[Paper Review] 0-concordance of knotted surfaces and Alexander ideals

Jason Joseph|arXiv (Cornell University)|Nov 29, 2019

Geometric and Algebraic Topology19 references4 citations

TL;DR

This paper establishes that the Alexander ideal of a knotted surface in $S^4$ is a 0-concordance invariant, inducing a homomorphism from the 0-concordance monoid $\mathscr{C}_0$ to the ideal class monoid of $\mathbb{Z}[t^{\pm 1}]$. The key result is that any surface knot with a nonprincipal Alexander ideal cannot be 0-slice or invertible in $\mathscr{C}_0$, providing the first obstruction to 0-concordance beyond the classical Fox-Milnor condition and proving the existence of infinitely many linearly independent 0-concordance classes.

ABSTRACT

In this paper we provide a new obstruction to 0-concordance of knotted surfaces in $S^4$ in terms of Alexander ideals. We use this to prove the existence of infinitely many linearly independent 0-concordance classes and to provide the first proof that the submonoid of 2-knots is not a group. The main result is that the Alexander ideal induces a homomorphism from the 0-concordance monoid $\mathscr{C}_0$ of oriented surface knots in $S^4$ to the ideal class monoid of $\mathbb{Z}[t^{\pm1}]$. Consequently, any surface knot with nonprincipal Alexander ideal is not 0-slice and in fact, not invertible in $\mathscr{C}_0$. Many examples are given. We also characterize which ideals are the ideals of surface knots, generalizing a theorem of Kinoshita, and generalize the knot determinant to the case of nonprincipal ideals. Lastly, we show that under a mild condition on the knot group, the peripheral subgroup of a knotted surface is also a 0-concordance invariant.

Motivation & Objective

• To establish a new obstruction to 0-concordance of knotted surfaces in $S^4$ using the Alexander ideal.
• To prove that the submonoid of 2-knots under connected sum is not a group by showing non-invertibility of knots with nonprincipal Alexander ideals.
• To characterize which ideals arise as Alexander ideals of surface knots, generalizing Kinoshita’s theorem.
• To show that the peripheral subgroup is a 0-concordance invariant under mild group-theoretic conditions.
• To extend the knot determinant to nonprincipal ideals and study its behavior under twist spinning.

Proposed method

• The paper defines a homomorphism $\Delta: \mathscr{C}_0 \to \mathcal{I}(\mathbb{Z}[t^{\pm 1}])$ by assigning to each surface knot its Alexander ideal.
• It proves that during a ribbon concordance, the Alexander ideal changes by multiplication by a principal ideal, using a factoring of 0-concordances into two opposing ribbon concordances.
• The key lemma establishes that the Alexander ideal of a connected sum is the product of the individual ideals, ensuring compatibility with the monoid structure.
• The paper analyzes twist spins of classical knots, showing that $\tau^nK$ has nonprincipal Alexander ideal if $|\Delta_K(-1)| \neq 1$.
• It characterizes the image of the Alexander ideal map by generalizing Kinoshita’s theorem to arbitrary ideals in $\mathbb{Z}[t^{\pm 1}]$.
• It proves that the peripheral subgroup is a 0-concordance invariant when the knot group satisfies a primality condition on its commutator subgroup.

Experimental results

Research questions

• RQ1 Does every 0-slice 2-knot admit a ribbon concordance to the unknot?
• RQ2 Is the submonoid of 2-knots in $\mathscr{C}_0$ closed under inversion?
• RQ3 Can the Alexander ideal detect non-0-sliceness even when the Alexander polynomial is trivial or principal?
• RQ4 Are there infinitely many linearly independent 0-concordance classes of surface knots?
• RQ5 Is the peripheral subgroup preserved under 0-concordance for knots with prime knot groups?

Key findings

• The Alexander ideal induces a well-defined homomorphism $\Delta: \mathscr{C}_0 \to \mathcal{I}(\mathbb{Z}[t^{\pm 1}])$, making it a 0-concordance invariant.
• Any surface knot with a nonprincipal Alexander ideal is not 0-slice and has no inverse in $\mathscr{C}_0$, proving the submonoid of 2-knots is not a group.
• For any classical knot $K$ with $|\Delta_K(-1)| \neq 1$, there exist infinitely many $n \in \mathbb{Z}$ such that the $n$-twist spin $\tau^nK$ has nonprincipal Alexander ideal.
• The determinant of a 2-bridge knot is an invariant of 0-concordance for its 2-twist spins, so $\tau^2K$ and $\tau^2J$ are 0-concordant only if $|\Delta_K(-1)| = |\Delta_J(-1)|$.
• Under the condition that the knot group is prime (e.g., 2-twist spins of $(2,p)$-torus knots for odd prime $p$), the peripheral subgroup is a 0-concordance invariant.
• The paper generalizes the knot determinant to nonprincipal ideals, showing that the ideal class of the Alexander ideal captures arithmetic obstructions to 0-concordance.