{"id":1704,"date":"2025-11-09T07:03:43","date_gmt":"2025-11-09T02:03:43","guid":{"rendered":"https:\/\/paknews.centers.pk\/unpicking-how-to-measure-the-complexity-of-knots\/"},"modified":"2025-11-09T07:03:43","modified_gmt":"2025-11-09T02:03:43","slug":"unpicking-how-to-measure-the-complexity-of-knots","status":"publish","type":"post","link":"https:\/\/paknews.centers.pk\/ur\/unpicking-how-to-measure-the-complexity-of-knots\/","title":{"rendered":"Unpicking How to Measure the Complexity of Knots"},"content":{"rendered":"


\n<\/p>\n

\n

The duo kept their program running in the background for over a decade. During that time, a couple of computers from their ragtag collection succumbed to overheating and even flames. \u201cThere was one that actually sent out sparks,\u201d Brittenham said. \u201cThat was kind of fun.\u201d (Those machines, he added, were \u201chonorably retired.\u201d)<\/p>\n

Then, in the fall of 2024, a paper about a failed attempt to use machine learning<\/a> to disprove the additivity conjecture caught Brittenham and Hermiller\u2019s attention. Perhaps, they thought, machine learning wasn\u2019t the best approach for this particular problem: If a counterexample to the additivity conjecture was out there, it would be \u201ca needle in a haystack,\u201d Hermiller said. \u201cThat\u2019s not quite what things like machine learning are about. They\u2019re about trying to find patterns in things.\u201d<\/p>\n

But it reinforced a suspicion the pair already had\u2014that maybe their more carefully honed sneakernet could find the needle.<\/p>\n

The Tie That Binds<\/h2>\n

Brittenham and Hermiller realized they could make use of the unknotting sequences they\u2019d uncovered to look for potential counterexamples to the additivity conjecture.<\/p>\n

Imagine again that you have two knots whose unknotting numbers are 2 and 3, and you\u2019re trying to unknot their connect sum. After one crossing change, you get a new knot. If the additivity conjecture is to be believed, then the original knot\u2019s unknotting number should be 5, and this new knot\u2019s should be 4.<\/p>\n

But what if this new knot\u2019s unknotting number is already known to be 3? That implies that the original knot can be untied in just four steps, breaking the conjecture.<\/p>\n

\u201cWe get these middle knots,\u201d Brittenham said. \u201cWhat can we learn from them?\u201d<\/p>\n

He and Hermiller already had the perfect tool for the occasion humming away on their suite of laptops: the database they\u2019d spent the previous decade developing, with its upper bounds on the unknotting numbers of thousands of knots.<\/p>\n