July’s topics:

Unknotting conjecture disproved.

Matthew Sparkes contributed “Complex knots can actually be easier to untie than simple ones” to New Scientist, July 15, 2025. The title is actually misleading. Sparkes is covering new research on the unknotting number (precise definition below), which measures how hard it is to untie a knot. The news: Some complex knots have a lower unknotting number than predicted. Thus, a more accurate headline might be “Some complex knots are easier to untie than expected.”

To explain the unknotting number, we need a few details. Typically, a knot is illustrated by its projection onto a plane. When two points on the knot lie above the same point in the plane, this is called a “crossing.” At each crossing, one strand crosses over, the other under. (It is easy to tell from the illustration which is which.)

The crossing number of a knot is the smallest number of crossings occurring in any projection of the knot. This is the most basic knot invariant; in knot tables, the entries are grouped by their crossing numbers and distinguished by subscripts. For example, the seven different knots with seven crossings (a lovely coincidence) are labeled $7_1, 7_2, \dots, 7_7$. To compute the unknotting number, we change crossings, meaning we interchange the two strands of the knot that lie above that point. A crossing change gives a different knot, perhaps simpler, perhaps more complicated. The exact location where a crossing is changed can make a big difference in the topology of the result.

Crossing changes in different places yield different results. Here, we show two different crossing changes of the knot $7_2$. The crossings to be changed are highlighted. The crossing change on the left leads to the unknot. The one on the right produces the non-trivial knot $5_2$. Image credit: Tony Phillips.

Now we can define the unknotting number. It’s the smallest number of crossing changes that will result in a completely unknotted circle. For example, the knot $7_1$, also known as the right-handed $(2,7)$ torus knot, has unknotting number 3.

The torus knot $7_1$ has unknotting number 3. After changing the three highlighted crossings, the knot can be unfolded into an unknotted circle. (The mirror image of this figure would illustrate the same property for ${\bar 7_1}$, the left-handed $(2,7)$ torus knot.) It can be proved that no smaller number of crossing changes does the job. Image credit: Tony Phillips.

The new research concerns the behavior of the unknotting number under the operation of connected sum. The connected sum of two knots is the knot formed by building a 2-stranded bridge between them.

The connected sum of a right trefoil and a left trefoil is a square knot. Image credit: Tony Phillips.

A longstanding conjecture in knot theory is that the unknotting number of a connected sum is the sum of the unknotting numbers of the components. This sounds reasonable, and in many cases is true. But not always: In “Unknotting number is not additive under connected sum,” Mark Brittenham and Susan Hermiller (University of Nebraska) report the existence of infinitely many counterexamples, and present one in detail. They prove that the unknotting number of the connected sum of $7_1$ and its mirror image ${\bar 7_1}$ is less than or equal to 5, whereas the two separate unknotting numbers add up to 6.

The (2,7) torus knot and its mirror image have both been cut and the loose ends connected to form a single knot.
The connected sum of the knot $7_1$ and its mirror image ${\bar 7_1}$. Brittenham and Hermiller prove that it has unknotting number less than or equal to 5, even though the two components, $7_1$ and ${\bar 7_1}$, each have unknotting number 3. Image credit: Tony Phillips.

The proof has many steps. It takes the authors 18 consecutive diagrams, some quite gnarly, to explain how the connected sum knot can be manipulated to manifest the appropriate sequence of crossings.

The unknotting number sounds like a fairly elementary concept, but it can be elusive. According to Brittenham and Hermiller, there are still several 10-crossing knots whose unknotting numbers are unknown. Also, remarkably, it is still not known whether the crossing number itself is additive under connected sum. The question had been open for 100 years when Colin Adams mentioned it as a “big unsolved question” in The Knot Book (2004).

Primes and partitions.

Earlier this summer, several news stories popped up touting new research on detecting prime numbers by William Craig, Jan-Willem van Ittersum, and Ken Ono.

Prime numbers are whole numbers that are only divisible by themselves and 1. The first few are 2, 3, 5, 7, 11, and 13. Euclid proved that there are infinitely many primes. But precisely how the primes are distributed among the integers is still murky.

So it was mathematical news when Craig, van Ittersum, and Ono produced an infinite family of functions that take in a whole number $n$ and spit out $0$ if and only if $n$ is a prime. Their paper, “Integer partitions detect the primes,” was published in PNAS last September.

