© | Dror Bar-Natan: Classes: 2015-16: Math 475 - Problem Solving Seminar: | (21) |
Next: Quiz #5 - "Modify the Problem" / "Choose an Effective Notation"
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**Reading.** Section 1.6 of Larson's textbook.

**Next Quiz.** Thursday March 3, on that section, this handout, and more.

**Dror's Favourite** "Choose an Effective Notation" problem is more a
computer problem than a math problem, yet it is pretty and it will make us
look at some pretty (and significant!) mathematics.

The *Kauffman Bracket* $\langle D\rangle$ of a knot diagram $D$
is defined by the following recursive relations:

(where $A$ is some variable).

**Remark.** The celebrated "Jones Polynomial" of $D$ is very easily
defined from $\langle D\rangle$,
\[ J(D) = \left.(-A^3)^{w(D)}\frac{\langle D\rangle}{-A^2-A^{-2}}\right|_{A\to q^{1/4}}, \]
where $w(D)$ is the *writhe* of $D$, the difference between the
number of right-handed and left-handed crossings in $D$. Yet this matters
not for us today.

**Homework Problem** (not to be submitted, yet fame and glory for an
excellent solution). Write a very short computer program that will be
able to compute the Kauffman bracket (and hence the Jones polynomial)
of the following three knots:

**Problem 1** (Larson's 1.6.2).

- Of all the rectangles which can be inscribed in a given circle, which has the greatest area?
- Maximize $\sin\alpha+\sin\beta+\sin\gamma$, where $\alpha,\beta,\gamma$ are the angles of a triangle.
- Of all the triangles of a fixed perimeter, which has the greatest area?
- Of all the parallelepipeds of volume 1, which has the smallest surface area?
- Of all the $n$-gons which can be inscribed in a given circle, which has the greatest area?
- Dror adds: of all the $n$-gons with a given perimeter, which has the greatest area?

**Problem 2** (Larson's 1.6.4). Let $P$ be a point on the graph
$G$ of $y=f(x)$, where $f$ is a cubic polynomial. Assume the tangent
to the curve at $P$ intersects $G$ again at a point $Q$. Let $A$ be the
area bound by $G$ and the segment $PQ$, and let $B$ be the area defined
in exactly the same manner, except starting with $Q$ rather than with
$P$. What is the relationship between $A$ and $B$?

**Problem 3.** A function on the plane has the property that the sum
of its values on the corners of any square (of any size and orientation)
is zero. Prove that the function is the zero function.

**Problem 4.** Prove: You cannot colour the points of the plane
with just three colours, so that no two points of distance 1 will be
coloured with the same colour. What if you had four colours available?

**Dror's Favourite** "Exploit Symmetry" problem: Two players
alternate placing $1\times 1$, $2\times 1$, and $4\times 1$ lego pieces
on a circular table of diameter 100, with no overlaps. If a player cannot
place a further piece, (s)he looses. Would you rather be the first to
move of the second?

**Dror's Favourite** "Explore Symmetry" topic: See http://drorbn.net/Treehouse: