Category Archives: algebraic-topology

## Computing the Alexander polynomial: a guide

Given a knot $K$, which is an embedding $S^1\to \R^3$, we have see how to compute the fundamental group of $K$, defined as $\pi_1(\R^3 – K)$. For example, we have computed the fundamental group of the trefoil knot and the fundamental group of the cinquefoil knot. The fundamental group of the trefoil can be given […]

Let $\mathcal{A}$ be a small category and $\mathbf{B}\mathcal{A}$ its geometric realisation. It is evident that $\mathbf{B}\mathcal{A}$ and $\mathbf{B}\mathcal{A}^\circ$ are homotopy equivalent, and in fact homeomorphic. However, can we find functors that realise this equivalence? This post summarises some informal notes I have written on this following D. Quillen’s paper Higher Algebraic […]