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Invisible wrote:
> At one time, it was briefly theorised that maybe the ~105 elements of
> the periodic table were each a little tangle of energy, and different
> kinds of tangling gave rise to different chemical properties. This
> sparked a great deal of interest in knot theory. Later this idea was
> abandoned, and knot theory became unpopular again. But some people still
> study it.
Correct:
http://en.wikipedia.org/wiki/History_of_knot_theory
> It is useful to think of "the knot", which is an invariant, unchanging
> thing, and "projections" of the knot.
Wikipedia claims these are "knot diagrams".
> Aside from stretching and shrinking the string, a knot projection can be
> "changed" in three fundamental ways. These ways are named after a
> mathematician who's name is beyond my ability to spell or pronounce.
> Remember we're talking about knot projections, that is, 2D drawings of a
> particular configuration of a knot.
>
> - A type I move involves taking a strand, making a small loop with it,
> and poking it over the top of another strand. (Or, alternatively, under
> it.) This makes the projection slightly more complicated (there are now
> more crossings), but does not alter the knot itself. The reverse
> process, i.e., drawing a loop back over (or under) a strand, simplifies
> the projection. (Note that a loop *around* a strand cannot be
> so-simplified. Only a loop completely over or completely under.)
>
> - A type II move involves taking a straight strand and twisting it to
> form a loop. You have now added one new crossing to the projection.
> Alternatively, untwisting the strand to eliminate a loop, thus reducing
> the number of crossings. (You can only do this if nothing passes through
> the loop, of course.)
>
> - A type III move involves moving a strand from one side of an unrelated
> crossing to the other.
>
> These three moves, then, alter the 2D projection of a knot without
> actually changing its fundamental structure --- i.e., which knot it is.
Wrong:
http://en.wikipedia.org/wiki/Reidemeister_move
> The details of specific invariants escape me now. Suffice it to say that
> several mathematicians have come up with polynomials that can be
> constructed from a knot projection. The polynomial itself doesn't
> calculate anything interesting, but its formula is derived from the knot
> projection. The algebraic properties of polynomials are used in such a
> way that the three moves change the subterms, but when you simplify it
> all into standard polynomial form, you get the same answer.
>
> I remember one of these polynomials involves recursively splitting the
> knot projection into simpler ones, eventually building a trivial
> polynomial for each part, and then combining these polynomials back
> together according to how the original parts where connected. Shift some
> algebra, and at the end every projection of a given knot comes up with
> the exact same polynomial.
Wrong:
http://en.wikipedia.org/wiki/Alexander_polynomial
http://en.wikipedia.org/wiki/Jones_polynomial
http://en.wikipedia.org/wiki/Alexander-Conway_polynomial
http://en.wikipedia.org/wiki/HOMFLY_polynomial
> One rather entertaining way goes something like this. (I've probably
> screwed up the algorithm; this is from memory.)
>
> - Pick a starting point on the string, and draw an arrow representing a
> direction. Doesn't matter what you pick, but stick to it.
>
> - Trace your way around the knot. Each time you reach a crossing, number
> it, starting from 1. If the strand you're on goes over the top, use a
> positive number. If it goes under, assign a negative number.
>
> - Write down a list of all the pairs of numbers at each crossing.
>
> - Throw away the lowest number in each pair (ignoring sign).
>
> It is possible to completely reconstruct the know from the list you're
> left with.
Wrong:
http://en.wikipedia.org/wiki/Dowker_notation
(Note particularly that my algorithm is wrong, and that the notation is
ambiguous in a precise way.)
> An alternative way to describe knots is by "braid theory".
>
> A "braid" is a series of vertical strands. Initially, they are all
> parallel. If you say "+3", that means that strand 3 and strand 4 swap
> places, with strand 3 going over the top of strand 4. Alternatively,
> "-3" means the same swap, but strand 4 going over the top.
>
> In this way, you can say "-3, +5, +2". This describes a sequence of
> strand swaps, starting from the top and working downwards. Something
> like this:
>
> 1 2 3 4 5 6
> | | | | | |
> | | \ / | |
> | | / | |
> | | / \ | |
> | | | | | |
> | | | | \ /
> | | | | \
> | | | | / \
> | | | | | |
> | \ / | | |
> | \ | | |
> | / \ | | |
> | | | | | |
> 1 2 3 4 5 6
>
> So that's a braid. Now if you imagine taking this and bending it over so
> that the ends at the top connect with the ends at the bottom, this would
> make a closed loop. In fact, in this case, the result would be *several*
> closed loops. The 1 strand would be an unknot, not connected to anything
> else. Strands 5 and 6 would become a single strand, which can then be
> untwizzled to make an unknot. And strands 2, 3 and 4 would be connected;
> off the top of my head, I'm not sure if this would be a nontrivial knot.
>
> Such a collection of possibly-connected knots is called a "link". In
> general, "closing" a braid (i.e., connecting its ends together) produces
> a link. Sometimes the whole link consists of one knot (i.e., one
> continuous strand), and sometimes several knots that can be seperated.
> And occasionally, several knots connected such that you can't seperate
> them without cutting.
>
> The fun part, of course, is the algebraic structure of a braid.
