knot theorists needed... for unknotting stuff
John Steinberger
I have recently stumbled into an unknotting algorithm of a very simple
diagrammatic kind, but I have been unable to prove it will always
unknot a projection of the unknot. It would be really exciting though
if it did because the algorithm runs in O(n^3) (n = the number of
crossings in the projection) whereas the algorithms developped so far
for unknotting run in exponential times that are not real-life
implementable. Also I should mention that my algorithm has happily
weathered the supposedly nasty unknot projections that discouraged
people in the past from using diagrammatic approaches (some of these
can be seen at http://f2.org/maths/kt/unknoteq.html).
The algorithm runs like this: define a (p,q) move on a knot projection
to be a move taking a p-crossing over(or under-)strand to a position
where it has q crossings (still all over or under, accordingly). For
instance a -I Reidemeister move (which removes a single loop) is a
(1,0) move, acting on the overstrand of a single crossing. My
conjecture is that a non-trivial unknot projection always contains a
(p,q) move with p>q (which we denote by a '-(p,q)' move). And my
discovery is that when present in a diagram, these moves can be
located very quickly--in O(n^2) or less.
I've been thinking about it for a while but I cannot either prove or
disprove the algorithm--even in much simpler cases (such as requiring
the unknot's Seifert surface be a flattish, untwisted pancake) I have
not been able to prove a -(p,q) move is always present. Which is
either discouraging for the conjecture or just means that I have
watched one too many Pepsi commercials.
Any inspiration and insight would be greatly appreciated!
Johnny
John Conway
I made this conjecture a million years ago, along with what I
regarded as the obvious generalization, to an algorithm for finding
all the projections of an arbitrary knot that have minimal crossing
number. Let me describe the latter. My terms are (p,q)-PASS for
this move:
--|----##### -------*****
--|----#####------- to -------*****---|---
--|----***** -------*****
where there are p horizontal strings on the left, and q on the right,
(this one is a (3,1)-OVERPASS - there's also a (p,q)-UNDERPASS).
There's also a (p,q)-TWIST, which takes you from
__ __ ______ ______ __ __
\ / ##### ##### \ /
\ ##### to ##### \
__/ \__#####______ ______#####__/ \__
(this one's a (2,2)-TWIST - in general there'd be p strings on the
left, with a single twist before the move and no twist after, and q on
the right, with a single twist after and none before).
The conjecture was originally that if you start from any projection
of the knot (in which term I include links) and apply all possible
(p,q)-passes and (p,q)-twists for which p >= q, then you'll get all
minimal projections.
My evidence for this conjecture was pretty overwhelming, since I
used them to prepare my knot tables. However, the above conjecture is
false. I disproved it first by finding a new move - the WRISTWATCH
move, which was needed to related two diagrams for a certain 10-crossing
link, and which I then realised was even needed for unlinks! I wasn't
too worried by this, since it seemed that the conjecture could be cured
by adding these wristwatch moves.
Let me briefly describe them. Imagine an arm wearing a wristwatch:
/\ /\ /\
/ / /
#######
-------#######-------
-------#######-------
-------#######-------
#######
\ \ \
\/ \/ \/
whose armband consists of a number of strings going around the arm,
while there are also p strings going along the surface of the arm to
the left of the watch, and q similar strings to the right.
Now suppose one's wearing two such watches, one "outside" the
other, and initially to the left of it. Then the wristwatch move
slides this watch rightawrds along the arm, over the other one.
The modified conjecture stated that all minimal projections were
obtainable from an given projection by applying just those passes,
twists and wristwatch moves that never increased the crossing number.
Of course it was slight less credible than the previous one had been,
because it wasn't at all obvious that there wouldn't be more "new"
moves like the wristwatch one, so when I wrote my paper I refrained
from publicly making this conjecture. There's a sort of sotto voce
reference to it, though in a sentence when I say that a certain
algorithm had been applied to all the knots in the table, as evidence
that forms I said were distinct reall were.
However, that conjecture was wrong. There were two 10-crossing
knots in the table, which had been thought to be distinct ever since
Tait first enumerated knots in the 19th century, and which I listed as
distinct since the above algorithm didn't distinguish them, but which
were later found by Perko to be the same. They are related by yet
another "new" move studied by him, to which I gave the name
'Perkolation". I won't attempt to describe it here. A few other
moves were found later, that seemed to be independent of the passes,
twists, watch-moves and Perkolation, and I know longer believe that
there's any algorithm of this kind for finding all minimal
presentations - in other words, I think there will be an infinity
of new and essentially different such moves.
Strictly speaking, this says nothing about the unknot version of
the conjecture. However, my (very well-informed) guess is that will
be false too. I know some intriguing unlinks that disprove certain
natural generalizations, and my guess is that there will be some
unknots in the 20-30 crossing range that disprove this one. But
they're very hard to find!
John Conwa
John Steinberger
I had problems completely picturing the WRISTWATCH move--could you
tell me what 10-crossing link it was? (I think Adams' book lists the
10-crossing links, though I'm not sure.)
What are the "intriguing unlinks that disprove certain
natural generalizations"? I'm very curious--about the generalizations
too. If it's inconvenient to describe the links maybe you can post
some GIFs or JPEGs. I actually don't know of any simple knot drawing
software just good for drawing nice-looking 2D-sketches. Though I
might convince my friend Fred Curtis to work on that, so that we can
all post our knot diagrams easily in HTML tags instead of "-/\|"
stuff.
I'm sure the counterexamples, if they exist, are hard to find. Can I
ask what makes you guess 20-30 crossings? Just a gut feeling or what?
I keep on being frustrated by my inability to prove the conjecture for
the (seemingly obvious) case in which only one side of the Seifert
surface is facing the viewer, so that the Seifert surface basically
looks like one big multilayered, multi-tongued pancake. It seems so
obviously true that a -(p,q) move would be present yet I have not been
able to pin it down formally. If anyone could shed a light I would be
*eternally* thankful.
Thanks all!
John