Nontrivial Knot Research Articles

Uniqueness of Free Actions on S3 Respecting a Knot

In this paper we consider free actions of finite cyclic groups on the pair (S3, K), where K is a knot in S3. That is, we look at periodic diffeo-morphisms f of (S3, K) such that fn is fixed point free, for all n less than the order of f. Note that such actions are always orientation preserving. We will show that if K is a non-trivial prime knot then, up to conjugacy, (S3, K) has at most one free finite cyclic group action of a given order. In addition, if all of the companions of K are prime, then all of the free periodic diffeo-morphisms of (S3, K) are conjugate to elements of one cyclic group which acts freely on (S3, K). More specifically, we prove the following two theorems.THEOREM 1. Let K be a non-trivial prime knot. If f and g are free periodic diffeomorphisms of (S3, K) of the same order, then f is conjugate to a power of g.

The finiteness theorem for symmetries of knots and 3-manifolds with non-trivial characteristic decompositions

In 1961, fox asked what knots have infinitely many symmetries. This question can actually be split into two questions. First, what knots have finite group actions of arbitrarily large orders? Second, what knots have infinitely many non-conjugate group actions of a given order? It was previously shown by the author that torus knots are the only knots with finite group actions of arbitrarily large orders. Now, we show that no non-trivial knots have infinitely many non-conjugate group actions of a given order. Thus we have completely answered the question posed by Fox. In addition we show here that any 3-manifold with a non-trivial characteristic decomposition has only finitely many non-conjugate finite group actions. These results are in contrast to the fact that there exist examples of knot complements which have infinitely many non-homotopic finite cyclic actions of a given order.

Least area Seifert surfaces and periodic knots

It is shown that if a nontrivial knot K in S 3 has genus g and is invariant under a smooth action of a cyclic group C m of order m which fixes a simple closed curve disjoint from K, then K is the boundary of an invariant Seifert surface of genus g. As a corollary one has that m⩽2 g+1. The proof utilizes the theory of least area surfaces in 3-manifolds.

Classification of Homotopy Torus Knot Spaces

The existence of nontrivial homotopy torus knot spaces is established as a corollary to the Theorem. Let $p$ and $q$ be two integers with $p > 1,q > 1$, and $(p,q) = 1$. Let $\mathfrak {M}$ be a maximal set of topologically distinct compact orientable irreducible $3$-mainfolds with fundamental group presented by $\langle a,b|{a^p}{b^q}\rangle$. Then card $(\mathfrak {M}) = 1/2\Phi (pq)$, where $\Phi$ denotes Euler’s function.

Classification of homotopy torus knot spaces

The existence of nontrivial homotopy torus knot spaces is established as a corollary to the Theorem. Let p p and q q be two integers with p > 1 , q > 1 p > 1,q > 1 , and ( p , q ) = 1 (p,q) = 1 . Let M \mathfrak {M} be a maximal set of topologically distinct compact orientable irreducible 3 3 -mainfolds with fundamental group presented by ⟨ a , b | a p b q ⟩ \langle a,b|{a^p}{b^q}\rangle . Then card ( M ) = 1 / 2 Φ ( p q ) (\mathfrak {M}) = 1/2\Phi (pq) , where Φ \Phi denotes Euler’s function.

More on complements of minimal spanning surfaces

W. R. Alford in volume 91 of the Annals of Mathematics has shown the existence of a knot which has two minimal spanning surfaces whose complements in S are not homeomorphic. The trefoil knot is a companion to the knot. This paper shows that any nontrivial knot k is a companion to a knot K which has at least two minimal spanning surfaces. Introduction. In [1], W. R. Alford exhibited a knot k and two minimal spanning surfaces Sx and S2 for k such that S 3 — S{ are not homeomorphic. The knot was formed by sending the torus T containing the knot I in Fig. 1 faithfully to a regular neighborhood of the trefoil knot. In a later paper [2], Alford and C. B. Schaufele constructed knots with 2 really distinct minimal spanning surfaces; the surfaces do not have homeomorphic complements. The examples were constructed by sending the torus T containing the knot I in Fig. 1 faithfully to a regular neighborhood of the sum of m nice knots. The selection of the knots was strongly influenced by their algebraic properties. The purpose of this paper is to show that any nontrivial knot is a companion to a knot K which has at least two minimal surfaces. The knot K is the image of the knot I in T in Fig. 1 under a faithful homeomorphism of the solid torus T to a regular neighborhood V of the knot I. The Alexander polynomial of K is (2 — t) • (2t — 1) [4] for any nontrivial k used. Thus K had genus at least one. The spanning surfaces for K have genus one, so K has genus one. The surfaces. The surfaces for K are constructed as in [2]. The knot I is spanned by a singular disk in T as shown in Fig. 2. Only one side is shown; the singularities are in heavy lines. Received by the editors June 4,1971 and, in revised form, November 8,1971. AMS (MOS) subject classifications (1970). Primary 55A25; Secondary 57A10.

LOCAL KNOTTING OF SUBMANIFOLDS

In this paper the author investigates the modification of the fundamental group of the complement of a submanifold of codimension 2 by knotting in a neighborhood of one of its points. With the aid of such knotting he constructs closed nonorientable surfaces in R4 with finite noncommutative groups. In the appendix he constructs a nontrivial knot such that, knotting by means of it does not change the type of the simplest imbedding RP2 → R4. Figures: 6. Bibliography: 10 items.

On the existence of incompressible surfaces in certain $3$-manifolds.

If M is the closure of the complement of a regular neighborhood of a nontrivial knot in S3 then there exists a nonsingular torus T embedded in M, which is incompressible (i.e. the inclusion i: T->M induces a monomorphism i*: 7r1(T)-7r1(M)). If F is any orientable closed incompressible surface embedded in M then ri(M) contains ri(F) as a subgroup. L. Neuwirth [3, Question T] asks whether the converse is true: If 7rw(M) contains the group a of a closed (orientable) surface of genus g>1, does there exist a nonsingular closed surface F of genus g whose fundamental group is injected monomorphically into 7rw(M) by inclusion? As a partial answer we show that not for every such Cw7ri(M) there exists an incompressible FCM. The question remains open whether M contains incompressible closed surfaces of genus > 1. We show that for torus knots M does not contain such surfaces, by showing that 7ri(M) does not contain subgroups ~.