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Let $K$ be a knot in an L-space $Y$ with a Dehn surgery to a surface bundle over $S^1$. We prove that $K$ is rationally fibered, that is, the knot complement admits a fibration over $S^1$. As part of the proof, we show that if $K\subset Y$ has a Dehn surgery to $S^1 \times S^2$, then $K$ is rationally fibered. In the case that $K$ admits some $S^1 \times S^2$ surgery, $K$ is Floer simple, that is, the rank of $\hat{HFK}(Y,K)$ is equal to the order of $H_1(Y)$. By combining the latter two facts, we deduce that the induced contact structure on the ambient manifold $Y$ is tight. In a different direction, we show that if $K$ is a knot in an L-space $Y$, then any Thurston norm minimizing rational Seifert surface for $K$ extends to a Thurston norm minimizing surface in the manifold obtained by the null surgery on $K$ (i.e., the unique surgery on $K$ with $b_1>0$)., 25 pages, 1 figure; v2: minor revisions throughout. This is the version to appear in Transactions of the AMS

We show that some hyperbolic 3-manifolds which are tessellated by copies of the regular ideal hyperbolic tetrahedron embed geodesically in a complete, finite volume, hyperbolic 4-manifold. This allows us to prove that the complement of the figure-eight knot geometrically bounds a complete, finite volume hyperbolic 4-manifold. This the first example of geometrically bounding hyperbolic knot complement and, amongst known examples of geometrically bounding manifolds, the one with the smallest volume., 9 pages, 4 figures, typos corrected, improved exposition of tetrahedral manifolds. Added Proposition 3.3, which gives necessary and sufficient conditions for M_T to be a manifold, and Remark 4.4, which shows that the figure-eight knot bounds a 4-manifold of minimal volume. Updated bibliography

Let M M be a cusped hyperbolic 3 3 -manifold, e.g. a knot complement. Thurston showed that the space of deformations of its fundamental group in P G L ( 2 , C ) \mathrm {PGL}(2,\mathbf {C}) (up to conjugation) is of complex dimension the number ν \nu of cusps near the hyperbolic representation. It seems natural to ask whether some representations remain discrete after deformation. The answer is generically not. A simple reason for it lies inside the cusps: the degeneracy of the peripheral representation (i.e. representations of fundamental groups of the ν \nu peripheral tori). They indeed generically become non-discrete, except for a countable set. This last set corresponds to hyperbolic Dehn surgeries on M M , for which the peripheral representation is no more faithful. We work here in the framework of P G L ( n , C ) \mathrm {PGL}(n,\mathbf {C}) . The hyperbolic structure lifts, via the n n -dimensional irreducible representation, to a representation ρ g e o m \rho _{\mathrm {geom}} . We know from the work of Menal-Ferrer and Porti that the space of deformations of ρ geom \rho _{\textrm {geom}} has complex dimension ( n − 1 ) ν (n-1)\nu . We prove here that, unlike the P G L ( 2 ) \mathrm {PGL}(2) -case, the generic behaviour becomes the discreteness (and faithfulness) of the peripheral representation: in a neighbourhood of the geometric representation, the non-discrete peripheral representations are contained in a real analytic subvariety of codimension ≥ 1 \geq 1 .

We construct a class of knots with the CI ∗ {}^* property, that is, π 1 ( M ( n ) ∣ ∂ M ( n ) ) ≠ { e } \pi _1(M(n)\mid \partial M(n))\neq \{e\} for some n > 0 n>0 . It follows that the infinite cyclic covering of such a knot cannot be embedded in any compact 3-manifold.

In the article, we study Fintushel-Stern's knot surgery four-manifold E(n) K and its monodromy factorization. For fibered knots we provide a smooth classification of knot surgery 4-manifolds up to twisted fiber sums. We then show that other constructions of 4-manifolds with the same Seiberg-Witten invariants are in fact diffeomorphic.

Abby Thompson proved that if a link $K$ is in thin position but not in bridge position then the knot complement contains an essential meridional planar surface, and she asked whether some thin level surface must be essential. This note is to give a positive answer to this question, showing that the if a link is in thin position but not bridge position then a thinnest level surface is essential. A theorem of Rieck and Sedgwick follows as a consequence, which says that thin position of a connected sum of small knots comes in the obvious way.

