Maybe I'm dumb, but they have two knots that have a number of 3, one is the mirror of the other. They were hoping that it would add up to six, but it only adds to 5.
Wouldn't this mean that there is a sort of "negative" number implied here? That one knot is +2/+1 and that the other knot is +2/-1, and that their measure (the unknotting number) is only the sum of the abs()?
Maybe I'm really dumb, but it should be obvious that replacing a section of rope in one knot with another, is intuitively not going to simply "add the unknotting numbers"
All I know is a triple fisherman's is nearly impossible to untie in 5.6mm UHWMPE after taking a whip on a sling made out of it. It's sort of comforting have the rock hard knot; it'll break the cordelette before untying. Interestingly, an unweighted one is pretty simple to untie!
A lovely thing in math is that a counterexample, especially if it leads to infinitely more counterexamples in a particular class, can teach you more about the problem. I find this article hopeful. It made me excited about knot theory for the first time in a while.
Speaking of knots, and not to hijack this post: I am interested in learning, say, 10 most useful knots that could be useful in most situations: joining two ropes, attaching a rope to a tree branch, etc. etc. Is there a youtube channel people would recommend I watch to pick them up?
Every Quanta story posted here seems to be 'simple math thing is unexpectedly difficult' or 'elegant solution to difficult math problem is unexpectedly simple.'
> The duo kept their program running in the background for over a decade
Were they not searching for counter examples to the sum conjecture during this time? Or how did they program not identify this simple 3+3 example sooner?
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[ 3.0 ms ] story [ 35.9 ms ] threadWouldn't this mean that there is a sort of "negative" number implied here? That one knot is +2/+1 and that the other knot is +2/-1, and that their measure (the unknotting number) is only the sum of the abs()?
Specifically tying a knot with opposite chirality to one existing on a line can cause both knots to capsize and roll out.
One would not take it as given that three knots plus three knots would yield six knots in this scenario.
Were they not searching for counter examples to the sum conjecture during this time? Or how did they program not identify this simple 3+3 example sooner?