[Guest]

This is as much a cry for help as a puzzle, but it is a puzzle of sorts, I just don't have the answer (though I'm pretty sure there must be one), so I'll stick it here and see if a moderator decides otherwise.

I've been having a browse around t'internet to try to find an elegant proof of why antiknots don't exist. No success, just a few mentions of some that are, alas, not given. I bet some braindenner knows, though.

For those who don't know, an antiknot is easy enough to envisage. Imagine a simple knot tied in a piece of string. With a little manipulation it can be moved up and down the string. An antiknot is another knot which, if you move the two together, will cancel out the first knot and produce a straight piece of string (if you think of the string as a closed loop it's a bit more sound since the knots can't escape off the end and disappear that way, though I guess an infinitely long string would also be fine). Experience tells us that there is no such thing as an antiknot. But can you come up with a reasonably easy to understand proof of this?

[Guest]

but it seems to me that as a knot was moving down a string, it would deal with any other knot as just another point on the string (albeit a more complicated point, but still just a point nonetheless). IF the two knots could interact at more than one point simultaneously, then I think it could be possible to have them undo each other, maybe, although I haven't found one yet in limited attempts. But I believe that's beyond the scope of the original question.

[bonanova]

This is as much a cry for help as a puzzle, but it is a puzzle of sorts, I just don't have the answer (though I'm pretty sure there must be one), so I'll stick it here and see if a moderator decides otherwise.

I've been having a browse around t'internet to try to find an elegant proof of why antiknots don't exist. No success, just a few mentions of some that are, alas, not given. I bet some braindenner knows, though.

For those who don't know, an antiknot is easy enough to envisage. Imagine a simple knot tied in a piece of string. With a little manipulation it can be moved up and down the string. An antiknot is another knot which, if you move the two together, will cancel out the first knot and produce a straight piece of string (if you think of the string as a closed loop it's a bit more sound since the knots can't escape off the end and disappear that way, though I guess an infinitely long string would also be fine). Experience tells us that there is no such thing as an antiknot. But can you come up with a reasonably easy to understand proof of this?

Start with the ends of the string fastened to opposite walls of a room. Or a loop, but I like the fastened ends idea.

Knots in the rope are of two types. Knots of one type can be made starting from a straight string without use of an end - for example pulling a loop of string through your fingers and tying a knot in the resulting double strand. Clearly this type of knot can also be removed from the string, without use of an end, by reversing the motions used to "tie" it. The other type of knot requires the use of an end of the string; knots of this type are impossible to tie in our case because the ends are not available.

The first type of knot could be argued to possess an anti knot, as two of them could be brought together and removed. But that's specious - they could be removed individually without bringing them together. So let's turn to the type of knot that needs an end to create. If two such knots - one of them being the anti-knot to the other - could annihilate each other, without involving an end, then by reversing the annihilating motions they could be re-created without using an end. But by definition the only type of knot that can be created without use of an end is the first type, and this contradicts the condition that we are only considering knots of the second type.

That's the best I can do off the top of my head.

[Guest]

Start with the ends of the string fastened to opposite walls of a room. Or a loop, but I like the fastened ends idea.

Knots in the rope are of two types. Knots of one type can be made starting from a straight string without use of an end - for example pulling a loop of string through your fingers and tying a knot in the resulting double strand. Clearly this type of knot can also be removed from the string, without use of an end, by reversing the motions used to "tie" it. The other type of knot requires the use of an end of the string; knots of this type are impossible to tie in our case because the ends are not available.

The first type of knot could be argued to possess an anti knot, as two of them could be brought together and removed. But that's specious - they could be removed individually without bringing them together. So let's turn to the type of knot that needs an end to create. If two such knots - one of them being the anti-knot to the other - could annihilate each other, without involving an end, then by reversing the annihilating motions they could be re-created without using an end. But by definition the only type of knot that can be created without use of an end is the first type, and this contradicts the condition that we are only considering knots of the second type.

That's the best I can do off the top of my head.

Not bad, but suppose you could create a knot of the first type and split it in two such that each half could not be removed without reconnecting it to the other? Then you would have a knot/antiknot pair. If you could prove that was impossible you'd be there...

[Guest]

This is as much a cry for help as a puzzle, but it is a puzzle of sorts, I just don't have the answer (though I'm pretty sure there must be one), so I'll stick it here and see if a moderator decides otherwise.

I've been having a browse around t'internet to try to find an elegant proof of why antiknots don't exist. No success, just a few mentions of some that are, alas, not given. I bet some braindenner knows, though.

I saw a "proof" that went something like this.

Suppose the antiknot exists, and consider an infinite sequence of knot/antiknot pairs +k -k +k -k +k -k +k -k ... = 0.

Now, you could equivalently say +k + (-k +k) + (-k +k) + (-k +k) + (-k +k) ... = k. So, the only knot with an antiknot is the "un"knot k=0.

However, you can use this same logic to prove 1=0, so I can't say that I'm really satisfied by it.

I much prefer Bonanova's notion that you can't undo a knot without using an end. I'm not sure how that disproves the existence of an annihilator though...

[bonanova]

Not bad, but suppose you could create a knot of the first type and split it in two such that each half could not be removed without reconnecting it to the other? Then you would have a knot/antiknot pair. If you could prove that was impossible you'd be there...

