You might enjoy these BBC broadcast of the lives of 10 mathematicians
Brief History of Mathematics

To save some time, rather than lecture on the result that any triangulation of the torus requires at least 14 triangles I wrote up the argument (more is actually proven). Restriction on triangulation of a surface.

Info on Knot theory
Torus Knots

Some topics I hope to cover:

I - Quick introduction to the foundations, e.g. point set topology.
(a) Definitions
(b) Quotient, products, Hausdorff, connectedness.
(c) I may spend 20 minutes on Fractals.
(d) Ambient Isotopy - this is relevant to knot theory.

II- Euler Characteristic
(a) Definition and outline of invariance.
(b) Platonic solids

(c) Polyhedral decompositions of a 2-sphere.
e.g we will prove propositions such as: Each face must have at least 3 edges, Each vertex belongs to at least 3 edges, There cannot be 7 edges. III- Classification of surfaces.

IV- The fundamental group

The Browder fixed point theorem.

. V - Covering spaces

VI - Knot theory.
Knot group of a torus knot.
Knot group of a clover leaf knot
Alexander polynomials and Skein Relations.
Jones Polynomial
Kaufman bracket
Kaufman polynomial
Seifert Surface

VII- Coloring of surfaces
Harwood's theorem on coloring of surfaces.

Much of the material can be found in Chapters 2,3,4,and 5 of

Prof. Thompson's topology notes

There is also chapter 1 of the book by Alan Hatcher
Algebraic Topology, Cambridge University Press.

Some other references:

Algebraic Topology, An Introduction, W. S. Massey, Springer-Verlag
I will use Massey for classification of surfaces.

Algebraic Topology, A First Course, Greenberg and Harper, HarperCollins Canada.

Here is the picture I was attempting to draw illustrating the ambient isotopy between linked and unlinked handles.
unlinking

Proof that the free product is a group

Torus#P

Here is the reference to the lecture on classification of surfaces. The above picture of the homeomorphism between the torus#P and the Klein bottle # P appears on the last page of these notes.
Classification of surfaces

Here is a link to a site that has some cool stuff about non-orientable surfaces. Note the section on the relations between music and the Mobius band.

Non orientable surfaces.

Here is a link with pictures of the Platonic solids (the first 5 in the list).

Solids