Peter D. Horn
NSF Postdoc
Research
My research interests are low-dimensional topology, knot theory, and the pertinent group theory. I am here at Columbia to study what implications Heegaard Floer homology has in knot concordance.
Knot Concordance and Homology Cobordism
joint with Tim Cochran, Bridget Franklin and Matt Hedden
We consider the question: "If the zero-framed surgeries on two oriented knots in $S^3$ are integral homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?" We show that this question has a negative answer in the smooth category, even for topologically slice knots. To show this we first prove that the zero-framed surgery on $K$ is $\mathbb{Z}$-homology cobordant to the zero-framed surgery on many of its winding number one satellites $P(K)$. Then we prove that in many cases the $\tau$- and $s$-invariants of $K$ and $P(K)$ differ. Consequently neither $\tau$ nor $s$ is an invariant of the smooth homology cobordism class of the zero-framed surgery. We also show, that a natural rational version of this question has a negative answer in both the topological and smooth categories, by proving similar results for $K$ and its $(p,1)$-cables.
available at arXiv: 1102.5730
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Higher-order Signature Cocycles for Subgroups of Mapping Class Groups and Homology Cylinders
joint with Tim Cochran and Shelly Harvey
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 $S$.
available at arXiv: 1003.4977
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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 $K$ detects more structure of minimal genus Seifert surfaces for $K$. 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 $L^2$-signatures bound this invariant from below.
available at arXiv: 0901.2095
or the published version
in Proceedings of the American Mathematical Society, 138, (2010), 2209-2215.
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Higher-order Analogues of the Slice Genus of a Knot
For certain classes of knots we define geometric invariants called higher-order genera. Each of these invariants is a refinement of the slice genus of a knot. We find lower bounds for the higher-order genera in terms of certain von Neumann $\rho$-invariants, which we call higher-order signatures. The higher-order genera offer a refinement of the Grope filtration of the knot concordance group.
available at arXiv: 0807.0434
or the published version
in International Mathematics Research Notices, 2011, no. 5, 1091-1106.
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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 in a paper of Cochran, Orr and Teichner. Our main result is that successive quotients at each
stage in this filtration have 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.
available at arXiv: 0804.2661
or the published version
in Commentarii Mathematici Helvetici, 85, (2010), no. 4, 751-773.
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The First-order Genus of a Knot
We introduce a geometric invariant of knots in the three-sphere, called the first-order genus, that is derived from certain 2-complexes called gropes, and we show it is computable for many examples. In computing this invariant, we draw some interesting conclusions about the structure of a general Seifert surface for some knots.
available at arXiv: 0712.1010
or the published version (local link)
in Mathematical Proceedings of the Cambridge Philosophical Society, 146, (2009), 135-149.
(link to publisher's site)
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