When I was young I read the following puzzle in (I think) the
Mathematical Intelligencer, and I've always been wondering about the
answer.  Consider an idealization of a rope (having diameter 1 (one
unit length), and being incompressible but infinitely flexible: part
of the puzzle was to model this, I think).  If you tie a simple knot in
such a rope it becomes a certain amount shorter.  The puzzle was to
determine the exact length that it shortens and to prove that it is the
correct answer.  So my question: is there anything on the web or in the
literature about this puzzle?  And in particular: how much shorter does
the rope get?

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Request for Question Clarification by mathtalk-ga on 26 Dec 2002 12:42 PST

Hi, wondering-ga:

First, thanks again for chiming in on the machine-readable math
theorems question.  Your comments really helped to round out the
answer.

Now, I'd like to suggest a way of modelling the length of the
"incompressible but infinitely flexible" rope.  Imagine a simple
smooth curve, which represents the "core" of the rope.  Around each
point on the curve, a disk centered at that point which has diameter
1, lying in the plane normal to the curve.  The length of the rope is
the length of the "core" curve, and the constraint is that disks
around two distinct points of the curve may not intersect at points
interior to either disk.

One implication of this is that it puts a lower bound on the radius of
curvature of the "rope" (or more precisely, the core curve).  A circle
of diameter of one could be "wrapped" by disks in the manner
prescribed, but any smaller radius of curvature would induce
"overlap".

So the problem can be formulated as a sort of minimum length problem. 
Consider a straight curve between (0,0,0) and say (10,0,0).  This has
length 10, but no knot.  Now find the shortest curve connecting
(0,0,0) and (10,0,0) which has a "half-hitch" pattern, subject to the
non-intersecting unit diameter disks conditions outlined above, and
subtract 10 from its length.  The difference is "the length of the
knot."

The difficulty lies in coming up with a mathematical representation of
a half-hitch knot.  I can see doing a parametric B-spline to
approximate the curve to any desired degree of accuracy, but this
would only produce a numerical value.

Perhaps there's a more elegant approach to find here.

regards, mathtalk-ga

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Clarification of Question by wondering-ga on 27 Dec 2002 05:27 PST

Dear mathtalk-ga,

I'm happy that you liked my additions to your answer.  I was pleasantly
surprised that you knew Mizar (my favorite proof checking system).  In
theoretical computer science PVS is the big thing, and sometimes I feel
that Mizar there is considered to be a bit fringe.

About the perpendicular disks: this is the way I thought about it too.
An alternative would be to have a unit sphere around every point of the
curve and then to require that once spheres have become disjoint, they
are not allowed to overlap again.  I think I once convinced myself that
for the simple knot those two choices would amount to the same thing...

About putting the curve between 0 and 10: I think that the ropes going
out of the knot might not need to be in the same line.  Therefore I think
one maybe should "let the 10 go to infinity", to get the correct answer.

About your approximation with splines: my expectation is that the shape
of the shortest knot would consist of lines, circle arcs, and maybe helixes.
Something like vinods-ga was describing, but then more subtle (I don't know
exactly how he visualizes his one-and-a-half circle).  I expect that
numerical experimentation would show what those arcs would be.  Of course
that would not be a formal proof that it is the optimal knot, but it would
give a precise number...

So I never did the numerical experimentation.  Maybe I will myself, if this
question doesn't get an answer that points to someone that already has done it.

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Clarification of Question by wondering-ga on 30 Dec 2002 02:55 PST

What I even don't know is what the shortest knot looks like topologically.
Suppose we take the "perpendicular disk" model, and connect two points on
the "inner curve" by a unit line segment when their disks touch.  That way
we get a surface that consists of bands of unit width.  I don't even know
what the topology of that surface for the shortest simple knot is (in the
sense of how the "border lines", "mid-lines" and "end segments" connect as
a graph).

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Request for Question Clarification by mathtalk-ga on 08 Jan 2003 04:56 PST

It seems that a minimal half-hitch knot, while it might be tied in
either a "left-handed" or "right-handed" orientation, will have a 180
degree rotational symmetry, as the rope ends point in opposite
directions.

So it looks to me like the knot can be divided into two identical
pieces, each a sort of "shephard's crook", whose circular ends join
seamlesslly to make the completed half-hitch.

regards, mathtalk-ga

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Clarification of Question by wondering-ga on 09 Jan 2003 03:25 PST

Yes, I also believe that the "solution" will have this symmetry
(although I have *no* idea how
to prove that).  But I don't see how it would help solve the problem:
the two copies of your
"shephard's crook" have to "fit around each other", and I don't see
what that would mean for
their shape...

Hi,

Just wanted to take a crack at this one! A simple knot geometrically
is the sum of one circle and half a circle (that goes through the
first circle and comes out straight). So it should be (circumference
of the circle) 2Pi + (the circumference of the 1/2 circle) Pi = 3Pi
since r = 1 unit. This is the length of the knot.

Maybe right, maybe wrong. However, I'm very curious to know if i am
right! :)

warm regards
vinods-ga (Google Answers Researcher)

Hi, wondering-ga:

Just to follow up on a few of your observations, conjectures, and
questions:

1) I agree that to get the length of the knot, it would in general be
necessary (given how I set up the minimization problem) to pass to the
limit as the endpoints move to infinity.

2) I think the disjoint spheres approach you suggest (to formulate the
incompressible but infinitely flexible rope concept) differs from the
disjoint disks condition, esp. in regard to "local" constraints.  For
example, if I understand the condition, so long as spheres "remain"
overlapping, there is no restriction on "how" they overlap.  Hence a
right angle turn in the rope (or at least one of arbitrarily small
turning radius) would be allowed.

