Topics from today's class:
- The bracket polynomial
- The X polynomial
- The Jones polynomial
- 6.3, 6.4, 6.5, 6.7, 6.8
Read for next class:
Write one sentence about the reading assignment. You cannot repeat what another student has posted.
The span, the difference between the highest and lowest exponent in the polynomial, is an invarient for the bracket polynomial. This means that for a given knot k the bracket polynomial for any projection will always have the same span.
We can view calculating the bracket polynomial as a linear combination of end-states, which can be calculated without having to deal with so many intermediate knots.
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I was a little bit confused about how to label A and B.
Not sure why there are n+2 regions in any alternating projection.
The discussion on p. 157 about how to compute the bracket polynomial of a link from its states confused me a bit. I am curious to learn more about how to use polynomials to discover characteristics about knots.
If a knot has more than one reduced alternating projections, are their properties of the polynomials the same??