Let K be a knot in a closed orientable irreducible 3-manifold M. Suppose M admits a genus 1 Heegaard splitting and we denote by H the splitting torus. We say H is a 1-genus 1-bridge splitting of (M, K) if H intersects K transversely in two points, and divides (M, K) into two pairs of a solid torus and a boundary parallel arc in it. It is known that a 1-genus 1-bridge splitting of a satellite knot admits a satellite diagram disjoint from an essential loop on the splitting torus. If M = S 3 and the slope of the loop is longitudinal in one of the solid tori, then K is obtained by twisting a component of a 2-bridge link along the other component. We give a criterion for determining whether a given 1-genus 1-bridge splitting of a knot admits a satellite diagram of a given slope or not. As an application, we show there exist counter examples for a conjecture of Ait Nouh and Yasuhara.

There is a positive constant c 1 c_1 such that for any diagram D \mathcal {D} representing the unknot, there is a sequence of at most 2 c 1 n 2^{c_1 n} Reidemeister moves that will convert it to a trivial knot diagram, where n n is the number of crossings in D \mathcal {D} . A similar result holds for elementary moves on a polygonal knot K K embedded in the 1-skeleton of the interior of a compact, orientable, triangulated P L PL 3-manifold M M . There is a positive constant c 2 c_2 such that for each t ≥ 1 t \geq 1 , if M M consists of t t tetrahedra and K K is unknotted, then there is a sequence of at most 2 c 2 t 2^{c_2 t} elementary moves in M M which transforms K K to a triangle contained inside one tetrahedron of M M . We obtain explicit values for c 1 c_1 and c 2 c_2 .

It is shown that a hyperbolic knot in S3 admits at most one nonintegral Dehn surgery producing a toroidal manifold. Let K be a knot in the 3-sphere S and M = MK the complement of an open regular neighborhood of K in S. As usual, the set of slopes on the torus ∂M (i.e. the set of isotopy classes of essential simple loops on ∂M) is parameterized by {m/n : m,n ∈ Z, n > 0, (m,n) = 1} ∪ {1/0}, so that 1/0 is the meridian slope and 0/1 is the longitude slope. A slope m/n is called non-integral if n ≥ 2. The manifold obtained by Dehn surgery on S along the knot K (equivalently, Dehn filling on M along the torus ∂M) with slope m/n, is denoted by M(m/n). Now suppose that K ⊂ S is a hyperbolic knot, i.e. the interior of M has a complete hyperbolic metric of finite volume. A basic question in Dehn surgery theory is: when can a surgery on K produce a non-hyperbolic 3-manifold? A special case of this question is: when can a surgery on K produce a toroidal 3-manifold, i.e. a 3-manifold which contains an (embedded) incompressible torus? In [GL1], Gordon and Luecke showed that if m/n is a non-integral slope and M(m/n) contains an incompressible torus, then n = 2. In this paper, we show that there is at most one such surgery slope. Theorem 1. For a hyperbolic knot in S, there is at most one surgery with nonintegral slope producing a manifold containing an incompressible torus. There are examples of hyperbolic knots in S which admit toroidal surgeries with non-integral slopes [EM]. The best known example is the (−2, 3, 7)-pretzel knot, on which the surgery with slope 37/2 gives a toroidal 3-manifold. It is a conjecture ([Go1], [K, Problem 1.77]) that for a hyperbolic knot in S, there is at most one surgery with non-integral slope producing a non-hyperbolic 3-manifold, and further, if there is such a surgery, it must be a toroidal surgery and the knot must belong to the collection of examples given in [EM]. Theorem 1 provides some supporting evidence for this conjecture. We now go on to the proof of Theorem 1. Our argument is based on applications of results and combinatorial techniques developed in [CGLS], [GL1], [Go2]. Received by the editors May 20, 1997 and, in revised form, August 3, 1998. 1991 Mathematics Subject Classification. Primary 57N10, 57M25. The first author was partially supported by NSF grant DMS 9626550. The first and second authors were supported in part by Research at MSRI NSF grant #DMS 9022140. c ©2000 American Mathematical Society 1869 1870 C. McA. GORDON, Y-Q. WU, AND X. ZHANG Recall that the distance between two slopes m1/n1 and m2/n2 is defined as the number ∆ = ∆(m1/n1,m2/n2) = |m1n2 − n1m2|, which is equal to the minimal geometric intersection number between simple loops representing the two slopes on ∂M . Suppose that there are two slopes m1/2 and m2/2 such that both M(m1/2) and M(m2/2) contain incompressible tori. It follows from [Go2, Theorem 1.1] that there are exactly four hyperbolic manifolds that admit two toroidal Dehn fillings at distance more than 5, but for homological reasons, only one of these is the complement of a knot in S, namely the figure 8 knot complement, and by [Th] every non-integral surgery on this manifold is hyperbolic. Hence we have ∆(m1/2,m2/2) = |2m1 − 2m2| = 2|m1 − m2| ≤ 5. Note that both m1 and m2 are odd integers. So |m1 − m2| is even and thus must be equal to 2. Hence the distance between the two slopes is exactly 4. Our task here is to show that this is impossible. Note, however, that 4 can be realized as the distance between integral toroidal surgery slopes for a hyperbolic knot in S. For instance the slopes 16 and 20 are both toroidal surgery slopes for the (−2, 3, 7)-pretzel knot. The reason that distance 4 is impossible in our situation is mainly due, as we will see, to the fact that the first homology of M(mi/2) with Z2 coefficients is trivial for i = 1, 2. By [GL1] and [GL2] (see [GL1, Theorem 1.2]), for i = 1, 2, there is an incompressible torus Ti in M(mi/2) such that M ∩ Ti = Ti is an incompressible, ∂-incompressible, twice punctured torus properly embedded in M with each component of ∂Ti having slope mi/2 in ∂M . Note that Ti separates M since M(mi/2) has finite first homology. By an isotopy of Ti, we may assume that T1 and T2 intersect transversely, and T1 ∩ T2 has the minimal number of components. So T1 ∩ T2 is a set of finitely many circle components and arc components properly embedded in Ti, i = 1, 2. Furthermore, no circle component of T1 ∩ T2 bounds a disk in Ti and no arc component of T1 ∩ T2 is boundary parallel in Ti, i = 1, 2, since Ti is incompressible and ∂-incompressible. We shall use the indices i and j to denote 1 or 2, with the convention that, when they are used together, {i, j} = {1, 2} as a set. Let Vi denote the solid torus that is attached to M in forming M(mi/2). The torus Ti intersects Vi in two disks Bi(1) and Bi(2), which cut Vi into two 2-handles, which we denote by Hi(1) and Hi(2). Correspondingly, ∂Ti = ∂Bi(1) ∪ ∂Bi(2) cuts ∂M into two annuli Ai(1) and Ai(2), where Ai(k) ⊂ ∂Hi(k). Each ∂Tj∩Ai(k) consists of exactly 8 essential arcs on Ai(k). The torus Ti separates M(mi/2) into two submanifolds which we denote by Xi(1) and Xi(2), with Hi(k) ⊂ Xi(k). Correspondingly Ti separates M into two pieces, denoted by Xi(1) and Xi(2). Thus Xi(k) = M ∩ Xi(k). Note that Xi(k) is obtained by attaching the 2-handle Hi(k) to Xi(k) along the annulus Ai(k), k = 1, 2. Also note that Fi(k) = ∂Xi(k) = Ti ∪Ai(k) is a closed surface of genus two, k = 1, 2. Lemma 2. For i = 1, 2, k = 1, 2, we have H1(Xi(k), Ti; Z2) = 0. Proof. Since H1(M(mi/2); Z2) = 0, and Ti is connected, we have 0 = H1(M(mi/2), Ti; Z2) = H1(Xi(1), Ti; Z2)⊕H1(Xi(2), Ti; Z2); hence each H1(Xi(k), Ti; Z2) = 0. Now, as in [CGLS, 2.5], and [Go2], we construct two graphs Γ1 and Γ2 in T1 and T2 respectively by taking the arc components of T1∩T2 as edges and Ti− int(Ti) = TOROIDAL SURGERY ON HYPERBOLIC KNOTS 1871 1 2 1 2 1 2 1 2 Bi(k)

Let Σ \Sigma be the image of a topological embedding of S n − 2 {S^{n - 2}} into S n {S^n} . In this paper the homotopy groups of the complement S n − Σ {S^n} - \Sigma are studied. In contrast with the situation in the smooth and piecewise-linear categories, it is shown that the first nonstandard homotopy group of the complement of such a topological knot can occur in any dimension in the range 1 through n − 2 n - 2 . If the first nonstandard homotopy group of the complement occurs above the middle dimension, then the end of S n − Σ {S^n} - \Sigma must have a nontrivial homotopy group in the dual dimension. The complement has the proper homotopy type of S 1 × R n − 1 {S^1} \times {\mathbb {R}^{n - 1}} if both the complement and the end of the complement have standard homotopy groups in every dimension below the middle dimension. A new form of duality for noncompact manifolds is developed. The duality theorem is the main technical tool used in the paper.

