Lagmental Vicfred

A Knot Group Comes from the Complement by Vicfred

tags: mathematics, CW complexes, fixed points, knot theory

The fundamental group of three-space minus a knot is an invariant computable from a diagram. The point is to make the formal expression readable enough to audit line by line.

Notation

A CW complex is assembled by attaching disks \(D^n\) along maps from their boundaries \(S^{n-1}\). Cellular chains convert the attaching data into algebra.

$$ G_K=\pi_1(S^3\setminus K) $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ G_{\text{trefoil}}\cong\langle a,b\mid a^2=b^3\rangle $$

Stress the formula

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ \langle x_1,\ldots,x_m\mid x_jx_ix_j^{-1}=x_k\ \text{at each crossing}\rangle\qquad\text{(Wirtinger presentation)} $$

Interpretation

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad G_K=\pi_1(S^3\setminus K),\\[5pt] \mathsf{C}\;&:\quad G_{\text{trefoil}}\cong\langle a,b\mid a^2=b^3\rangle. \end{aligned} $$

Limit of the argument

Euler characteristic is homotopy invariant for finite CW complexes, but equal Euler characteristics do not imply homotopy equivalence.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] G_{\text{trefoil}}\cong\langle a,b\mid a^2=b^3\rangle \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Thu 07 April 2016. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.