PL and smooth knots and links.  Wild knots. Reidemeister moves.  
3-colorings.  Oriented and invertible knots. Chiral knots.  Toric knots.  
Linking number.  Framed knots and preferred framing.  Pretzel and 
rational knots.  Unknotting number, crossing number.  Alternating 
diagrams. Tietze theorem, Wirtinger presentation.  n-colorings.  
Connected sum of links, satellite links. Spheres and tori in the 
3-sphere.  Genus of a knot. Prime decomposition. Kauffman bracket and 
polynomial.  Jones polynomial. Applications to alternating knots. The 
fundamental quandle and associated invariants.

The course will cover also some other topics among the following ones:

Bridge number. Slice and ribbon knots. Seifert surfaces: definition and 
existence. The Seifert form.  The universal abelian covering of a link 
complement. Boundary links. Homology boundary links. The cut number of a 
3-manifold. Presentations of modules over a ring with unity. Alexander 
ideals and Alexander polynomials of finitely presented modules. Group 
rings. Fox calculus. Applications of the Fox calculus to Alexander 
polynomials. Braid groups. Alexander Theorem and Markov Theorem. Conway 
algebras.  HOMFLY-PT and Alexander-Conway polynomials. Dehn-Lickorish 
theorem.  Rational and integer surgery. Lickorish-Wallace theorem.  
Branched covers.  Hilden-Montesinos theorem.