3a) The topological classification of knots holds a bit of a surprise.
 The characterization of "homeomorphism" classes among knots depends
not on the mappings of the rope L (which is always an interval) but
rather on the three dimensional complementary space R^3\L (space
"outside" the rope).  It is this structure which, from a topological
point of view, must be considered to define a knot.

3b) Common sense suggests that the "shortest" knot would be the
simplest, the so-called half-hitch.  Two of these make up a square
knot or a 'granny' knot, depending on the arrangement, and three may
be found in a sailor's hitch.

3c) The "connecting web" surface which you propose may be complex, for
a fully "tightened" knot.  At some points along the length of the
knot, the unit disk contacts may be attained with respect to more than
one "neighbor".  As the disks in contact will typically not be
coplanar (because the rope sections are not expected to be parallel),
it is even conceivable that there might be more than three disks, each
in contact with the other (unlikely, but conceivable; consider for
example four disks inscribed in the faces of a regular tetrahedron).

I suspect that a bit of numerical experimentation is called for at
this point, to develop better insight into the most compact
presentation of the half-hitch.

regards, mathtalk-ga

Dear mathtalk-ga,

Some remarks on your remarks:

Ad 2) I didn't mean to say that the two characterizations are equivalent,
but I do think that they will give the same "tightest half-hitch" (to
use your terminology).  If you pull a rope tight, the right angle turn
won't be there anyway.

Ad 3c) I'm not sure you can get three disks all in contact with each
other if the disks have to be part of a "rope", i.e., if they're locally
the cross-section of non-intersecting cylinders (if the three disks all
touch, it seems the cylinders will "run into each other").  But I agree that
you can't rule out multiply connected disks: for instance, if you wind a
rope around another rope, you'll get this (it will touch the central disk on
many sides).  For all I know the "tightest half-hitch" might contain such
a configuration.  However, in such a situation the surface that I defined
still makes sense, I hope.

I agree that we can endlessly talk about this on this web page, but that
numerical experimentation is probably a better idea.  Only I don't have
a very clear idea what a good implementation strategy for that would be :-)

The statement of the question is wrong. 

I am not a researcher of Google, but an expert on knots in mathematics.

Hi, just4look-ga:

I'd be interested to read any amplification on this remark you care to
make.  As the original poster noted, making a precise mathematical
formulation out of the informal specification of an "incompressible
but infinitely flexible rope" is part of the challenge.  Your brief
comment leaves me wondering whether it is the translation from
informal to formal terms that "is wrong" or if a more fundamental
aspect of the question statement concerns you.

regards, mathtalk-ga

Well, it's just a glimpse rather than a fully worked out solution. But
it seems to me that we can "decompose" the parameters of the rope's
position into a finite number of parameters.  Symmetry helps by
cutting down on the number of parameters.

Consider a stretch of the rope in which it is not constrained by
touching itself along there.  One can specify boundary conditions,
consisting of the unit disks which are the "ends" of that length of
rope, and then ask what minimal path the rope follows between the
specified "endpoints".  I believe at least a semi-analytical solution
can be given.

Starting from a symmetric arrangement in which the half-hitch is so
loosely "drawn" as to have no points of contact, the rope may be
"shortened" by varying the parameters of the endpoint disks, i.e. the
centers and unit normals, until (by symmetric action) a symmetric
finite set of points of contact are created.  New endpoint disks may
then be "inserted" at the corresponding points of contact, suitably
constrained to prevent "self-intersection" of the rope.

This idea shares much with my earlier idea (of approximating the rope
with parametric B-splines), but it incorporates subsections of "rope"
which might be argued to represent portions of an exact solution.

regards, mathtalk-ga

What you're after is called the rope-length of a knot.  This has
recieved a bit of attention recently in mathematics and there is some
information on this although not enough to satisfy your question.

This paper contains everything I know about rope-length of knots:

http://torus.math.uiuc.edu/jms/Papers/thick/ropelen.pdf

Basically what it is saying is this: the amount your rope "shortens"
is
very similar to what they study in this paper, called the
"rope-length"
of a knot.  They aren't exactly the same, but they are very close
ideas. The
difference between the ideas is this: if you take your knot that is
tied in
a string that is attached to two different "walls", and cut the two
loose ends and glue them together (cutting out all the slack in the
rope until the knot is "tight") then the length of this "closed" knot
is the rope-length of the knot.
In other words, the rope-length is the shortest you can make such a
"closed"
knot.

Anyhow, in this paper they prove that rope-length seems to be similar
to a
concept called the "genus" of the knot, which is similar to the
"crossing number" of the knot.  Specifically, they get the lower
bound:

Length of knot >= 2pi*(2+sqrt(2*genus-1))

Here "genus" is the "genus of the knot".  This is a measure of
complexity of a knot.  Specifically, if you want to know what this
number is: every closed
knot is the boundary of a "Seifert surface".  This is a surface that
is orientable, which has a single boundary component, the knot. 
Surfaces have
been classified and they are classified by a number (an integer)
called their
genus.  The genus is always greater than or equal to 0, so we define
the genus of a knot to be the surface bounding the knot of minimal
genus.  This is a difficult number to compute, unfortunately, but
there are some tables of computations I believe in Kawauchi's knot
theory book.

Good luck answering this question as it is quite complicated -- it
turns out the study of knots is much easier from a totally different
perspective.  Knots really belong in a field of mathematics called
"hyperbolic geometry".

If you want to find a reference to the link between genus and crossing
number of a knot probably the best sources would be either Kawauchi's
book or Rolfsen's book. Take care