MATH2141 MATH2151 Geometric Topology.
Epiphany 2015
Lecturer: Andrew Lobb
Schedule:
Tuesdays 3pm-4pm CG84
Thursdays 4pm-5pm CG84
Books
Dirk Schuetz has kindly provided a copy of his typed lecture notes for this course. We shall be following Dirk's lecture notes by-and-large, although we may well emphasize things a little differently or give alternative derivations and examples. These notes should be your first port of call. After this, there are a number of good books out there covering much of our material; you should look out for the books written by
Adams;
Gilbert and Porter;
Cromwell;
Rolfsen.
Lectures
Here's a brief outline of what I intend to cover in each week's lectures.
Week |
Content |
1 |
Knots and links intuitively. The Reidemeister moves. Tri-colorings. |
2 |
Crossing number, composition of knots, alternating knots. |
3 |
Writhe, linking number, the bracket and X polynomials. |
4 |
Alexander polynomial, absolute polynomial, beginnings of the topology of surfaces. |
5 |
Simplicial complexes, triangulations, Euler characteristic. |
6 |
Surfaces with boundary, classification theorem, Seifert surfaces. |
7 |
Seifert surfaces continued, genus of a knot, winding number. |
8 |
Vector fields in the plane, singularities, index, tangent curves. |
9 |
Vector fields on surfaces, hairy ball theorem, Poincare-Hopf theorem. |
Assignments
Your homework is handed in at 5:30pm on Thursdays. Solutions will be posted shortly afterwards. Not every question is to be handed in for marking - this is indicated in the table below. We'll do some of the other questions in the tutorials and problems classes.
Exercise Sheet |
Date due |
Questions to hand in |
Solutions |
22nd January |
1,5,8. |
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29th January |
3,4. |
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5th February |
3,10. |
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12th February |
1(a), 3 (for "Jones" please read "X"). |
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19th February |
2 (for "Jones" read "X"), 6 (for "Jones" read "X"). |
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26th February |
2(a), 3. |
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5th March |
1, 4. |
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12th March |
2, 4. |
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No date. |
For exam practice. |