Let $M$ be an irreducible, compact, connected, orientable 3-manifold whose boundary is a torus. We show that if $M$ is hyperbolic, then it admits at most six finite/cyclic fillings of maximal distance 5. Further, the distance of a finite/cyclic filling to a cyclic filling is at most 2. If $M$ has a non-boundary-parallel, incompressible torus and is not a generalized 1-iterated torus knot complement, then there are at most three finite/cyclic fillings of maximal distance 1. Further, if $M$ has a non-boundary-parallel, incompressible torus and is not a generalized 1- or 2-iterated torus knot complement and if $M$ admits a cyclic filling of odd order, then $M$ does not admit any other finite/cyclic filling. Relations between finite/cyclic fillings and other exceptional fillings are also discussed., Comment: 7 pages

This paper deals with the question of which knot surgeries on S 3 {S^3} can yield 3 3 -manifolds homeomorphic to, or with the same fundamental group as, the Poincaré homology 3 3 -sphere.

The depth one foliations with a single compact leaf are classified for twist knots and pretzel knots ( 3 , − 3 , 2 n + 1 ) (3, - 3,2n + 1) .

We showed in [8] that a locally flat knot is topologically unknotted if and only if the Lusternik-Schnirelmann category of the complement is one. In this paper we will show that the complement of a ribbon knot is of category two. 1. REVIEW OF KNOWN RESULTS The category cat X of a space X is the least integer n so that X may be covered by the n + 1 open subsets each of which is contractible to a point in X; if there is no such integer, catX = oc. In particular, catX = 0 if and only if X is itself contractible. So, cat Sn = 1 . This definition is due to Fox [4], although we reduced the number by one from his definition. Originally Lustemik-Schnirelmann defined the category by the closed covering and these two notions coincide with each other for ANR's (cf. [5, ?3]). One of the most important properties is that cat X is a homotopy type invariant of X. Using this we can easily prove that cat X 1. We have immediately for example that cat(Sn, x ... x Snk) = k. Moreover, a closed manifold M is a homotopy sphere if and only if cat M = 1. In fact, if cat M = 1 then M is a homology sphere by the property (BGW) and Poincar6 duality theorem. So, it suffices to apply the lemma (cf. [4, ?23]: If X is an arcwise connected and locally 1-connected compact metrizable space and cat X < 1, then n1 (X) is a free group. A corresponding theorem in the case of knot complement is Theorem 1 ([8]). A locally flat knot Sn D Sm is topologically unknotted if and only if cat(Sn Sm) = 1. By the facts known in geometric topology we have only to show that Received by the editors September 26, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 55M30, 57Q45. ? 1992 American Mathematical Society 0002-9939/92 $1.00 + $.25 per page

A classification is given for the composite knots and the Dehn surgery on these knots which yield Seifert fibered surgery manifolds. We prove that if a knot K is the composition of two torus knots, then some (unique) integral surgery on K yields a Seifert fibered manifold, and conversely if the surgery manifold of a composite knot K is Seifert fibered, then K is the composition of two torus knots and the surgery must be integral surgery, which is uniquely determined. In this paper we classify the composite knots and the Dehn surgeries on these knots which yield Seifert fibered surgery manifolds. In [8] Moser conjectured that surgery on a nontorus knot could not yield a Seifert fibered manifold, in particular, could not yield a lens space. Many counterexamples to Moser's conjecture have been found. Baily and Rolfsen [2] showed that the lens space L(23, 7) could be obtained by -23-surgery on the (11, 2)-cable knot about the trefoil knot; Simon discovered similar examples. Fintushel and Stern [4] constructed infinitely many noniterated torus knots, upon which certain surgery yields lens spaces. More recently, Gordon [6] classified the surgery manifolds of all iterated torus knots, Berge [3] and Gabai [5] have independently constructed an infinite collection of knots in solid tori such that certain surgery on them in the solid torus yield D2 x S , and therefore yield lens spaces when the knots are considered to be in S . Since cable knots are prime knots and the surgery manifold of a composite knot contains an incompressible torus (see, for example, [6]), none of the above-mentioned knots is a composite knot. Indeed, no nontrivial surgery on a composite knot may yield a lens space. It is then natural to ask if any such surgery manifold is Seifert fibered. In this paper, we show that if a knot K is the composition of two torus knots, then some (unique) integral surgery on K yields a Seifert fibered manifold, and conversely if a surgery manifold of a composite knot K is Seifert fibered, then the surgery Received by the editors January 4, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M05, 57M25.

