Knot knock

August 5, 2008 at 8:16 am | Posted in Math Learning | Leave a comment

Knot table to Eight crossing from Lickorish,1997

Do you know how hard to draw the knot picture?!…
You must try it!!
Look at the eight crossings knot table…

  • First, You should know how to tie a knot, thus you can know the upper segment and lower part!
  • Second, Upper segment is a real black line, and the lower part must anticipate an empty.
    ※You should keep all basic knot form deeply in your brain.
  • Third, Draw it with your pen. 3-dimension memory and 2-dimension drawing. Keep practice, practice and practice or you cannot draw it as well.(Actually speaking I still cannot do it well now)
  • Fourth, Reverse what you draw…to show the different positive picture and negative picture like.

This is What I did before…A never succeed work><

After going to Taipei, I open on the book: W.B. Raymond Lickorish. An Introduction to Knot Theory, GTM;175. Springer-Verlag New York(1997).To turn on my memory of Knot…It was Chun Chung tell me the good one. Seems a hard reading book for a undergraduate guy…I always confuse that Why I can easily understand what knot is after Chun Chung easily explained it, but cannot understand what the book show us. Alas~O if there I can take part in a course of he to tell me what the knot is…Not just a book.

Alwxander Polynomial Table from Lickrish,1997,p.59

Take something for example, look at the definition1.1 to 1.4, and compare with knot table to Eight Crossing(table one). To make the definition clearly had been a hard work for me…

Definition1.1: A link L of m components is a subset of S3 or of R3, that consists of m disjoint, piecewise linear, simple closed curves. A link of one component is a Knot.

Definition1.2: Links L1 and L2 in S3 are equivalent if there is an orientation-preserving piecewise linear homeomorphism H: S3 → S3 such that h(L1)=(L2)

Note Definition: A knot is said to be the unknot if it bounds an embedded piecewise linear discs in S3.

Definition1.3: A knot K is a prime knot if it is not the unknot, and K=K1+K2 implies that K1 or K2 is the unknot.

Definition1.4 Suppose that L is a two-component oriented link with components L1 and L2. The linking number lk(L,L) of L1 and L2 is half the sum of the sign, in a diagram for L, of the crossings at which one strand is from L1 and the other is from L2

My learning problems always stubs my brain..

Here: Who knows the “link”?! In mathworld shows that ” Formally, a link is one or more disjointly embedded circles in three space. More informally, a link is an assembly of knots with mutual entaglements”( The definitions of knot and link are much easier on the page with picture( course, I understand it now, but it quite hard for me to state the definition. Cause I found at least 3 definition of Knot or link…

And: Is there anybody can show me the knot that been bounded an embedded piecewise linear discs in S3 (in definition1.2) ?….It sounds like noodles in a big pan. God~There must no people read this book like me…@@

OK~take it easy: Who knows the number of Prime knots after see the definition? We can distinguish the prime knot from definion1.3.. also can know the number of Prime knots not equal to the prime number from the knot table. As for the definition1.4…How does the “half the sum of the sign” work?…It just says that something related in homology theory. How can I catch the meanings of it from such few words?!

After looking over these, how you think about to know other important invariant-related topics like Alexander polynomial, Jones polynomial and so on in knot theory? It should be another hard work to overcome.  Of course, you can see what the stupid student’s story about learning math, especially in complex geometry , Lie algebra, knots and those with Quantization on the blog from now on.  Thus for the important part of knot,Alexander polynomial and Jones polynomial, you will see what the  learning problem I met.

Jones Polynomial Table from Lickorish,1997


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