Ribbon Knots Research Articles

A loop theorem for duality spaces and fibred ribbon knots

In this paper we prove a version of the loop theorem for surfaces in the boundary of a 3-dimensional duality space, i.e. a space which resembles a 3manifold only in that it satisfies the appropriate form of Poincar6-Lefschetz duality over some field of untwisted coefficients. Our motivation comes from the fact that such spaces occur as the infinite cyclic coverings of certain 4manifolds which arise in the study of knot concordance, and as the main application of our theorem we show that if a fibred knot in the 3-sphere is a ribbon knot, then its monodromy extends over a handlebody. We approach the loop theorem via the study of planar coverings of a surface, as in the original paper of Papakyriakopoulos [11] and the subsequent work of Maskit [9]. w contains a simple geometric treatment of these matters. The main result of w is that a duality space actually satisfies duality with (twisted) coefficient module the quotient of the fundamental group ring by any power of the augmentation ideal. In w 4, the results of w167 2 and 3, together with an algebraic lemma on the intersection of the powers of the augmentation ideal of a group ring, are used to prove the loop theorem for 3-dimensional duality spaces. (For the reader's convenience a proof of the algebraic temma is included as an appendix.) w contains the application to fibred ribbon knots mentioned above. In w the result of w is used to obtain a limited amount of information on some questions about knots in the boundaries of contractible 4-manifolds. In w we apply our methods to another aspect of knot concordance, and show that for any concordance with a rationally anisotropic fibred knot (see [7]) at one end, the inclusion of the complement of the knot into the complement of the concordance induces an injection of fundamental groups. For torus knots, this question was raised by Scharlemann [14].

Ribbon knots and ribbon disks

Ribbon knots and ribbon disks

Trivializing ribbon links by Kirby moves

In this note it is shown that any ribbon link is a sublink of a ribbon link for which surgery on the longitudes gives a connected sum of copies of S1 x S2. In particular there are many links for which the analogue of the knot theoretic Property R fails, and sublinks of homology boundary links need not be homology boundary links. Higher dimensional analogues of these results are also given and it is shown that if n ≥ 2 the group of a ν-component ribbon n-link has a presentation of deficiency ν. Hence there are high dimensional slice knots which are not ribbon knots.

Examples of Higher-Dimensional Slice Knots which are not Ribbon Knots

A definition for ribbon n-knot is given which reduces to the previously studied definitions for $n = 1$ and $n = 2$. It is shown that for each $n \geqslant 2$ there is a slice n-knot which is not a ribbon n-knot.

Examples of higher-dimensional slice knots which are not ribbon knots

A definition for ribbon n-knot is given which reduces to the previously studied definitions for n = 1 n = 1 and n = 2 n = 2 . It is shown that for each n ⩾ 2 n \geqslant 2 there is a slice n-knot which is not a ribbon n-knot.

The Minkowski units of ribbon knots

2. Ribbon projections. Let k be a ribbon knot in R3, let 12 be the unit square, and let f: 12 -R3 be a ribbon whose boundary is k. By definition of a ribbon, the singular set S= {xflff(x) xx} of f consists of an even number 2n of pairwise disjoint arcs. Half of these arcs, say C1, * * , C., have the property that both their endpoints lie on 2, but they are otherwise disjoint from J2. For each Ci, i= 1, * , n there is a unique arc C in S such that f(Ci) =f(Ci) and C! fI2 The arcs C* separate f2 into n +1 components Xo, * * , Xn, and the endpoints of the arcs Ci separate 12 into a number of component arcs. At least two of these component arcs, say E1, * * *, Em, have the property that they join the two endpoints of some arc Ci. For each i=1, = * , n, let pi be an interior point of Ci and let p be the unique point of C' such that f(p) =f(pi). Also let qi be an interior point of Ei for each i= 1, * , m. Then each component Xi of I2-Ufo1 Ci has a certain number r of pi's and qi's designated on its boundary and a certain number s of p!'s designated on its interior.