What are Mathematical Knots?

In mathematical knot theory, a knot is defined as a closed, non-self-intersecting curve embedded in three-dimensional space (ℝ³). Unlike everyday knots in ropes, mathematical knots have their ends joined together so they cannot be undone.

The simplest knot is the unknot (or trivial knot), which is essentially a simple loop. Knot theory studies how knots can be deformed, classified, and distinguished from one another through continuous transformations (without cutting or passing through themselves).

Unsolved Problem: Additivity of the Unknotting Number

Background and Definition

The unknotting number \( u(K) \) of a knot \( K \) is the minimum number of crossing changes required to transform \( K \) into the unknot.

A fundamental operation in knot theory is the connected sum of two knots \( K_1 \) and \( K_2 \), denoted \( K_1 \# K_2 \). This involves cutting each knot and connecting them to form a composite knot.

The Conjecture and Its Recent Disproof

For nearly 90 years, mathematicians believed that the unknotting number was additive under connected sum:

\[ u(K_1 \# K_2) = u(K_1) + u(K_2) \]

This conjecture, first proposed by Hilmar Wendt in 1937, was intuitively appealing: combining two knots should make them at least as hard to untie as the sum of their complexities.

However, in a groundbreaking 2025 preprint, Mark Brittenham and Susan Hermiller (University of Nebraska–Lincoln) disproved this conjecture. They demonstrated that for infinitely many pairs of knots:

\[ u(K_1 \# K_2) < u(K_1) + u(K_2) \]

Specifically, they showed that for the knot \(7_1\) and its mirror image \(\overline{7_1}\):

\( u(7_1 \# \overline{7_1}) \leq 5 \)
\( u(7_1) + u(\overline{7_1}) = 3 + 3 = 6 \)

This means the composite knot is easier to untie than the sum of its parts.

Why This Problem Remains Open

While Brittenham and Hermiller found infinitely many counterexamples, the full characterization of when additivity holds or fails is still unknown. Key open questions include:

  • Which knots violate additivity?
  • How large can the gap be between \( u(K_1) + u(K_2) \) and \( u(K_1 \# K_2) \)?
  • Are there knots where additivity still holds?
  • What is the computational complexity of determining unknotting numbers?

Other Unsolved Problems in Knot Theory

Additivity of Crossing Number

Is the crossing number additive under connected sum? This problem has been open for over 100 years and relates to knot factorization.

The Slice-Ribbon Conjecture

Are all slice knots also ribbon knots? This problem connects 3D knots to 4D topology.

The Geodesic Problem

Do all hyperbolic knots have a shortest geodesic? This is important for hyperbolic knot invariants.