METU Department of Mathematics Math 559 Low-dimensional Topology Spring 2009

CONTENT OF MIDTERM I

Theoretical questions (to formulate a theorem, or to give a definition)

(1) Alexander theorem, Markov theorem

(2) Reidemeister moves, Markov moves

(3) linking number, writhe

(4) Seifert matrix, Seifert form

(5) properties of the Alexander polynomial

Practical questions

(1) Find the linking number for a given oriented link and the writhe for a given knot diagram;

(2) find a sequence of Reidemeister moves between equivalent knot diagrams; find a sequence of Markov moves for equivalent braid presentation of a knot;

(3) give a braid presentation of a given knot; sketch a knot diagram of a given braid;

(4) recognize braid presentations of the same knot; recognize a braid of connected sum of knots;

(5) sketch a 2-bridge diagram and a 4-plat for a given 2-bridge knot (e.g., K_{2,n}, W h^{±}(n), etc.)

(6) determine the twist-sequence of a 4-plat presenting a given braid b ∈ B_{3} and find the corresponding rational number;

(7) find the Euler characteristic of a spanning surface; find the genus of a Seifert surface;

(8) find the intersection indices between given curves on a surface; find an intersection matrix for a span of a knot;

(9) find the framing of curves on a spanning surface find a Seifert matrix;

(10) find the Alexander-Conway polynomial using various methods (from a Seifert matrix, by Alexander’s method, via Burau presentation, using skein relations);

(11) find the determinant, and the signature of a knot (or a link).