Let's try. Knots are of the first type [does not need an end] or the second type [needs an end]

[1] You have a knot of the first type.

[2] You split the knot into two knots of the second type.

[3] You reverse the motions in step [2] to produce a knot of the first type, which is then removed.

Step [2] is impossible by definition. Knots of the second type need an end to create.

[Guest]

Let's try. Knots are of the first type [does not need an end] or the second type [needs an end]

[1] You have a knot of the first type.

[2] You split the knot into two knots of the second type.

[3] You reverse the motions in step [2] to produce a knot of the first type, which is then removed.

Step [2] is impossible by definition. Knots of the second type need an end to create.

Ahh... but there's still the assumption that you need an end to create a knot of the second type and that it cannot also be created as part of a knot/antiknot pair.

[soop]

I seem to have just done it with my phone charger. It's obviously not looped or anything.

I started with a loop, and tied a knot like you would when tying your shoe. Then I made another, pulling the first knot through the loop of the second. Then bit by bit, I pulled the second knot through and around the first, and that undid, leaving me with one loop. Then by reversing, you can pull the knot off the end of the loop.

I'm not entirely sure if the first didn't just undo as a secondary process of me moving the knot around (in that you can do it without disturbing the first), but I'll let someone else figure that out.

*edit* nah, just retried, and it didn't work

Edited October 7, 2008 by soop

[Guest]

I saw a "proof" that went something like this.

Suppose the antiknot exists, and consider an infinite sequence of knot/antiknot pairs +k -k +k -k +k -k +k -k ... = 0.

Now, you could equivalently say +k + (-k +k) + (-k +k) + (-k +k) + (-k +k) ... = k. So, the only knot with an antiknot is the "un"knot k=0.

However, you can use this same logic to prove 1=0, so I can't say that I'm really satisfied by it.

I much prefer Bonanova's notion that you can't undo a knot without using an end. I'm not sure how that disproves the existence of an annihilator though...

Apparently... while using this sort of argument to prove that 1=0 is invalid, in the field of knots (and some other fields) it is a valid argument - one known as the Mazur Swindle.

[Guest]

My comments may seem to be pointlessly finicky, so by way of illustration here's something to compare my knotty problem with:

Consider a flexible rod with fixed ends. You cannot add a twist to the rod without turning one end. Without turning either end all you can do is add a localised twist (as in the middle picture).

If we apply bonanova's logic to this, a localised twist cannot be split into two proper twists, since it did not involve turning either end.

But clearly you can split the localised twist into a twist and "anti-twist" (which in this case is merely a twist in the opposite direction).

So if the ends are fixed, you cannot create either a twist or an anti-twist on their own, but you can create the two together.

It remains to be shown that the same does not apply to knots.

[Guest]

Apparently... while using this sort of argument to prove that 1=0 is invalid, in the field of knots (and some other fields) it is a valid argument - one known as the Mazur Swindle.

While this may be correct, I have great difficulty in seeing why it should be. The idea that knots can be made progressively smaller in order to make an infinite sum of them convergent seems highly dodgy to me. The twist example above, while seeming pretty trivial, gives perhaps a useful test of any proof of the non-existence of antiknots. If the proof could be equally well applied to twists, there's obviously something wrong with it, since anti-twists clearly do exist. Can anyone tell me why infinite sums of knots make more sense than infinite sums of twists?

[Guest]

While this may be correct, I have great difficulty in seeing why it should be. The idea that knots can be made progressively smaller in order to make an infinite sum of them convergent seems highly dodgy to me. The twist example above, while seeming pretty trivial, gives perhaps a useful test of any proof of the non-existence of antiknots. If the proof could be equally well applied to twists, there's obviously something wrong with it, since anti-twists clearly do exist. Can anyone tell me why infinite sums of knots make more sense than infinite sums of twists?

Dunno. I suspect you'd need a proper topologist to answer that question. But the world of twists is a much simpler one than the world of knots - there is only one type of twist (and its inverse) but lots (an infinite amount?) of unique 'prime' knots.

The progressively smaller knot argument does sound a bit dodgy... probably an over-simplification of some deeper theorem.

[Guest]

And a few more moments thought makes me think that twists don't exist... I think you could just change your frame of reference to show that a twist is isomorphic to an untwisted rope.

That may, of course, be nonesense.

[Guest]

If your string was a closed loop a twist would certainly exist I think. A string with a twist and anti-twist would of course be isomorphic to one with no twist, though that's kind of the point I think, also with the knots.

As far as the bewildering variety of knots is concerned, we could just look at the simplest sort:

If we could prove there was no anti-knot to that I for one would be just as happy

[Guest]

As far as the bewildering variety of knots is concerned, we could just look at the simplest sort:

If we could prove there was no anti-knot to that I for one would be just as happy

While that might be a knot to a sailor, it is not a knot to a mathematician. Yarrr. That be a braid matey.

[Guest]

While that might be a knot to a sailor, it is not a knot to a mathematician. Yarrr. That be a braid matey.

Ahoy there! Never trust a scurvy landlubber who can't tell his sheet from a rope and his head from a toilet say I. Yo ho ho.