I borrowed A Survey of Knot Theory (by Akio Kawauchi) and Knot
Theory & its Applications (by Kunio Murasugi) from the local college
library this weekend, and learned a lot. I had hoped to present a
simple explanation of various knot polynomials, but I'm not sure that's
possible. Bridge representations are fairly easy to explain, though.
Each knot or link has a minimal bridge number. A bridge
is a section that goes over other sections. The above representation
of the Borromean Rings has three bridges. Many knots require
only two bridges, though. In 1949, mathematician H. Schubert presented
what is now known as Schubert's normal form. The following
knot is S(19,5).
Again, the above knot is S(19,5). The 19 comes from the length of
each bridge. I started with a line, then divided it so that I could
number the ends with 0 and 19. I also numbered the undercrossings,
first 1-18 and then 20-37. Then I copied this bridge elsewhere, upside
down. That explains the 19 in S(19, 5). What about the 5?
Well, start with 0, and start travelling. Note the first number you
meet after passing under a bridge. You'll find the sequence of numbers
is 0, 5, 10, 15, 20, 25, 30, 35, 2, 7, 12, 17, 22, 27, 32, 37, 4,
9, 14, 19. At every bridge crossing, your number will increase
by 5, modulo 38 (A clock face can be considered modulo 12.
5:00 + 13 hours = 6:00). The bridges always alternate. Drawing
S(89,34) would need two bridges of length 89, and a winding path that aimed
at 0, 34, 68, 102, 136, 170, 26, 60, etc. Here is what S(19,5) looks
like with a minimal crossings diagram:
A fantastic diagram of many simple knots is available here
from the excellent KnotPlot
site. The following is my picture of Knot 8_{5} (this
is the catalog number). This knot requires three bridges. Is
there a standardized way to represent 3-bridge knots? This is an
unsolved problem in mathematics.
The Schubert normal form for 2-bridge knots by their Catalog Number
4_{1} = S(5,2)
5_{2} = S(7,3)
6_{1} = S(9,4)
6_{2} = S(11,4)
6_{3} = S(13,5)
7_{2} = S(11,5)
7_{3} = S(13,4)
7_{4} = S(15,4)
7_{5} = S(17,7)
7_{6} = S(19,7)
7_{7} = S(21,8)
8_{1} = S(13,6)
8_{2} = S(17,6)
8_{3} = S(17,4)
8_{4} = S(19,5)
8_{6} = S(23,10)
8_{7} = S(23,9)
8_{8} = S(25,9)
8_{9} = S(25,7)
8_{11} = S(27,10)
8_{12} = S(29,12)
8_{13} = S(29,11)
8_{14} = S(31,12)
9_{2} = S(15,7)
9_{3} = S(19,6)
9_{4} = S(21,5)
9_{5} = S(23,6)
9_{6} = S(27,5)
9_{7} = S(29,13)
9_{8} = S(31,11)
9_{9} = S(31,9)
9_{10} = S(33,10)
9_{11} = S(33,14)
9_{12} = S(35,13)
9_{13} = S(37,10)
9_{14} = S(37,14)
9_{15} = S(39,16)
9_{17} = S(39,14)
9_{18} = S(41,17)
9_{19} = S(41,16)
9_{20} = S(41,15)
9_{21} = S(43,18)
9_{23} = S(45,19)
9_{26} = S(47,18)
9_{27} = S(49,19)
9_{31} = S(55,21)
Pictures of all of these knots can be seen in the CRC
Encyclopedia of Mathematics under the knot
listing. Two knots not listed above are S(3,2) and S(3,1).
I show pictures of them below. Can you figure out what these knots
are? Send answer.
From these, you can see that S(a,b) = S(c,d) is possible. What are
the rules for (a,b,c,d)? What is the minimal crossing number for
S(2003, 541)? Is a smaller bridge possible for S(2003, 541)?
It would be interesting for a computer to crank through all 2-bridge knots
with bridges of length 1000 or less, calculate the Alexander, Jones, and
HOMFLY polynomials for each, and sort through them for equivalencies.
Why so many polynomials? The following two knots have identical Jones
polynomials (-t^{-3} + 2t^{-2} - 3t^{-1} + 5 -
4t + 4t^{2} - 3t^{3} + 2t^{4} - t^{5})
and identical Alexander polynomials (2t^{-2} - 6t^{-1}
+ 9 - 6t + 2t^{2}). They are different knots, though.
For more about knots, try searching on 'Alexander Polynomial'. Or
visit any of the links I've mentioned.