knots intro

${\mathbb{S}}^1=\{x,y\in \mathbb{R}|x^2+y^2=1\}$

A knot is a path in 3d space: $k:{\mathbb{S}}^1\to {\mathbb{R}}^3$.

Two knots $k,k^{\prime }$ are considered equivalent if there exists a function $F:[0,1]\times {\mathbb{R}}^3\to {\mathbb{R}}^3$ with $F(0,\_){|}_{{\mathbb{S}}^1}=k$ and $F(1,\_){|}_{{\mathbb{S}}^1}=k^{\prime }$

Fundamental group of a space $X$ at $p\in X$: ${\pi }_1(X,p)=\{[\gamma ]|\gamma \in P(X)\}$, met $P(X)$ alle paden die op p beginnen en eindigen.

A knot invariant is an object $I(k)$ such that for two knots $k,k^{\prime }$ which are equivalent, it holds that $I(k)\cong I(k^{\prime })$.

Example of a knot invariant: ${\pi }_1({\mathbb{R}}^3\backslash k({\mathbb{S}}^1),p)$

Category of pointed spaces ($To{p}_{\ast }$):

Objects: $O{b}_{To{p}_{\ast }}=(X,x)$ met $x\in X$

Morphisms tussen $(X,x)$, $(Y,y)$: $Ho{m}_{(X,x),(Y,y)}=f:X\to Y$ met $f(x)=y$.

Category of groups ($Grp$):

Objects: groups

Morphisms: group homomorphisms

een groepshomomorfisme tussen $G,H$ is een functie $\varphi :G\to H$ met $\varphi (a{\cdot }_{G}b)=\varphi (a){\cdot }_H\varphi (b)\forall a,b\in G$

${\pi }_1$ as a functor

On objects: ${\pi }_1:O{b}_{To{p}_{\ast }}\to O{b}_{Grp}{\pi }_1(X,x):$ fundamentaal groep

On morphisms: ${\pi }_1:Ho{m}_{(X,x),(Y,y)}\to Ho{m}_{{\pi }_1(X,x),{\pi }_1(Y,y)}:f\mapsto ([\gamma ]\mapsto [f\circ \gamma ])$