> Sometimes when you move a twist up or down the sequence, it changes the
> resulting link when the braid is closed. And sometimes it doesn't.
> Teasing how the relationships for this can get quite interesting.
>
> Best of all: any possible knot or link can be represented as a braid.
> (Although working out how usually isn't easy.)
Wrong:
http://en.wikipedia.org/wiki/Braid_theory
> Another method for constructing knots is to use "tangles".
>
> A "tangle" is a section of string or strings that have 4 ends. The ends
> are locked in place and can't move, and you can't loop the strands over
> those ends. If you imagine drawing a square with one end bolted to each
> corner of the square and the strands inside not allowed to leav the
> confines of the square, that's roughly what a tangle is.
>
> Again we have an algebra of tangle construction here. (I may well be
> getting some lefts/rights mixed up here, but the ideas are essentially
> correct.)
>
> We start with the "0 tangle". This is where the two top corners are
> linked, and the two bottom corners are linked, and the strands aren't
> tangled up in any way.
>
> Then we have the "1 tangle". This is where you take the 0 tangle and
> swap round the two right ends, such that the strand from the bottom-left
> corner passes over the one from the top-left corner. The "-1 tangle" is
> identical, but twisted the opposite way. (I.e., the top-left thread is
> on top.)
>
> You can "add" two tangles together by placing them side by side, and
> connecting the two right-hand ends of the left tangle to the two
> left-hand ends of the right tangle.
>
> If, for example, you add a 0 tangle to a 0 tangle, you get a new 0
> tangle. If you add a 1 tangle to a -1 tangle, you also get... a tangle
> where one thread moves over the other, and then back again. Performing a
> type-I move, this becomes the 0 tangle again.
>
> So, 0 + 0 = 0 and (+1) + (-1) = 0. That's cute. But if you add a 1
> tangle to a 2 tangle, you get a tangle where the two threads cross over
> each other twice in the same direction - the "2 tangle". (A "-2 tangle"
> is defined similarly, but with the twist in the opposite direction.)
>
> So, an N tangle is the 0 tangle twisted N times clockwise, and a -N
> tangle is twisted N times anticlockwise. (Assuming you look at it from
> the right direction.)
>
> There is also an "infinity tangle", which is like the 0 tangle, rotated
> This involves placing one above the over, and joining the corners in
> that direction instead.
>
> Here, however, we find that tangle algebra doesn't work *quite* like
> number arithmetic; if you multiply the 0 tangle by the 0 tangle, you get
> something that isn't even a tangle; it's like a 0 tangle with a trivial
> knot floating in the middle of it. And a 1 tangle multiplied by a 1
> tangle gives you something that can't be described any other way. (In
> particular, *not* a 1 tangle!)
>
> By pairing up the ends of a tangle (either vertically or horizontally -
> it makes a difference) you can again construct any possible knot. And
> again there's an interesting algebra of operations which are equivilent
> and those which are not.
Wrong:
http://en.wikipedia.org/wiki/Tangle_theory
> In a similar way, you can "add" regular knots. You take two knots, cut
> them both, and join the cut parts. The thing is, depending on exactly
> where you cut them, and how you join the ends up, you can make several
> different knots in any addition operation. So for general knots,
> "adding" isn't very precisely defined. (At least, if you just specify
> two knots and that they be added, the result is not well-defined. You
> need to specify lots of extra info to make it well-defined. Even the
> knot projection might make a difference.)
Wrong:
http://en.wikipedia.org/wiki/Knot_sum
(There are only two possible results from a knot sum, and it doesn't
matter where the join is, only the relative orientation of the two knots.)
> There are other ways of making knots too. The trefoil is a "toriodal
> knot". That is, you can generate it by marking a point on a circle, and
> rotating that circle while sweeping it around a perpendicular circle.
> (In other words, tracing a path on the surface of a torus.) By varying
> the number of rotations of one circle for each rotation of the other,
> you can build various different knots, of which the trefoil is just one.
> (There is an infinite set of parameters that generate any given toriodal
> knot, however.)
Correct:
http://en.wikipedia.org/wiki/Torus_knot
> Related to this are "cable knots". This involves sweeping a circle not
> along a circle but along another knot. Sometimes the result can be
> unravelled to be isomorphic to the original knot; sometimes it can't.
> (In other words, sometimes it's a genuinely new knot.) You can also
> generate links this way by not rotating the circle as it is swept;
> sometimes these links are seperable, and sometimes they aren't.
Wrong:
http://en.wikipedia.org/wiki/Cable_knot
http://en.wikipedia.org/wiki/Satellite_knot
> What about knots in 4D instead of just 3D? What would that be like?
>
> Well, it turns out to be pretty boring, actually. In 4D, and knot
> composed of a 1D strand can actually be completely untangled. That is,
> every knot is equivilent to the trivial knot in 4D. That's not terribly
> interesting.
>
> What you *can* do, however, is construct knots out of a 2D "ribbon"
> rather than a 1D "string". The result is a family of knots that only
> exist in 4D, but it's really *far* too mind-bending to think about.
> (Projecting back into 3D can look pretty though...)
Correct:
http://en.wikipedia.org/wiki/Knot_theory#Higher_dimensions
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