3
$\begingroup$

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.

$\endgroup$
1

1 Answer 1

6
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ Thanks! Gordian knot would be a fitting reference, but perhaps calling it the Gordian group would have been less confusing. $\endgroup$ Oct 14, 2022 at 18:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.