A two-generator cable knot exterior is a genus two handlebody with a single two-handle attached. Knot theorists have long studied knots via the fundamental group of the complement. These groups are finitely generated, and a naive measure of the complexity of such a group is the rank, i.e., the minimal number of generators needed to present the group. Dehn's lemma and homology show that only the unknot has a rank one (i.e., cyclic) group, so in this sense the "simpliest" non- trivial knots are those whose groups are of rank two, the so-called two-generator knots. It can be very difficult to determine the rank of a group (MKS, B); cor- respondingly it can be very difficult to recognize a 2-generator knot. Indeed, in R. Kirby's famous 1978 problem list (K, 1.10), L. Moser asks if there exists a geometric characterization of this algebraic condition on knots. A way to construct two-generator knots geometrically was given by Clark in a 1980 paper (C) and is reproduced now. Begin with a genus g Heegaard splitting of the 3-sphere, expressing one of the two handlebodies as a single zero- handle with g one-handles attached. Removing the zero-handle and a single one-handle produces the exterior of a knot K in the 3-sphere. The cores of the remaining one-handles are said to be a set of unknotting tunnels for K. The cardinality of a minimal set of unknotting tunnels for a given knot K is then defined to be the tunnel number of K and K is said to be tunnel t. Dually, the exterior of a tunnel t knot is constructed by attaching t two-handles to genus t + 1 handelbody. Such a space deformation retracts to a wedge of t+1 circles with t discs attached, so the group of a tunnel t knot has a presentation with t + 1 generators and t relators. Thus, tunnel one knots are two-generator. Scharlemann conjectured the converse in 1984 (S), and the result here is the first step in that direction.

We show that if a co-dimension two knot is deform-spun from a lower-dimensional co-dimension 2 knot, there are constraints on the Alexander polynomials. In particular this shows, for all n, that not all co-dimension 2 knots in S^n are deform-spun from knots in S^{n-1}., Comment: 6 Pages, 1 Figure. to appear in Proceedings of the AMS

For a given hyperbolic knot, the third author defined a function whose imaginary part gives the hyperbolic volume of the knot complement. We show that, for a twist knot, the function actually gives the complex volume of the knot complement using Zickert's and Neumann's theory of the extended Bloch groups and the complex volumes.

We get a necessary condition under which a nonsimple knot (i.e. a satellite knot) admits a nontrivial surgery producing a lens space. Interesting corollaries are: ( 1) if a lens space can be obtained from a nontrivial surgery on a nonsimple knot, then the order of its fundamental group is not smaller than 23; (2) any nonsimple knot admits at most one nontrivial surgery which produces a lens space. Some topologists are interested in the problem of doing surgery on a knot complement to get a lens space. This note is devoted to providing more information on this problem. Our main result is the following: Theorem 1. If a lens space can be obtainedfrom a nontrivial surgery on a nonsimple knot (i.e. a satellite knot) J(K), then K is a torus knot and J is a closed braid in a tubular neighborhood of K. There are two interesting corollaries of Theorem 1. D. Gabai proved that one cannot get a lens space with fundamental group of order 1 (i.e. S3) from a nontrivial surgery on a nonsimple knot (in fact nonsimple knots have property P). J. Bailey and D. Rolfsen [BR] (also R. Fintushel and R. Stern [FS] and C. Gordon [G]) proved that some lens space with fundamental group of order 23 can be obtained from a nontrivial surgery on a nonsimple knot. The first corollary says that 23 is the smallest possible. Corollary 1. If a lens space can be obtained from a nontrivial surgery on a nonsimple knot (i.e. a satellite knot), then the order of its fundamental group is not smaller than 23. M. Culler, C. M. Gordon, J. Luecke, and P. Shalen [CGLS] proved that there are at most two nontrivial surgeries on a nontorus knot which produce lens spaces. Corollary 2 below says that a nontorus knot which admits two nontrivial surgeries producing lens spaces must be a hyperbolic knot. (Berge found infinitely many such nontorus knots.) Received by the editors February 4, 1988, and in revised form, August 17. 1988. 1980 Alathernatics S/ihject Classification (I1985 Rewision). Primary 57N 10. (? 1989 American Mathematical Society 0002-9939/89 $ 1.00 + $.25 per page

