Satellites of infinite rank in the smooth concordance group. (English) Zbl 1473.57013
This article offers several conjectures on how satellite operations on knots interact with the concordance group.
Two knots \(K_1\), \(K_2\) are said to be (smoothly) concordant if they cobound an (smoothly) embedded cyclinder in \(S^3\times[0,1]\). The equivalence classes of knots under concordance form a group \(\mathcal{C}\) under the connected sum operation. For any knot \(P\) in the solid torus \(S^1\times D^2\) there is a satellite operation which induces a map \(P:\mathcal{C}\to\mathcal{C}\), which usually is not a group homomorphism.
The central conjecture of the article is that if \(P:\mathcal{C}\to\mathcal{C}\) is not constant, then the image of \(P:\mathcal{C}\to\mathcal{C}\) generates a subgroup of infinite rank in \(C\). The conjecture is proved for the case where \(P\) has a non-zero winding number.
The main result of the article is a condition that is sufficient for the conjecture to hold for a given knot \(P\) with winding number zero. This criterion is based on rational linking numbers of lifts of \(\partial D^2\) to a double branched cover. The result is established using techniques from \(SO(3)\) gauge theory, such as instanton cobordism obstructions derived from instanton moduli spaces and Chern-Simons invariants. Some examples are included, which highlight how the criterion can be tested in practice.
Two knots \(K_1\), \(K_2\) are said to be (smoothly) concordant if they cobound an (smoothly) embedded cyclinder in \(S^3\times[0,1]\). The equivalence classes of knots under concordance form a group \(\mathcal{C}\) under the connected sum operation. For any knot \(P\) in the solid torus \(S^1\times D^2\) there is a satellite operation which induces a map \(P:\mathcal{C}\to\mathcal{C}\), which usually is not a group homomorphism.
The central conjecture of the article is that if \(P:\mathcal{C}\to\mathcal{C}\) is not constant, then the image of \(P:\mathcal{C}\to\mathcal{C}\) generates a subgroup of infinite rank in \(C\). The conjecture is proved for the case where \(P\) has a non-zero winding number.
The main result of the article is a condition that is sufficient for the conjecture to hold for a given knot \(P\) with winding number zero. This criterion is based on rational linking numbers of lifts of \(\partial D^2\) to a double branched cover. The result is established using techniques from \(SO(3)\) gauge theory, such as instanton cobordism obstructions derived from instanton moduli spaces and Chern-Simons invariants. Some examples are included, which highlight how the criterion can be tested in practice.
Reviewer: Benjamin Bode (Osaka)
MSC:
| 57K10 | Knot theory |
Software:
khocaReferences:
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