Cobordism of disk knots
نویسنده
چکیده
We study cobordisms and cobordisms rel boundary of PL locally-flat disk knots Dn−2 ↪→ Dn. Any two disk knots are cobordant if the cobordisms are not required to fix the boundary sphere knots , and any two even-dimensional disk knots with isotopic boundary knots are cobordant rel boundary. However, the cobordism rel boundary theory of odd-dimensional disk knots is more subtle. Generalizing results of J. Levine on the cobordism of sphere knots, we define disk knot Seifert matrices and show that two higher-dimensional disk knots with isotopic boundaries are cobordant rel boundary if and only if their disk knot Seifert matrices are algebraically cobordant. We also ask which algebraic cobordism classes can be realized given a fixed boundary knot and provide a complete classification when the boundary knot has no 2-torsion in its middle-dimensional Alexander module. In the course of this classification, we establish a close connection between the Blanchfield pairing of a disk knot and the Farber-Levine torsion pairing of its boundary knot (in fact, for disk knots satisfying certain connectivity assumptions, the disk knot Blanchfield pairing will determine the boundary Farber-Levine pairing). In addition, we study the dependence of disk knot Seifert matrices on choices of Seifert surface, demonstrating that all such Seifert matrices are rationally S-equivalent, but not necessarily integrally S-equivalent. 2000 Mathematics Subject Classification: Primary 57Q45; Secondary 57Q60, 11E39, 11E81
منابع مشابه
Cobordism of Knots on Surfaces
We introduce a relation of cobordism for knots in thickened surfaces and study cobordism invariants of such knots. AMS Classification 57M27
متن کاملGrope Cobordism of Classical Knots
Motivated by the lower central series of a group, we define the notion of a grope cobordism between two knots in a 3-manifold. Just like an iterated group commutator, each grope cobordism has a type that can be described by a rooted unitrivalent tree. By filtering these trees in different ways, we show how the Goussarov-Habiro approach to finite type invariants of knots is closely related to ou...
متن کاملHomology Cobordism and Classical Knot Invariants
In this paper we define and investigate Z2–homology cobordism invariants of Z2–homology 3–spheres which turn out to be related to classical invariants of knots. As an application we show that many lens spaces have infinite order in the Z2–homology cobordism group and we prove a lower bound for the slice genus of a knot on which integral surgery yields a given Z2– homology sphere. We also give s...
متن کاملInvolutive Heegaard Floer Homology
Using the conjugation symmetry on Heegaard Floer complexes, we define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to Z4-equivariant Seiberg-Witten Floer homology. Further, we obtain two new invariants of homology cobordism, d and d̄, and two invariants of smooth knot concordance, V 0 and V 0. We also develop a formula for the involutive Heeg...
متن کاملGrope Cobordism and Feynman Diagrams
We explain how the usual algebras of Feynman diagrams behave under the grope degree introduced in [CT]. We show that the Kontsevich integral rationally classifies grope cobordisms of knots in 3-space when the “class” is used to organize gropes. This implies that the grope cobordism equivalence relations are highly nontrivial in dimension 3. We also show that the class is not a useful organizing...
متن کامل