The classification of equivariant concordance classes of high-dimensional codimension two knots invariant under a cyclic action, T, of order m has previously been reported on by Cappell and Shaneson [CS2]. They give an algebraic solution in terms of their algebraic k-theoretic Γ \Gamma -groups. This work gives an alternative description by generalizing the well-known Seifert linking forms of knot theory to the equivariant case. This allows explicit algorithmic computations by means of the procedures and invariants of algebraic number theory (see the subsequent work [St], particularly Theorem 6.13). Following Levine [L3], we define bilinear forms on the middle-dimensional homology of an equivariant Seifert surface B i ( x , y ) = L ( x , i + ( T ∗ i y ) ) {B_i}(x,y) = L(x,{i_ + }(T_{\ast }^iy)) , for i = 1 , ⋯ , m i = 1, \cdots ,m . Our first result (2.5) is that an invariant knot is equivariantly concordant to an invariant trivial knot if and only if there is a subspace of half the rank on which the B i {B_i} vanish simultaneously. We then introduce the concepts of equivariant isometric structure and algebraic concordance which mirror the preceding geometric ideas. The resulting equivalence classes form a group under direct sum which has infinitely many elements of each of the possible orders (two, four and infinite), at least for odd periods. The central computation (3.4) gives an isomorphism of the equivariant concordance group with the subgroup of the algebraic knot concordance group whose Alexander polynomial, Δ \Delta , satisfies the classical relation | ∏ i = 1 m Δ ( λ i ) | = 1 \left | {\prod \nolimits _{i = \,1}^m {\Delta \left ( {{\lambda ^i}} \right )} } \right |\, = \,1\, , where λ \lambda is a primitive mth root of unity. This condition assures that the m-fold cover of the knot complement is also a homology circle, permitting the geometric realization of each equivariant isometric structure. Finally, we make an explicit computation of the Browder-Livesay desuspension invariant for knots invariant under an involution and also elucidate the connection of our methods with the results of [CS2] by explicitly describing a homomorphism from the group of equivariant isometric structures to the appropriate Γ \Gamma -group.

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.

Cowsik and Swarup (CS) have shown that the homology groups of the infinite cyclic cover of a knot inject into the inverse limit of the homology groups of the branched cyclic covers. They also give conditions under which the injection is an isomorphism. We prove an analogous result for the fundamental group and generalize it to the case of links. The relationship between the homology of the infinite cyclic cover of a knot complement and that of the inverse limit of the branched cyclic covers has been investigated in (CS, Dl, D2). We would like to extend some of these results to homotopy. Let K be a knot in S3, A the infinite cyclic cover of A = S3 — K, and Xn the n-fold cyclic cover of A. We write £" for the branched n-fold cover of S3 branched over K. This gives the following commutative diagram: X -» Xk

The free genus of an untwisted doubled knot in S3 can be arbi- trarily large. Every knot K in S3 bounds a surface F for which S3 — F is a solid handlebody. Such a surface is called a free Seifert surface and the minimal genus of such a surface is called the free genus of the knot, denoted g/(K). This note presents a proof of a statement made in Problem 1.20 of (K): the free genus of an untwisted double of a knot in S3 can be arbitrarily large. Problem 1.20a asks for relations between the free genus of a knot and its other invariants. Since an untwisted double of a knot has trivial Alexander polynomial, classical abelian invariants will provide no upper bounds on the free genus of a knot. Yoav Moriah (Mo) has proved a similar result for twisted doubles. In essence he used dihedral representations of the knot group to study the free genus. These representations do not exist for untwisted doubles. Symmetric representations will be used here. Connectivity arguments permit us to avoid detailed calculations of homology groups. The use of irregular covers instead of regular ones will decrease the complexity of the covers involved. A reference for the techniques of knot theory used here is (R). The covering space theory is contained in (Ma), and is summarized at the end of this paper. Homology groups are always with integer coefficients.

In this paper we shall show that the unknotting numbers of the ( l , 2 k l ± 1 ) (l,2kl \pm 1) -torus knots are at least ( k ( l 2 − 1 ) − 2 ) / 2 (k({l^2} - 1) - 2)/2 for l l odd and ( k l 2 − 2 ) / 2 (k{l^2} - 2)/2 for l l even, where l l is an integer greater than one and k k is a positive integer.

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.

Let L be an oriented tame link in the three sphere S 3 {S^3} . We study the Murasugi signature, σ ( L ) \sigma (L) , and the nullity, η ( L ) \eta (L) . It is shown that σ ( L ) \sigma (L) is a locally flat topological concordance invariant and that η ( L ) \eta (L) is a topological concordance invariant (no local flatness assumption here). Known results about the signature are re-proved (in some cases generalized) using branched coverings.

The complement of a ribbon n n -disc in the ( n + 2 ) (n + 2) -ball has a 2 2 -dimensional spine which shares some of the combinatorial properties of classical knot complement spines. It is an open question whether such 2 2 -complexes are always aspherical. To any ribbon disc we associate a labelled oriented tree, from which the homotopy type of the complement can be recovered, and we prove asphericity in certain special cases described by conditions on this tree. Our main result is that the complement is aspherical whenever the associated tree has diameter at most 3 3 .

We show that a knot exists with the property that there exists a killer of the knot group which is not the image of the meridian under any automorphism.

We propose the conjecture that every automorphism of a knot group preserves the meridian up to inverse and conjugation. We establish the conjecture for all composite knots, all torus knots, most cable knots, and at most one exception for hyperbolic knots; moreover we prove that the Property P Conjecture implies our conjecture. We also investigate hyperbolic knots in more detail, and give an example of figure-eight knot group and its automorphisms.

