相关论文: Knot Concordance and Torsion

We prove that if the order of the first homology of the 2-fold branched cover of a knot K in the 3-sphere is given by pm where p is a prime congruent to 3 mod 4 and gcd(p,m) =1, then K is of infinite order in the knot concordance group.…

Let $M_K$ be the 2-fold branched cover of a knot $K in $S^3$. If $H_1(M_K) = {\bf Z}_3 \oplus {\bf Z}_{3^{2i}} \oplus G$ where 3 does not divide the order of $G$ then $K$ is not of order 4 in the concordance group. This obstruction detects…

By a recent result of Livingston, it is known that if a knot has a prime power branched cyclic cover that is not a homology sphere, then there is an infinite family of non-concordant knots having the same Seifert form as the knot. In this…

This thesis develops some general calculational techniques for finding the orders of knots in the topological concordance group C. The techniques currently available in the literature are either too theoretical, applying to only a small…

We show that, for any prime p, a knot K in the 3-sphere is determined by its p-fold cyclic unbranched covering. We also investigate when the m-fold cyclic unbranched covering of a knot coincides with the n-fold cyclic unbranched covering of…

We define a nontrivial mod 2 valued additive concordance invariant defined on the torsion subgroup of the knot concordance group using involutive knot Floer package. For knots not contained in its kernel, we prove that their iterated…

If a knot K has Seifert matrix V_K and has a prime power cyclic branched cover that is not a homology sphere, then there is an infinite family of non-concordant knots having Seifert matrix V_K.

For some families of two-bridge knots, including double-twist knots with genus at least four, we determine precisely the set of integers $n>1$ such that the fundamental group of the $n$-fold cyclic branched cover of the 3-sphere along these…

Kirby and Lickorish showed that every knot in the 3-sphere is concordant to a prime knot, equivalently, every concordance class contains a prime knot. We prove here that their result can be strengthened: Every knot in the 3-sphere is…

The concordance group of knots in the three-sphere contains an infinite subgroup generated by elements of order two, each one of which is represented by a knot K with the property that for every n > 0, the n-fold cyclic cover of S^3…

We generalize the Manolescu-Owens smooth concordance invariant delta(K) of knots K in the 3-sphere to invariants delta_{p^n}(K) obtained by considering covers of order p^n, with p prime. Our main result shows that for any odd prime p, the…

By studying the Heegaard Floer homology of the preimage of a knot K in S^3 inside its double branched cover, we develop simple obstructions to K having finite order in the classical smooth concordance group. As an application, we prove that…

For any integer n > 2, the n-fold cyclic branched cover M of an alternating prime knot K in the 3-sphere determines K, meaning that if K is a knot in the 3-sphere that is not equivalent to K then its n-fold cyclic branched cover cannot be…

The concordance genus of a knot K is the minimum three-genus among all knots concordant to K. For prime knots of 10 or fewer crossings there have been three knots for which the concordance genus was unknown. Those three cases are now…

We discuss an infinite class of metabelian Von Neumann rho-invariants. Each one is a homomorphism from the monoid of knots to the real line. In general they are not well defined on the concordance group. Nonetheless, we show that they pass…

We prove that a prime knot K is not determined by its p-fold cyclic branched cover for at most two odd primes p. Moreover, we show that for a given odd prime p, the p-fold cyclic branched cover of a prime knot K is the p-fold cyclic…

Levine defined the rational algebraic knot concordance group and proved that each nontrivial element is of order two, of order four, or of infinite order. The determination of the order of an element depends on a p-adic analysis for all…

We show that the 3-fold cyclic branched cover of any genus 2 two-bridge knot $K_{[-2q,2s,-2t,2l]}$ is an L-space and its fundamental group is not left-orderable. Therefore the family of 3-fold cyclic branched cover of any genus 2 two-bridge…

Many well studied knots can be realized as positive braid knots where the braid word contains a positive full twist; we say that such knots are twist positive. Some important families of knots are twist positive, including torus knots,…

The existence of topologically slice knots that are of infinite order in the knot concordance group followed from Freedman's work on topological surgery and Donaldson's gauge theoretic approach to 4-manifolds. Here, as an application of…