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40
Link concordance and higherorder Blanchfield duality
, 2008
"... In 1997, T. Cochran, K. Orr, and P. Teichner [13] defined a filtration of the classical knot concordance group C, · · · ⊆ Fn ⊆ · · · ⊆ F1 ⊆ F0.5 ⊆ F0 ⊆ C. The filtration is important because of its strong connection to the classification of topological 4manifolds. Here we introduce new techn ..."
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Cited by 30 (9 self)
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In 1997, T. Cochran, K. Orr, and P. Teichner [13] defined a filtration of the classical knot concordance group C, · · · ⊆ Fn ⊆ · · · ⊆ F1 ⊆ F0.5 ⊆ F0 ⊆ C. The filtration is important because of its strong connection to the classification of topological 4manifolds. Here we introduce new techniques for studying C and use them to prove that, for each n ∈ N0, the group Fn/Fn.5 has infinite rank. We establish the same result for the corresponding filtration of the smooth concordance group. We also resolve a longstanding question as to whether certain natural families of knots, first considered by CassonGordon, and Gilmer, contain slice knots.
A survey of twisted Alexander polynomials
, 2009
"... We give a short introduction to the theory of twisted Alexander polynomials of a 3–manifold associated to a representation of its fundamental group. We summarize their formal properties and we explain their relationship to twisted Reidemeister torsion. We then give a survey of the many applications ..."
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Cited by 27 (16 self)
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We give a short introduction to the theory of twisted Alexander polynomials of a 3–manifold associated to a representation of its fundamental group. We summarize their formal properties and we explain their relationship to twisted Reidemeister torsion. We then give a survey of the many applications of twisted invariants to the study of topological problems. We conclude with a short summary of the theory of higher order Alexander polynomials.
Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation
"... Abstract. Given a knot complement X and its p–fold cyclic cover Xp → X, we identify twisted polynomials associated to GL1(F[t ±1]) representations of π1(Xp) with twisted polynomials associated to related GLp(F[t ±1]) representations of π1(X) which factor through metabelian representations. This prov ..."
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Cited by 17 (5 self)
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Abstract. Given a knot complement X and its p–fold cyclic cover Xp → X, we identify twisted polynomials associated to GL1(F[t ±1]) representations of π1(Xp) with twisted polynomials associated to related GLp(F[t ±1]) representations of π1(X) which factor through metabelian representations. This provides a simpler and faster algorithm to compute these polynomials, allowing us to prove that 16 (of 18 previously unknown) algebraically slice knots of 12 or fewer crossings are not slice. We also use this improved algorithm to prove that the 24 mutants of the pretzel knot P(3,7, 9, 11,15), corresponding to permutations of (7, 9, 11,15), represent distinct concordance classes. 1.
Link concordance, homology cobordism, and Hirzebruchtype intersection form defects from towers of iterated pcovers
, 2007
"... Abstract. We obtain new invariants of topological link concordance and homology cobordism of 3manifolds from Hirzebruchtype intersection form defects of towers of iterated pcovers. Our invariants can extract geometric information from an arbitrary depth of the derived series of the fundamental gr ..."
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Cited by 16 (2 self)
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Abstract. We obtain new invariants of topological link concordance and homology cobordism of 3manifolds from Hirzebruchtype intersection form defects of towers of iterated pcovers. Our invariants can extract geometric information from an arbitrary depth of the derived series of the fundamental group, and can detect torsion which is invisible via signature invariants. Applications illustrating these features include the following: (1) There are infinitely many homology equivalent rational 3spheres which are indistinguishable via multisignatures, ηinvariants, and L 2signatures but have distinct homology cobordism types. (2) There is an infinite family of 2torsion (amphichiral) knots, including the figure eight knot, with nonslice iterated Bing doubles. (3) There exist infinitely many torsion elements at any depth of the CochranOrrTeichner filtration of link concordance. 1. Introduction and
HIGHERORDER ALEXANDER INVARIANTS AND FILTRATIONS OF THE KNOT CONCORDANCE GROUP
"... Abstract. We establish certain “nontriviality ” results for several filtrations of the smooth and topological knot concordance groups. First, as regards the nsolvable filtration of the topological knot concordance group, C, defined by K. Orr, P. Teichner and the first author: 0 ⊂···⊂F (n.5) ⊂F (n) ..."
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Cited by 16 (6 self)
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Abstract. We establish certain “nontriviality ” results for several filtrations of the smooth and topological knot concordance groups. First, as regards the nsolvable filtration of the topological knot concordance group, C, defined by K. Orr, P. Teichner and the first author: 0 ⊂···⊂F (n.5) ⊂F (n) ⊂···⊂F (1.5) ⊂F (1.0) ⊂F (0.5) ⊂F (0) ⊂C, we refine the recent nontriviality results of Cochran and Teichner by including information on the Alexander modules. These results also extend those of C. Livingston and the second author. We exhibit similar structure in the closely related symmetric Grope filtration of C. We also show that the Grope filtration of the smooth concordance group is nontrivial using examples that cannot be distinguished by the OzsváthSzabó τinvariant nor by J. Rasmussen’s sinvariant. Our broader contribution is to establish, in “the relative case”, the key homological results whose analogues CochranOrrTeichner established in “the absolute case”. We say two knots K0 and K1 are concordant modulo nsolvability if K0#(−K1) ∈ F (n). Our main result is that, for any knot K whose classical Alexander polynomial has degree greater than 2, and for any positive integer n, there exist infinitely many knots Ki that are concordant to K modulo nsolvability, but are all distinct modulo n.5solvability. Moreover, the Ki and K share the same classical Seifert matrix and Alexander module as well as sharing the same higherorder Alexander modules and Seifert presentations up to order n − 1. 1.
