Disproof of Knot Unknotting Number Additivity Conjecture

Key Quotes

"One of the most basic questions about knots is which knots are the same and which ones are different from each other so what this means is if I tie myself in climbing one of the key properties of my figure eight knot is that if I tighten it it's still a figure eight knot and mathematicians want to capture that feature of knots so that if you pull on your rope a little bit or twist it around it really is still the same thing but what they don't want you to do is untie the knot completely so what you do to create a mathematical knot is you tie a knot in a piece of string and then you're going to just merge the ends of the string together so that you have a knotted up circle and you're not going to be able to untie it unless you cut something essentially"

Leila Sloman explains that a mathematical knot is a closed loop of string where the ends are joined, making it impossible to untie without cutting. This definition is crucial for mathematicians to distinguish between different knots, ensuring that manipulations like pulling or twisting do not alter the fundamental nature of the knot. Sloman highlights that this concept allows mathematicians to study the inherent properties of knots independent of their specific configuration.

"One of the most basic questions about knots is what happens to invariants when you do this connected sum so many of the more abstractly defined invariants have this property that if you have two knots and you connect sum them then the invariant associated to the connect sum is just the sum so that's very convenient and it makes it easy to compute the invariant for all these more complicated knots because you have to compute it only for the prime knots and then you'll know what it is for every other knot that you might conceive of"

Sloman describes the concept of "connected sum" in knot theory, where two knots are combined. She explains that for many mathematical "invariants" (properties that remain unchanged under deformation), the invariant of the combined knot is simply the sum of the invariants of the individual knots. Sloman notes that this additive property is convenient for mathematicians, as it allows them to calculate invariants for complex knots by first calculating them for simpler "prime knots."

"So the additivity conjecture is or was that the unknotting number is additive under connected sum so the unknotting number of the two knots combined is just the sum of the individual unknotting numbers"

Sloman introduces the "additivity conjecture" in knot theory, which proposed that the "unknotting number" (the minimum number of crossing changes needed to untie a knot) of a connected sum of two knots is equal to the sum of their individual unknotting numbers. This conjecture suggested a predictable relationship between the complexity of individual knots and the complexity of their combined form. Sloman frames this as a long-standing idea that mathematicians believed to be true.

"Last June two researchers from the university of nebraska mark brittenham and susan hermiller posted a paper on the archive announcing that the additivity conjecture is not true this conjecture was around for almost a century and there's been a few results indicating that it might be true but really nothing conclusive and then all of a sudden last year they just resolved the whole thing"

Sloman reports that Mark Brittenham and Susan Hermiller disproved the long-standing additivity conjecture regarding unknotting numbers. She emphasizes that this conjecture had been a significant idea in knot theory for nearly a century, with some evidence suggesting its validity. Sloman highlights the surprising and conclusive nature of their findings, which resolved the conjecture after such a long period of consideration.

"The main counterexample they present in their paper is called the 27 torus knot to tie this knot you just take two pieces of string wind them around each other a few times and then glue the ends this is among the simplest knots so the trefoil might be simpler but there's not much else"

Sloman explains that the primary counterexample used to disprove the additivity conjecture involves a knot called the "2/7 torus knot." She describes this knot as being formed by winding two pieces of string around each other multiple times before joining the ends. Sloman notes that this knot is considered one of the simplest, with only the trefoil knot being potentially simpler, indicating that the counterexample arises from relatively basic knot structures.

"It basically leaves things wide open if the additivity conjecture had turned out to be true people could have figured out the unknotting number of all sorts of knots by using this procedure of calculating it just for the prime knots and then just adding up the results"

Sloman discusses the implications of the additivity conjecture being false for the field of knot theory. She explains that if the conjecture had been true, mathematicians could have easily determined the unknotting numbers of many complex knots by calculating them for simpler "prime knots" and then summing the results. Sloman indicates that the disproven conjecture removes this straightforward method for calculating unknotting numbers across a wide range of knots.