Knot theory

10/06. Knots, links, isotopies. Dehn lemma, sphere lemma (without proof). Wirtinger presentation.
10/13. Linking number. Seifert form. Alexander polynomial as $\det(tS-S^T)$. $S$-equivalence.
10/20. Link diagrams. Reidemeister theorem. A quick look onto jet spaces.
10/27. Proof of Reidemeister theorem. Loops of Reidemeister moves.
11/03. Branched covers and their homology. Linking form on cyclic covers.
11/17. Infinite cyclic covers. Homology with twisted coefficients. Alexander modules. Twisted Alexander modules.
11/24. Blanchfield form. Classification of linking forms over $\mathbb{R}$. Signatures.
12/01. 4-genus. Slice knots. Topological versus smooth concordance. First obstructions to concordance.
12/08. Unknotting number. Algebraic unknotting number. Casson-Gordon invariants.
12/15. Braids. Braid group. Markov and Alexander theorem with Vogel's proof.

Prove Alexander duality theorem.
Prove that Wirtinger presentation is correct.
Find a representation of $\pi_1$ of a torus knot with two generators and one relation.
Compute the fundamental group of the complement of the unlink and of the Hopf link.
Show that the Borromean rings are non-trivial by proving that the third component is non-trivial in $\pi_1$ of the complement of the other two.
Compute the fundamental group of simple knots: trefoil, figure eight, $5_2$.
Prove that knot complement is a $K(\pi,1)$ space.
Prove that the genus is additive.
Show that the Alexander polynomial of a split link is zero. Is this true for boundary links?
Find a pair of links whose complement has the same fundamental group.
Compute the Alexander polynomial of the figure eight knot.
Compute the Alexander polynomial of the $T(p,q)$ torus knot using a presentation of the knot group.
Compute the Seifert form of a $P(p,q,r)$ pretzel knot.
Compute the Seifert form of a Whitehead double.
Prove that if the Alexander polynomial is $1$, all the signatures are $0$.
Give an example of a $\mathbb{Z}[t,t^{-1}]$-module which is not cyclic, but it is cyclic when tensored over $\mathbb{Q}$.
Show that Ehresmann's fibration theorem might fail if the domain is not compact.
Prove the skein relation for Alexander polynomials.

Winter semester 2025/26