Primary decomposition and the fractal nature of knot concordance
, 2009
"... Abstract. For each sequence P = (p1(t), p2(t),...) of polynomials we define a characteristic series of groups, called the derived series localized at P. Given a knot K in S 3, such a sequence of polynomials arises naturally as the orders of certain submodules of a sequence of higherorder Alexander ..."
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Cited by 14 (5 self)
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Abstract. For each sequence P = (p1(t), p2(t),...) of polynomials we define a characteristic series of groups, called the derived series localized at P. Given a knot K in S 3, such a sequence of polynomials arises naturally as the orders of certain submodules of a sequence of higherorder Alexander modules of K. These group series yield filtrations of the knot concordance group that refine the (n)solvable filtration. We show that the quotients of successive terms of these refined filtrations have infinite rank. These results also suggest higherorder analogues of the p(t)primary decomposition of the algebraic concordance group. We use these techniques to give evidence that the set of smooth concordance classes of knots is a fractal set. We also show that no CochranOrrTeichner knot is concordant to any CochranHarveyLeidy knot. 1.
New topologically slice knots
, 2005
"... In the early 1980’s Mike Freedman showed that all knots with trivial Alexander polynomial are topologically slice (with fundamental group Z). This paper contains the first new examples of topologically slice knots. In fact, we give a sufficient homological condition under which a knot is slice with ..."
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Cited by 13 (1 self)
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In the early 1980’s Mike Freedman showed that all knots with trivial Alexander polynomial are topologically slice (with fundamental group Z). This paper contains the first new examples of topologically slice knots. In fact, we give a sufficient homological condition under which a knot is slice with fundamental group Z ⋉ Z[1/2]. These two fundamental groups are known to be the only solvable ribbon groups. Our homological condition implies that the Alexander polynomial equals (t − 2)(t −1 − 2) but also contains information about the metabelian cover of the knot complement (since there are many nonslice knots with this Alexander polynomial).
Polynomial splittings of CassonGordon invariants
 Math. Proc. Cambridge Philos. Soc
"... Abstract. In this paper we prove that the Casson–Gordon invariants of the connected sum of two knots split when the Alexander polynomials of the knots are coprime. As one application, for any knot K, all but finitely many algebraically slice twisted doubles of K are linearly independent in the knot ..."
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Cited by 10 (2 self)
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Abstract. In this paper we prove that the Casson–Gordon invariants of the connected sum of two knots split when the Alexander polynomials of the knots are coprime. As one application, for any knot K, all but finitely many algebraically slice twisted doubles of K are linearly independent in the knot concordance group. 1.
Link concordance and generalized doubling operators
 Algebr. Geom. Topol
"... Abstract. We introduce a new technique for showing classical knots and links are not slice. As one application we show that the iterated Bing doubles of many algebraically slice knots are not topologically slice. Some of the proofs do not use the existence of the CheegerGromov bound, a deep analyti ..."
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Cited by 10 (2 self)
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Abstract. We introduce a new technique for showing classical knots and links are not slice. As one application we show that the iterated Bing doubles of many algebraically slice knots are not topologically slice. Some of the proofs do not use the existence of the CheegerGromov bound, a deep analytical tool used by CochranTeichner. Our main examples are boundary links that cannot be detected in the algebraic boundary link concordance group, nor by any ρ invariants associated to solvable representations into finite unitary groups. 1.
Knot concordance and blanchfield duality
 Oberwolfach Reports
"... Abstract. We introduce a new technique for showing classical knots and links are not slice. As one application we resolve a longstanding question as to whether certain natural families of knots contain topologically slice knots. We also present a simpler proof of the result of CochranTeichner that ..."
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Cited by 10 (7 self)
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Abstract. We introduce a new technique for showing classical knots and links are not slice. As one application we resolve a longstanding question as to whether certain natural families of knots contain topologically slice knots. We also present a simpler proof of the result of CochranTeichner that the successive quotients of the integral terms of the CochranOrrTeichner filtration of the knot concordance group have rank 1. For links we have similar results. We show that the iterated Bing doubles of many algebraically slice knots are not topologically slice. Some of the proofs do not use the existence of the CheegerGromov bound, a deep analytical tool used by CochranTeichner. Our main examples are actually boundary links but cannot be detected in the algebraic boundary link concordance group, nor by any ρ invariants associated to solvable representations into finite unitary groups. 1.