Peter D. Horn

Filtering smooth concordance classes of topologically slice knots

We propose and analyze a structure with which to organize the difference between a knot in S 3 bounding a topologically embedded 2‐disk in B 4 and it bounding a smoothly embedded disk. The n‐solvable filtration of the topological knot concordance group, due to Cochran‐Orr‐Teichner, may be complete in the sense that any knot in the intersection of its terms may well be topologically slice. However, the natural extension of this filtration to what is called the n‐solvable filtration of the smooth knot concordance group, is unsatisfactory because any topologically slice knot lies in every term of the filtration. To ameliorate this we investigate a new filtration,fBng, that is simultaneously a refinement of the n‐solvable filtration and a generalization of notions of positivity studied by Gompf and Cochran. We show that each Bn=BnC1 has infinite rank. But our primary interest is in the induced filtration,fTng, on the subgroup, T , of knots that are topologically slice. We prove that T=T0 is large, detected by gauge-theoretic invariants and the , s , ‐invariants, while the nontriviality of T0=T1 can be detected by certain d ‐invariants. All of these concordance obstructions vanish for knots in T1 . Nonetheless, going beyond this, our main result is that T1=T2 has positive rank. Moreover under a “weak homotopy-ribbon” condition, we show that each Tn=TnC1 has positive rank. These results suggest that, even among topologically slice knots, the fundamental group is responsible for a wide range of complexity. 57M25

The non-triviality of the Grope filtrations of the knot and link concordance groups

We consider the Grope filtration of the classical knot concordance group that was introduced by Cochran, Orr and Teichner. Our main result is that each successive quotient in this filtration has infinite rank. We also establish the analogous result for the Grope filtration of the concordance group of string links consisting of more than one component.

Higher-order signature cocycles for subgroups of mapping class groups and homology cylinders

We define families of invariants for elements of the mapping class group of S, a compact orientable surface. Fix any characteristic subgroup H of pi_1(S) and restrict to J(H), any subgroup of mapping classes that induce the identity modulo H. To any unitary representation, r of pi_1(S)/H we associate a higher-order rho_r-invariant and a signature 2-cocycle sigma_r. These signature cocycles are shown to be generalizations of the Meyer cocycle. In particular each rho_r is a quasimorphism and each sigma_r is a bounded 2-cocycle on J(H). In one of the simplest non-trivial cases, by varying r, we exhibit infinite families of linearly independent quasimorphisms and signature cocycles. We show that the rho_r restrict to homomorphisms on certain interesting subgroups. Many of these invariants extend naturally to the full mapping class group and some extend to the monoid of homology cylinders based on S.

Structure in the bipolar filtration of topologically slice knots

is the bipolar filtration of T . We show that T0=T1 has infinite rank, even modulo Alexander polynomial one knots. Recall that knots in T0 (a topologically slice 0‐ bipolar knot) necessarily have zero ‐, s ‐ and ‐invariants. Our invariants are detected using certain d ‐invariants associated to the 2‐fold branched covers. 57M25; 57N70

A higher-order genus invariant and knot Floer homology

It is known that knot Floer homology detects the genus and Alexander polynomial of a knot. We investigate whether knot Floer homology of

The first-order genus of a knot

K

On Computing the First Higher-Order Alexander Modules of Knots

detects more structure of minimal genus Seifert surfaces for

Knot concordance and homology cobordism

K

On the intersection ring of graph manifolds

. We define an invariant of algebraically slice, genus one knots and provide examples to show that knot Floer homology does not detect this invariant. Finally, we remark that certain metabelian

Higher-order Analogues of the Slice Genus of a Knot

L^2