Manifolds and cell complexes Knot theory Theorem

Wirtinger presentation

Builds on Knot Knot group Presentation of a group

Statement

Let $K$ be an oriented tame knot in $\R^3$ , with a diagram that has $n \ge 1$ crossings. The diagram breaks into $n$ arcs $a_1, \ldots, a_n$ , each running from one undercrossing to the next. Take a base point above the plane of the diagram. For each arc $a_i$ , let $x_i$ be the class of a loop from the base point that passes once under $a_i$ , by the right-hand rule. At a crossing, let $a_k$ be the over-arc, let $a_i$ be the incoming under-arc and let $a_j$ be the outgoing under-arc. Let $\varepsilon = \pm 1$ be the sign of the crossing. Write $r$ for the relation $x_j = x_k^{\varepsilon} x_i x_k^{-\varepsilon}$ at this crossing. Then the knot group of $K$ has the presentation $\pi_1(\R^3 \setminus K) \cong \langle x_1, \ldots, x_n \mid r_1, \ldots, r_n \rangle,$ with one relation $r_c$ for each crossing $c$ . Any one of the relations may be omitted.

Remarks

The proof applies the Seifert–van Kampen theorem to the complement, cut into the half-space above the diagram plane and small pieces near the crossings. The half-space above, with tunnels under the arcs, gives a free group on the $x_i$ , and each crossing adds one relation. The sign convention for $\varepsilon$ depends on the orientation chosen for the loops $x_i$ ; with the opposite choice, $\varepsilon$ becomes $-\varepsilon$ . For the trefoil with arcs $a_1, a_2, a_3$ , the relations read $x_3 = x_1 x_2 x_1^{-1}$ , $x_1 = x_2 x_3 x_2^{-1}$ and $x_2 = x_3 x_1 x_3^{-1}$ for one orientation. Eliminating $x_3$ and dropping the last relation gives $\langle x_1, x_2 \mid x_1 x_2 x_1 = x_2 x_1 x_2 \rangle$ . All generators are conjugate, because each relation conjugates one generator to another and the arcs are connected by the knot. The hypothesis $n \ge 1$ is needed: a diagram of the unknot with no crossings has one arc and no crossing, and its group is $\Z$ .

Going deeper

For a student

It is the standard way to compute knot groups and their invariants. Representations of the knot group into a finite group can be found from it by assigning elements to arcs. Tricolorability is the case of the symmetric group on three letters. The Alexander polynomial can be computed from it by Fox calculus. The presentation has deficiency one, with one more generator than essential relations.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Knot concept
Knot group concept
Presentation of a group concept
Wirtinger presentation statement

Tag	`P6GR8ZQCK3`
URI	`m2://P6GR8ZQCK3`
Kind	statement, theorem
Writing	W0 stub
MSC	57K10, 57M05
Status	This copy of the site has no build data. Run `m2 status P6GR8ZQCK3`.
About	Knot group
Uses	Knot, Knot group, Presentation of a group
Same as	Wikidata Q8027029 Wikipedia Wirtinger presentation
Changed	2 November 2026