Complex knots can actually be easier to untie than simple ones

Why untangle two small knots to disagge more difficult than untangle it? Surprisingly, mathematicians have discovered that larger and apparently more complex nodes created by joining two simplest together can sometimes be easier to undo, invalidating a conjecture posed almost 90 years ago.

“We were looking for a counterexample without really expecting to find one, because this conjecture has existed for so long,” said Mark Brittenham at the University of Nebraska in Lincoln. “At the back of our heads, we thought that the conjecture was probably true. It was very unexpected and very surprising.

Mathematicians like Brittenham Student forms by treating them like tangled curls with jotted ends. One of the most important concepts of the node theory is that each node has a number of non -Node, which is the number of times you have to break the chain, move another piece of the loop through space, then join the ends before reaching a circle without no passages at all – known as “UNKNOT”.

Calculation of disative numbers can be a very intensive task in calculation, and there are still nodes with as little as 10 crossings which have no solution. For this reason, it may be useful to decompose nodes into two simpler nodes or more to analyze them, with those who cannot be divided better under the name of first -rate nodes, similar to prime numbers.

But a long-standing mystery is whether the number of nones of the two nodes granted would give you the number of non-node of the larger knot. Intuitively, it might be logical that a combined node is at least as difficult to undo as the sum of its constituent parts, and in 1937, it was conjectured that the cancellation of the combined node could never be easier.

Now, Brittenham and Susan Hermiller, also at the University of Nebraska in Lincoln, have shown that there are cases where it is not true. “The conjecture has existed for 88 years and as people continue to find nothing wrong, people hope more true,” explains Hermiller. “First of all, we found one, then quickly, we found many pairs of nodes infinitely for which the connected sum had numbers of unsubscribe which were strictly lower than the sum of the number of non-laws of the two pieces.”

“We have shown that we do not understand as well the figures to unsubscribe almost well as we had done,” explains Brittenham. “There could be – even for nodes which are not connected sums – more effective means than we have never imagined to undress them. Our hope is that it really opened a new door so that researchers are starting to explore.”

Although the research and verification of counter-examples involved a combination of existing knowledge, intuition and computing power, the last step in the verification of the proof was carried out in a resolutely simpler and practical way: the node with a piece of rope and disentangle it physically to show that the expected number of derocreation of researchers was correct.

Andras Juhasz at the University of Oxford, who previously worked with the company of IA Deepmind to prove a different conjecture of the theory of nodes, says that he and society had tried without success to withdraw this last problem on the additive sets in the same way, but without luck.

“We spent at least a year or two trying to find a counterexample and without success, so we abandoned,” said Juhasz. “It is possible that to find counter-examples which are like a needle in a hay boot, the AI may not be the best tool. It was a counterexample difficult to find, I think, because we looked for hard enough.”

Although there are many practical applications for the theory of nodes, from cryptography to molecular biology, Nicholas Jackson at the University of Warwick, in the United Kingdom, hesitates to suggest that this new result can be used. “I suppose that we now understand a little more about the functioning of the three-dimensional circles than before,” he says. “One thing we didn’t really understand a few months ago is now a little better understood.”