Knot theory
Winter semester 2025/26
Plan of the lecture
- 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.
Classes
Problems for classes
- 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 $T(2,3)$, $T(2,2k+1)$.
Sample problems for the exam (diagrams of the knots are not presented, but will be supplied for the
real exam).
- 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 $\mathbb{R}P^3$?
Topics for the oral exam.
- 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.
Exam
- No mid-terms.
- Written exam, 3 hours, 5 problems.
- Compulsory oral exam.
Bibliography
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