A theory of concordance for non-spherical 3-knots

Consider a closed connected oriented 3-manifold embedded in the -sphere, which is called a -knot in this paper. For two such knots, we say that their Seifert forms are spin concordant, if they are algebraically concordant with respect to a diffeomorphism between the 3-manifolds which preserves their spin structures. Then we show that two simple fibered 3-knots are geometrically concordant if and only if they have spin concordant Seifert forms, provided that they have torsion free first homology groups. Some related results are also obtained.

     

All frame-spun knots are slice

Frame-spun knots are constructed by spinning a knot of lower dimension about a framed submanifold of . We show that all frame-spun knots are slice (null-cobordant).

     

Unknotting Tunnels and Seifert Surfaces

Let K be a knot with an unknotting tunnel and suppose thatK is not a 2-bridge knot. There is an invariant = p/q Q/2Z,with p odd, defined for the pair (K, ). The invariant has interesting geometric properties. It is oftenstraightforward to calculate; for example, for K a torus knotand an annulus-spanning arc, (K, ) = 1. Although is definedabstractly, it is naturally revealed when K is put in thinposition. If 1 then there is a minimal-genus Seifert surfaceF for K such that the tunnel can be slid and isotoped to lieon F. One consequence is that if (K, ) 1 then K > 1. Thisconfirms a conjecture of Goda and Teragaito for pairs (K, )with (K, ) 1. 2000 Mathematics Subject Classification 57M25,57M27.     

A Calculation of the Culler-shalen Seminorms Associated to Small Seifert Dehn Fillings

We determine the relationship between the multiplicities ofthe zeros of certain rational functions defined on the SL₂(C)and PSL₂C character varieties of knot exteriors. This leadsto a formula for the Culler–Shalen seminorms associatedto small Seifert Dehn fillings of such manifolds. 2000 MathematicsSubject Classification: 57M05, 57M27, 57R65.     

Generalized Seifert surfaces and signatures of colored links

In this paper, we use `generalized Seifert surfaces' to extend the Levine-Tristram signature to colored links in . This yields an integral valued function on the -dimensional torus, where is the number of colors of the link. The case corresponds to the Levine-Tristram signature. We show that many remarkable properties of the latter invariant extend to this -variable generalization: it vanishes for achiral colored links, it is `piecewise continuous', and the places of the jumps are determined by the Alexander invariants of the colored link. Using a -dimensional interpretation and the Atiyah-Singer -signature theorem, we also prove that this signature is invariant by colored concordance, and that it provides a lower bound for the `slice genus' of the colored link.

     

Nonzero degree maps between closed orientable three-manifolds

This paper adresses the following problem: Given a closed orientable three-manifold , are there at most finitely many closed orientable three-manifolds 1-dominated by ? We solve this question for the class of closed orientable graph manifolds. More precisely the main result of this paper asserts that any closed orientable graph manifold 1-dominates at most finitely many orientable closed three-manifolds satisfying the Poincaré-Thurston Geometrization Conjecture. To prove this result we state a more general theorem for Haken manifolds which says that any closed orientable three-manifold 1-dominates at most finitely many Haken manifolds whose Gromov simplicial volume is sufficiently close to that of .

     

SOME APPLICATIONS OF PLANAR GRAPH IN KNOT THEORY

The relationship between a link diagram and its corresponding planar graph is briefly reviewed.A necessary and sufficient condition is given to detect when a planar graph corresponds to a knot.The rela...     

Braiding a link with a fixed closed braid

We show that every oriented link diagram with a closed braid diagram as a sublink diagram can be deformed into a closed braid diagram by a deformation keeping the sublink diagram and, under a mild condition, the number of Seifert circles fixed. As an application, we give an upper bound for the braid index of the link obtained by reversing the orientation of its sublink by using only the information of an original link.     

Imbeddings of free actions on handlebodies

Fix a free, orientation-preserving action of a finite group on a -dimensional handlebody . Whenever acts freely preserving orientation on a connected -manifold , there is a -equivariant imbedding of into . There are choices of closed and Seifert-fibered for which the image of is a handlebody of a Heegaard splitting of . Provided that the genus of is at least , there are similar choices with closed and hyperbolic.

     

Generalized Neuwirth Groups and Seifert Fibered Manifolds

The topological properties of the generalized Neuwirth groups, _n^k are discussed. For examp, we demonstrate that the group, _n^k is the fundamental group of the Seifert fibered space _n^k. Moreover, discuss some other invariants and algebraic properties of the above groups.This work was supported by Polish grant (BW-5100–5–0259–9) and the Russian Foundation for Basic Research (grant number 98–01–00699).2000 Mathematics Subject Classification: 20F34, 57M05, 57M60