To understand the concepts behind their functions, we can focus on the first function in the family: $$(n^2-3n+2)M_1(n)-8M_2(n).$$

The coefficients $M_1(n), M_2(n)$ are MacMahon numbers, combinatorial constants defined in terms of partitions. A partition of a whole number $n$ is a way of writing $n$ as a sum of natural numbers; for example, $6+3$ and $4+2+2+1$ are partitions of $9$, and so is $9$ itself.

Note that within a partition, terms can repeat themselves, like $7=3+3+1$, or $7=2+2+1+1+1$, or even $7= 1 + 1 + 1 + 1 +1 +1 +1$. The MacMahon numbers are based on recording how many different numbers are used in a partition, and how often each of them occurs.

The first MacMahon number, $M_1(n)$, is the easiest. You start with the whole number $n$ and consider all the partitions of $n$ with exactly one size (i.e., partitions where all the terms are equal). For each partition, note the multiplicity—how many times the term appears in the partition. For $n=7$, there are only two partitions with one size. There is the partition $7$, where the multiplicity is 1, and there is $1 + 1 + 1 + 1 +1 +1 +1$, where the multiplicity is 7.

Now sum up the multiplicities: In this example, you get $1 + 7 = 8$. Thus, $$M_1(7) = 8.$$ To find $M_1(n)$ for a different integer $n$, repeat this process using all the single-size partitions of $n$.

The later MacMahon numbers are a little more complicated. To calculate $M_k(n)$, you consider all the partitions of $n$ with $k$ distinct sizes. But now you get $k$ different multiplicities for each partition; these get multiplied, and now those products are added.

An example will help us unwind this definition. Suppose $k=2$ and $n=7$. There are 11 partitions of $7$ with 2 part sizes. In the table below, you can see the partitions, the multiplicities of the two parts, and the products of those multiplicities.

\begin{array} {c|c|c|c} \text{\small Partition} & \text{\small Multiplicity} & \text{\small Multiplicity} & \text{\small Product of}\\ ~& \text{\small of part size 1}&\text{\small of part size 2}&\text{\small multiplicities}\\\hline 6+1 & 1 & 1 & 1\\ 5+1+1 & 1 & 2 & 2\\ 5+2 & 1 & 1& 1\\ 4+3 & 1 & 1 & 1\\ 4+1+1+1 & 1 & 3 & 3\\ 3+3+1 & 2 & 1 & 2\\ 3+2+2 & 1 &2 & 2\\ 3+1+1+1+1 & 1 & 4 & 4\\ 2+2+2+1 & 3 & 1 & 3\\ 2+2+1+1+1 & 2 & 3 & 6\\ 2+1+1+1+1+1 & 1 & 5 & 5 \end{array}

$M_2(7)$ is the sum of all the products of multiplicities: $M_2(7)=1+2+1+1+3+2+2+4+3+6+5=30$.

There are other definitions of the MacMahon numbers, but this is the one that Craig et al. used, and is the one most directly tied to partitions.

We are now in a position to test the authors’ first prime-detecting function on the number 7. Evaluated at $n=7$, the expression $(n^2-3n+2)M_1(n)-8M_2(n)$ gives $(49-21+2)\cdot 8-8\cdot 30 = 0,$ as promised, since 7 is a prime. To repeat, the authors prove that this expression gives zero for any one of the infinitely many prime numbers, and a non-zero value (in fact, strictly positive) for any composite number. To sample just the beginning of this infinite sequence of prime detections, here are the values of $(n^2-3n+2)M_1(n)-8 M_2(n)$ on whole numbers from 2 to 49.

Values of $(n^2-3n+2)M_1(n)-8M_2(n)$ for $2 \leq n \leq 49$.

The authors give four other prime-detecting expressions, all linear combinations of $M_1(n), \dots, M_7(n)$. The coefficients of the linear combinations are integer-coefficient polynomials in $n$, and they conjecture that any prime-detecting expression of this (MacMahon number, integer-coefficient polynomial) type is essentially a linear combination of these five with integer-coefficient-polynomial coefficients.

Notably, the work brings together two usually separate parts of mathematics, combinatorics (partitions) and number theory (primes). The reasons for the connection are still mysterious, but clearly significant. “Clever,” “remarkable” and “mind-blowing” seem completely appropriate. My thanks to Matthew Russell (Texas A&M University) for help with this item.

—Tony Phillips, Stony Brook University

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