The decision problems in the title are solved using the solutions of the word and conjugacy problems obtained by Lyndon and Schupp, respectively, for certain classes of groups. 1. Introduction. Each tame knot K has a knot group G whose word problem is solvable by Waldhausen's result (o). Here we give additional results when K is also prime, alternating, and nontrivial. We show that the free product of G and a free group on one generator Xo is a group 77 which falls into one of the categories for which Lyndon (4) and Schupp (5) solved the word and conjugacy problems, re- spectively. (77 has a presentation satisfying the properties C(4) and T3 (4, p. 219).) We thereby obtain another solution of the word problem for G and a solution of the conjugacy problem for G. In the proof, the given knot is replaced with an equivalent one having a knot-diagram with special properties (referred to as a common knot-diagram). Presentations of G and of another group 77 are obtained from a knot-diagram, using Dehn's method (3, p. 157). When the knot-diagram is common, the presentation of 77 (as a factor group of a free group 7) satisfies C(4) and T3. An automorphism of P yields a second presentation of H in which one generator Xo is not mentioned in the defining relations. Adding the relation x0= 1 to this second presentation gives a presentation of G. Thus H is a free product as claimed. 2. Common knot-diagrams. Let K be a nontrivial polygonal knot in 3-space P3. Let K he the range of a continuous function / defined on (O, l), with/(0) =/(l). Assume K is in regular position with re- spect to the projection p which sends (x, y, z) to (x, y, 0). (Knot- theoretic terminology is taken from Crowell and Fox (2).) The knot

The fundamental group of the complement of a wild knot in a 3-sphere can be expressed as the colimit (direct limit) of a suitable family of groups and homomorphisms (Crowell [4]). To each group in the family we assign a Jacobian module, and in ?1 we prove that this assignment is functorial and preserves colimits. This is used in ?2 to show that the nullity of the Alexander module of a knot with one wild point is bounded above by its enclosure genus. This can be used in some cases to calculate the enclosure genus and the penetration index of the knot in a purely algebraic way. In ?3 we give examples to show that the upper bound of Theorem 2 is the best possible, along with some results concerning the penetration index of an arc relative to surfaces of genus r.

This paper deals with the computation of the second homotopy of a knot as a module over its fundamental group. The present note is a continuation of [18] whose notation and bibliography we use throughout. 4. Semidirect products. From (1) we conclude that LI is the semidirect product (12) I=1Ix(,Z where 9p:Z-*Aut LI can be described as follows: By (3), LI = ... *11* G(-') *11 G(0) *H G(1) *H ... (G(j) t G, j E Z) let q(r): Hl-I be the automorphism defined by gj -*gj+, for gj E Go). Thus LI can be thought of as the group of the (g, ti) (g E lI,j E Z) with multiplication (g, t r)(h, ts)= (9 (s)(g). h, t r#). Let P (resp. F) be the integral group ring of LI (resp. LI); then

It has been shown [6] that any closed, connected, orientable 3manifold can be constructed by removing a finite number, k, of mutually exclusive tame solid tori1 from the 3-sphere, S3, and then sewing them back in some possibly different manner. In particular, any homotopy 3-sphere can be obtained in this manner; thus it gives an approach to the Poincar6 Conjecture. We wish to consider the case k =1. To this end let T be a tame solid torus in SI and let M be a simply connected 3-manifold containing a solid torus S such that there is a homeomorphism h of S3-Int(T) onto M-Int(S). It is known [I ] that M is S' if T is unknotted or if T has the type of a trefoil knot. In [5] it is asserted that M is SI in every case. The proof given in [51 shows only that h must take some longitude of T onto a longitude of S-a fact that is not sufficient to prove this assertion. The purpose of this paper is to provide a proof that M is S3 in the case that T has the type of a torus knot. We first give a canonical description of the torus knot of type (p, q). We consider 5' as the one point compactification of E3 which we represent in cylindrical coordinates (r, 0, z). Let R be the torus R= { (r, 0, z): (r-2)2+z2= 1}. Let p and q be relatively prime positive integers. We assume, for convenience, that p>q; since it is known that the torus knots of types (p, q) and (q, p) are equivalent. For 1

In a recent paper Treybig shows that if two knot functions f , g f,g determine equivalent knots, then f , g f,g are the ends of a simple sequence x x of knot functions. In an effort to bound the length of x x in terms of f f and g g (1) a bound is found for the moves necessary in moving one polyhedral disk onto another in the interior of a tetrahedron and (2) it is shown that two polygonal knots K , L K,L in regular position can “essentially” be embedded as part of the 1 1 -skeleton of a triangulation T T of a tetrahedron, where (1) all 3 cells which are unions of elements of T T can be shelled and (2) the number of elements in T T is determined by K , L K,L . A number of “counting” lemmas are proved.

