The 0-concordance monoid is infinitely generated

Abstract

Under the relation of 0-concordance, the set of knotted 2-spheres in S 4 forms a commutative monoid M0 with the operation of connected sum. Sunukjian has recently shown that M0 contains a submonoid isomorphic to Z ≥0 . In this note, we show that M0 contains a submonoid isomorphic to (Z ≥0 ) ∞ . Our argument relates the 0-concordance monoid to linear independence of certain Seifert solids in the (spin) rational homology cobordism group.1991 Mathematics Subject Classification. 57Q45, 57Q60. 1 Here, by a linear relation in M0, we mean a 0-concordance between two connected sums #ciSi and #cjSj, with all coefficients non-negative (since M0 is a monoid). Linear independence is then defined in the obvious way.

0-concordance of knotted surfaces and Alexander ideals

“…The techniques of this paper provide a very different answer to the 0-concordance problem than was provided recently by several authors [Sun19], [DM19]. However, all of these approaches utilize the factoring of a 0-concordance into two ribbon concordances.…”

“…Also, the double branched cover of a knot in S 3 has a unique spin structure, so this agrees with the one induced by S 4 on the fiber of the 2-twist spin of the Stevedore. Thus this is an example where the techniques of [Sun19], [DM19] cannot obstruct 0-concordance, but Alexander ideals can. There are infinitely many slice 2-bridge knots with any given nonunit, square determinant, so all of their double branched covers share this property.…”

“…This established the existence of infinitely many 0-concordance classes of 2-knots. Dai and Miller proved that M and N as above must be spin rational homology cobordant, and utilized linear independence in the spin rational homology cobordism group to obtain infinitely many linearly independent 0-concordance classes of 2-knots [DM19].…”