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.

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.
Compute the unknotting number for $T(2,3)$, $T(2,2k+1)$.

Compute the Alexander polynomial for the $7_1$ knot.
Let $K$ be a knot and $\xi$ be a $p^k$-th root of unity with $p$ prime. Show that $\Delta(\xi)\neq 0$.
Prove that all roots of the Alexander polynomial of a torus knot are roots of unity. How does this interact with the previous problem?
Prove that for any knot the $4-$fold cyclic branched cover is a rational homology 3-sphere, i.e. the first homology group is torsion.
Draw the bifurcation diagram of the $(2,5)$ real cusp and explain what loop of Reidemeister moves is associated with it.
Show that the Whitehead link and the twisted Whitehead link have diffeomorphic complement. Hint: use surgery.
Compute the Alexander module of the figure eight knot.
Let $K$ be a knot and $\Delta$ its Alexander polynomial. Let $K'$ be the $(2,3)$-cable on $K$. Prove that $\Delta_{K'}(t)=\Delta_K(t^2)(t^2-t+1)$. Hint: decompose the infinite Abelian cyclic cover of $S^3\setminus K'$ and use Mayer-Vietoris.
Show that the figure eight knot is $2$-torsion in the concordance group.
What is the genus of the minimal genus cobordism between $5_1$ and $5_2$ knots?
Let $L$ be the Borromean rings and $L'$ be obtained from $L$ by replacing one component by a Whitehead link. Compute the fundamental group of $L'$.
Let $L$ be a 4-component link such that the components form a chain of positive Hopf links. For which framing the surgery on $L$ is the $\mathbb{R}P^3$?

Seifert surfaces. Construction. Properties.
Constructions of covers via Seifert surfaces.
Reidemeister theorem. Sketches of proofs.
Algebraic invariant from the Seifert forms: signature, Alexander polynomial, Alexander module
Fundamental group of link complement, Wirtinger presentation.
Surgery. Basics of Kirby calculus.
Unknotting number. Simple criteria.
Four genus. Topological and smooth.
Fibered knots and links. Examples of non-fibered links, relation between monodromy and Seifert forms.

Winter semester 2025/26