitely many minimal spanning surfaces

We show that if k 1 {k_1} and k 2 {k_2} are nonfibered knots, then the composite knot K = k 1 # k 2 K = {k_1}\# {k_2} has an infinite collection of minimal spanning surfaces, no two of which are isotopic by an isotopy which leaves the knot K fixed. This result is then applied to show that whether or not a knot has a unique minimal spanning surface can depend on what definition of spanning surface equivalence is used.