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Register a new userLinear independence of rationally slice knots
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A knot in $S^3$ is rationally slice if it bounds a disk in a rational homology ball. We give an infinite family of rationally slice knots that are linearly independent in the knot concordance group. In particular, our examples are all infinite order. All previously known examples of rationally slice knots were order two.
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We classify the positive definite intersection forms that arise from smooth 4-manifolds with torsion-free homology bounded by positive integer surgeries on the right-handed trefoil. A similar, slightly less complete classification is given for the (2,5)-torus knot, and analogous results are obtained for integer surgeries on knots of slice genus at most two. The proofs use input from Yang--Mills instanton gauge theory and Heegaard Floer correction terms.
We give simple homological conditions for a rational homology 3-sphere Y to have infinite order in the rational homology cobordism group, and for a collection of rational homology spheres to be linearly independent. These translate immediately to statements about knot concordance when Y is the branched double cover of a knot, recovering some results of Livingston and Naik. The statements depend only on the homology groups of the 3-manifolds, but are proven through an analysis of correction terms and their behavior under connected sums.
We study an invariant of a 3-manifold which consists of Reidemeister torsion for linear representations which pass through a finite group. We show a Dehn surgery formula on this invariant and compute that of a Seifert manifold over $S^2$. As a consequence we obtain a necessary condition for a result of Dehn surgery along a knot to be Seifert fibered, which can be applied even in a case where abelian Reidemeister torsion gives no information.
We define the stabilizing number $operatorname{sn}(K)$ of a knot $K subset S^3$ as the minimal number $n$ of $S^2 times S^2$ connected summands required for $K$ to bound a nullhomotopic locally flat disc in $D^4 # n S^2 times S^2$. This quantity is defined when the Arf invariant of $K$ is zero. We show that $operatorname{sn}(K)$ is bounded below by signatures and Casson-Gordon invariants and bounded above by the topological $4$-genus $g_4^{operatorname{top}}(K)$. We provide an infinite family of examples with $operatorname{sn}(K)<g_4^{operatorname{top}}(K)$.
A Chebyshev knot is a knot which admits a parametrization of the form $ x(t)=T_a(t); y(t)=T_b(t) ; z(t)= T_c(t + phi), $ where $a,b,c$ are pairwise coprime, $T_n(t)$ is the Chebyshev polynomial of degree $n,$ and $phi in RR .$ Chebyshev knots are non compact analogues of the classical Lissajous knots. We show that there are infinitely many Chebyshev knots with $phi = 0.$ We also show that every knot is a Chebyshev knot.
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