**Notation**:   

$L/K$, finite extension of global fields   

$K^\times$, unit group of $K$   

$L^\times$, units group of $L$   

$\mathbb{A}\_L^\times$, ideles of $L$   

$N\_{L/K}$, the norm map


The **knot group** of an extension of global fields $L/K$ is defined as the quotient group of 'local norms' by 'global norms':
$$\mathfrak{K}(L/K):=\frac{K^\times \cap N\_{L/K}(\mathbb{A}\_L^\times)}{N\_{L/K}(L^\times)}.$$


**Question**: In what way is this related to knots?


**Remarks**: 1. The terminology *knot group* appears to have been introduced by Arnold Scholz, in the papers *Totale Normenreste, die keine Normen sind, als Erzeuger nichtabelscher Körpererweiterungen*. Part I (1936), Part II (1940). (I have not looked into these papers yet.)   

2. I don't think this is related to the usual analogy between knots and number fields (but I could be wrong). For example, the usual knot group of a knot $\mathcal{K}$ in $S^3$ is the fundamental group $\pi\_1(S^3\setminus \mathcal{K})$. The analog of this should be something like $\pi\_1(\text{Spec}(\mathcal{O}\_K)\setminus \mathfrak{p})$, where $\mathcal{O}\_K$ is the ring of integers of a global field $K$ and $\mathfrak{p}\subset \mathcal{O}\_K$ is a prime ideal.



Knot group of a field extension
Arnold Scholz was fond of a colorful language in mathematics. I don't think there's any connection to actual knots except perhaps for a faint reference to the Gordian knot, which is difficult to solve without a striking idea.


