Smooth concordance of links topologically concordant to the Hopf link

Abstract

It was shown by Jim Davis that a 2‐component link with Alexander polynomial one is topologically concordant to the Hopf link. In this paper, we show that there is a 2‐component link with Alexander polynomial one that has unknotted components and is not smoothly concordant to the Hopf link, answering a question of Jim Davis. We construct infinitely many concordance classes of such links, and show that they have the stronger property of not being smoothly concordant to the Hopf link with knots tied in the components.

Injectivity of satellite operators in knot concordance

“…If such a 2-component link were concordant to the positive Hopf link (equal in C * ), then Proposition 2.3 shows that the resulting operator would be equal to that of the Hopf link, which is the identity operator. However, as mentioned in Section 2, there are many such links (operators) that are not concordant to the Hopf link as evidenced by several recent papers [5,6,12]. Thus, there appear to exist a large number of distinct strong winding number ±1 operators whereinP is unknotted.…”

“…Since a Hopf link with a local knot tied in the first component corresponds to the 'connected-sum' operator (the identity operator for the Hopf link itself), and since the connected-sum operator is clearly injective, our main result for strong winding number 1 operators would follow. Therefore, it is important to note that there are many such links that are not concordant to the Hopf link as evidenced by several recent papers [5,12]. Thus, there exist a very large number of strong winding number ±1 operators that are (presumably) distinct from the trivial 'connected-sum' operator.…”

Concordance groups of links

“…Partly oriented links L 2 #H and L 3 #H. The band shown passing through the box marked K is tied in the knot K with zero framing (cf. [5]).…”

“…Thus it bounds a smoothly embedded surface F in D 4 which is either one disk and two Möbius bands, or a disk and an annulus, in each case with the marked component bounding the disk. The first possibility is ruled out by linking numbers as in Lemma 2.4, and the second is equivalent to existence of a concordance in the traditional sense, given by two properly embedded annuli in S 3 × I, between L i and H. This is ruled out in the case of L 3 since δ(C) = 0 implies C is not slice, and is ruled out in the case of L 2 by recent work of Cha-Kim-Ruberman-Strle [5].…”

Concordance of links with identical Alexander invariants

“…The knot case has been extensively studied in the literature; see, for example, [16,21] as early works. The case of links with unknotted components has been shown recently in [4].…”