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()?
drakythe•1h ago
I don't think it implies that one of the knots has a negative number. It proves that when you add knots together you can't just add together their unknotting numbers and expect to get a correct answer. The article mentions "unpredictability of the crossing change" as a source of this issue (if I am reading that statement correctly).
Basically the unknotting number combes from how the string crosses itself and when you add two (or more?) knots together you can't guarantee that the crossings will remain the same, which makes a kind of intuitive sense but is extremely frustrating when there isn't a solid mathematical formula that can account for that.
taeric•1h ago
I think you are reaching more for imaginary numbers?
I could see trying to fit this with surreal numbers, as well. Would be fitting, as I think Conway was big into knots?
Regardless, no, not dumb. Numerically modelling things is hard, it turns out. :D
ashivkum•45m ago
That would certainly be interesting, though I don't know of any matching definition in knot theory. There is a notion of "positive" and "negative" crossings, so you could define the positive and negative unknotting numbers by asking how many of each you have to swap. Unfortunately, in their example, all torus knots can be drawn with all positive or all negative crossings.
rep_movsd•38m ago
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"
AlotOfReading•26m ago
And yet it almost always works. There were no known counterexamples where it failed until this was published.
jlarocco•15m ago
I'm with the OP on this one. Intuitively (to me, anyway) I wouldn't expect it to work in general.
I'm surprised it took so long to find a counterexample, but it doesn't surprise me at all to hear it doesn't work.
exabrial•37m ago
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!
rkomorn•31m ago
It's not often I come across a comment that I wouldn't understand any less if it was in a language I don't speak but I think this one makes the cut.
Holy topic-specific terminology, Batman!
Etheryte•26m ago
Related: "How I, a non-developer, read the tutorial you, a developer, wrote for me, a beginner" [0].
> All I know is a [knot used to join a length of rope into a circle/loop, performed three times) is nearly impossible to untie [when tied with large-diameter polyethylene plastic rope] after [falling (while climbing) and being caught by the loop I made to take load] made out of it. It's sort of comforting have the rock hard knot; it'll break the [loop itself, structurally] before untying. Interestingly, [if you don't tighten the knot by dropping bodyweight from a height on it like I did, they're] pretty simple to untie!
gilleain•12m ago
All we need is a fisherman-toplogist, and we can perfect the incomprehensibility of the discussion on particular knots.
NoMoreNicksLeft•1h ago
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()?
drakythe•1h ago
Basically the unknotting number combes from how the string crosses itself and when you add two (or more?) knots together you can't guarantee that the crossings will remain the same, which makes a kind of intuitive sense but is extremely frustrating when there isn't a solid mathematical formula that can account for that.
taeric•1h ago
I could see trying to fit this with surreal numbers, as well. Would be fitting, as I think Conway was big into knots?
Regardless, no, not dumb. Numerically modelling things is hard, it turns out. :D
ashivkum•45m ago