Monotonicity of the crossing spectrum of knots and the additivity threshold for multi-crossing numbers

The multi-crossing (or $n$-crossing) number $c_n(K)$ of a knot $K$ generalizes the classical crossing number: it is the minimal number of crossings in a projection of $K$ in which every crossing is a single point through which $n$ strands pass, each strand bisecting the crossing. The crossing spectrum of $K$ is the sequence $\{c_2(K), c_3(K), c_4(K),\ldots\}$, which always stabilizes at 1. Adams asked whether this spectrum is monotone non-increasing, i.e., whether $c_n(K)\ge c_{n+1}(K)$ for every knot $K$ and every $n\ge2$; the question is restated in Adams, Capovilla-Searle, Freeman, Irvine, Petti, Vitek, Weber, and Zhang (2014), where only the odd and even subsequences are known to decrease, leaving adjacent-parity comparisons open. The same work showed that for $n\ge4$ the $n$-crossing number is not additive under connected sum — examples satisfy $c_n(K_1\#K_2)=c_n(K_1)+c_n(K_2)-1$ — and left open whether a deficit worse than $-1$ can occur.

Background

A knot $K$ is a smooth embedding of a circle in three-dimensional space, considered up to continuous deformation; a link is a disjoint union of knots. A diagram of $K$ is a projection onto a plane whose only singularities are transverse double points, called crossings, each annotated with which strand passes over. The classical crossing number $c_2(K)$ is the minimal number of crossings over all diagrams of $K$.

An $n$-crossing (or multi-crossing) is a generalization introduced by Adams: a singular point of a projection through which $n$ strands pass, each strand bisecting the crossing (entering and leaving at antipodal points), together with a height ordering of the strands at the crossing. An $n$-crossing projection has only $n$-crossings as singularities, and Adams proved that every knot and link has an $n$-crossing projection for every $n\ge2$. The $n$-crossing number $c_n(K)$ is the minimal number of crossings in an $n$-crossing projection of $K$; $c_2$ recovers the classical crossing number. Every knot even has an übercrossing projection, one with a single multi-crossing, and a petal projection, in which that single crossing has no nested loops; consequently $c_n(K)=1$ for some $n$ depending on $K$.

The crossing spectrum of $K$ is the sequence $\{c_2(K), c_3(K), c_4(K),\ldots\}$, which stabilizes at 1 and is itself a knot invariant: for example, it distinguishes the composite knots $3_1\#5_1$ and $3_1\#m5_1$ (where $m$ denotes the mirror image), which many classical invariants do not separate. The principal general lower-bound tool for $c_n$ comes from the Kauffman bracket polynomial $\langle K\rangle$, a one-variable polynomial invariant of unoriented link diagrams defined by local skein relations. Its span, the difference between its maximal and minimal degrees, satisfies the classical bound $\mathrm{span}\langle K\rangle\le 4c_2(K)$ proved independently by Kauffman, Murasugi, and Thistlethwaite, which Adams et al. [1] generalized to $\mathrm{span}\langle K\rangle\le(\lfloor n^2/2\rfloor+4n-8)\,c_n(K)$ for all $n\ge3$.

What is known about how the spectrum varies with $n$ is a pattern of interlaced chains. Local moves convert an $n$-crossing into an $(n+2)$-crossing at no cost, giving $c_{n+2}(K)\le c_n(K)$, and Adams et al. [1] prove $c_n(K)\ge c_{2n}(K)$ for all $n\ge2$. Combined with sharp initial steps — $c_3(K)\le c_2(K)-2$ for every nontrivial knot other than certain 2-braid knots (Adams), $c_4(K)\le c_2(K)-1$ for every nontrivial knot (Adams), and $c_5(K)\le c_3(K)-1$ for every nontrivial nonsplit link other than the Hopf link (Adams–Hoste–Palmer) — these give two decreasing chains $c_2>c_3\ge c_5\ge\cdots$ and $c_2>c_4\ge c_6\ge\cdots$. Hagedorn later proved $c_9(K)\le c_3(K)-2$ for every knot other than the unknot, trefoil, and figure-eight, together with partial results toward the conjecture $c_{13}(K)<c_5(K)$. However, no comparison between $c_{2k}$ and $c_{2k+1}$ is known in general, and no counterexample to full monotonicity has appeared in any computed spectrum.

The behavior under connected sum is the second structural question. The connected sum $K_1\#K_2$ of two knots is obtained by removing a small arc from each and gluing the exposed endpoints; the classical crossing number is additive, $c_2(K_1\#K_2)=c_2(K_1)+c_2(K_2)$. For multi-crossing numbers this fails: for every $n\ge4$ there are knots with $c_n(K_1\#K_2)=c_n(K_1)+c_n(K_2)-1$ (for instance $c_4(5_2\#m6_2)=3=2+2-1$ and $c_5(6_2\#6_2)=3=2+2-1$), while for $n=3$ no failure of additivity is known and additivity has been proved for large classes of knots (the Type I and Type II classes treated in [1]). Whether the deficit can be smaller than $-1$ is open.

Problem Statement

Determine the comparative structure of the multi-crossing numbers of knots, by answering the two open questions discussed in Adams–Capovilla-Searle–Freeman–Irvine–Petti–Vitek–Weber–Zhang (2014), the first of which is attributed there to Adams:

1. (Monotonicity of the crossing spectrum.) Is it true that for every knot $K$ and every integer $n\ge2$:

$$c_n(K)\ \ge\ c_{n+1}(K)?$$

Equivalently: prove that the crossing spectrum is non-increasing in $n$, or exhibit a knot $K$ and an $n\ge2$ with $c_{n+1}(K)>c_n(K)$. The corresponding statement for links may be treated as a generalization.

2. (Additivity threshold under connected sum.) Is it true that for all knots $K_1$ and $K_2$ and all $n\ge2$:

$$c_n(K_1\#K_2)\ \ge\ c_n(K_1)+c_n(K_2)-1?$$

Deficits of exactly $-1$ are known to occur for $n\ge4$ (for example, $c_4(5_2\#m6_2)=3=2+2-1$ and $c_5(6_2\#6_2)=3=2+2-1$), so the question is whether the deficit $c_n(K_1\#K_2)-c_n(K_1)-c_n(K_2)$ can equal $-2$ or be smaller, or whether $-1$ is the sharp universal lower bound.

Both questions concern one scientific objective — how the minimal number of crossings behaves as the crossing order $n$ increases and under the operation of connected sum — and each is posed in full generality, with no restriction on the knots, on $n$, or on the parity of $n$. The two questions are independent: neither implies the other.

References

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