The theorem of the title is proved by using the techniques of Weinbaum along with the prime decomposition of knots and an extension of some small cancellation techniques of Lyndon and Schupp.

The aim of this paper is to show that the com- mutator subgroup of the alternating knot group is the (proper or improper) free product of free groups with isomorphic subgroups amalgamated. 1. Main theorem. A surface SES3 is said to be algebraically unknotted if 7Ti(53 — S) is free. It was proved by Neuwirth (6)thatany orientable surface of minimal genus spanned by a fibred knot is al- ways algebraically unknotted. In this note, the following theorem will be proved.

We show that a theorem of Burde on the Alexander polynomial of a cyclically periodic knot K, such that zA2(K) = 1, may be extended to all cyclically periodic knots. A knot K in a homology 3-sphere I has cyclic period q if it is invariant under an automorphism of Y of order q which has fixed point set a circle disjoint from K. In [3] we gave new proofs of theorems of Burde [1] and Murasugi [5] on the Alexander polynomial ?11(K) of such a knot. A refinement of the second lemma of [3] enables us to remove the restriction A2(K) = 1 from Burde's theorem. We shall adopt the notation of [3]. THEOREM. Let K be a knot in an homology 3-sphere I which has cyclic period q, where q = pr for some prime p. Then either (i) iA,(K) 1_ modulo (p) or (ii) the degree of the qth roots of unity over the splitting fields of AI (K) is at most n, where A n(K) =# 1 but An+I(K) = 1. PROOF. The argument given in [3] shows that if (i) fails then the Q[t, t ']-module C = (G'/G") IDz Q (the rational homology of the infinite cyclic cover of the knot complement) has an automorphism of order exactly q. (For otherwise the (q/p)th power of the given periodic self-homeomorphism T of z induces the identity on G'/G". Applying Milnor duality and the Wang sequence with coefficients Z/pZ to the p-fold branched cyclic covering map from Z to Z, the results of 0-framed surgery on K in 2 and on the image of K in E/T(q/P), respectively, we conclude that (G'/G") 0 (Z/pZ) must be 0, and so L\,(K) 1 modulo (p).) Then for some irreducible factor 8 of /,(K), the 8-primary submodule of C has an automorphism T of order exactly q, so we may assume zA0(C) = 3'. Let L = Q[t, t-']/(8). If m 2 1 the kernel of the natural map from Aut(C/6m+ 'C) to Aut(C/SmC) is isomorphic to Hom(C/8m? 'C, V'C/18' 'C) and so is an L-vector space. In particular, it is torsion free. Therefore the image of T in Aut(C/SC) GL(n, L) has order exactly q. Let A = L[ ] be the subalgebra of M(n, L) generated by this element. Then dimL A s n by the Cayley-Hamilton theorem. On the other hand, the image of T in the semisimple algebra A/rad A again has order exactly q, as the kernel of the map on Received by the editors December 15, 1981. 1980 Mathematics Subject Classification. Primary 57M25.

Let Jk(*) = nrtr + • ■ • + asta, r > s, be the Jones polynomial of a knot if in S3. For an alternating knot, it is proved that r — s is bounded by the number of double points in any alternating projection of K. This upper bound is attained by many alternating knots, including 2-bridge knots, and therefore, for these knots, r — s gives the minimum number of double points among all alternating projections of K. If K is a special alternating knot, it is also proved that a3 = 1 and s is equal to the genus of K. Similar results hold for links. 1. Introduction. Let K be an oriented knot or link in S3 and let Jx(t) be the polynomial defined by V. Jones (5) which is now called the Jones polynomial of K. Jk (t) is an invariant of a knot or link type. It is not clear, however, to what extent Jk (t) is related to known algebraic or topological invariants in knot theory. Although it is shown (10) that Jk (t) determines the Arf invariant of a knot or a link, it is also known that Jk(i) does not determine the genus or the signature of K(8}. In this paper, we prove that for an alternating knot, Jk(t) provides some infor- mation that has never been obtained from other algebraic invariants. To be more precise, let K be an alternating knot in S3 and K an alternating projection of K. Let Jk(t) be the Jones polynomial of K.1 Write Jk(i) — artr + • • • + asts, where r > s and ar ^ 0 / as. Let h(K) be the number of double points in K. Then we can prove