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Worked Wirtinger presentations

Example

The crossing-free unknot code gives the presentation $⟨ x ∣ ⟩$ , hence its knot group is $Z$ . For the standard three-crossing trefoil diagram, label the under-arcs $a, b, c$ so the crossing relations are $c = a b a^{- 1}$ , $a = b c b^{- 1}$ , and $b = c a c^{- 1}$ . Eliminating $c$ gives $⟨ a, b ∣ a b a = b a b ⟩$ . The assignment $a \mapsto (12)$ and $b \mapsto (23)$ defines a surjection of this group onto the nonabelian group $S_{3}$ . For the four-crossing figure-eight code with under-arcs $x_{1}, x_{2}, x_{3}, x_{4}$ , the crossing equations are $R_{1} : x_{1} x_{2} = x_{3} x_{1}, R_{2} : x_{1} x_{4} = x_{4} x_{2}, R_{3} : x_{3} x_{2} = x_{2} x_{4}, R_{4} : x_{3} x_{4} = x_{1} x_{3} .$ The figure-eight code has over-arc labels $x_{1}, x_{4}, x_{2}, x_{3}$ at $R_{1}, R_{2}, R_{3}, R_{4}$ , respectively, and its oriented under-arc transitions are $x_{1} \to x_{2}$ at $R_{2}$ , $x_{2} \to x_{3}$ at $R_{1}$ , $x_{3} \to x_{4}$ at $R_{3}$ , and $x_{4} \to x_{1}$ at $R_{4}$ ; the four displayed local charts give their side orders and orientations. The transition cycle verifies that the code has one component and four under-arcs. The fourth equation follows from the first three: left-multiply $R_{3}$ by $x_{1}$ , substitute $R_{1}$ and then $R_{2}$ , and cancel the common right factor $x_{2}$ .

The first diagram has two oriented strands crossing three times between top and bottom; the two bottom endpoints are joined to the corresponding top endpoints by the outside arcs. Its under-arc labels are chosen so that the triples $(o, L, R)$ of over-arc, left under-arc, and right under-arc are $(b, c, a)$ , $(c, a, b)$ , and $(a, b, c)$ at crossings $1, 2, 3$ . The white gaps mark underpasses; the redrawn strand is over.

The panels below record the local crossing charts for the four figure-eight equations. In every panel the horizontal over-strand points right; the upper-right under-meridian is $x_{L}$ and the lower-left is $x_{R}$ . The upper side is the target and lower side the source. In each panel the fixed relation is $x_{L} = x_{o}^{- 1} x_{R} x_{o}$ , independently of the under-arrow direction.

Example references

Complete integer figure-eight drawing and its verified positive-meridian crossing table

Finite scan that outputs one meridian generator per under-arc and one relator per crossing

Identification of the output group with the knot complement group

Local crossing chart and oriented conjugacy relation

Cattabriga and Francaviglia, Knot Theory and Applications: Lecture Notes, §4.2

Charbonneau, “Computing the Wirtinger Presentation of a Knot Group from a Knot Diagram,” §4

Facts & Assumptions

Given: The crossing-free unknot, the standard oriented trefoil diagram with the stated under-arc labels, and the stated four-crossing figure-eight code.

[F1]

The constructor returns one meridian generator per under-arc and scans every crossing once. Wirtinger presentation from a finite diagram code

[F2]

Each crossing's positive meridians satisfy the conjugacy relation determined by its oriented chart. Local meridian conjugacy at a Wirtinger crossing

[F3]

The resulting presentation group is isomorphic to the knot complement group, with each generator mapped to its positive meridian. Correctness of the Wirtinger presentation

[F4]

A group presentation denotes a quotient of the free group on its generators by the normal closure of the listed relators. Group presentation by generators and relations

[F5]

The free group on one generator is isomorphic to the additive integers by exponent sum. Free group on a set of generators

[F6]

A quotient by a relator normal closure has the presentation's universal property. A homomorphism that kills a normal subgroup factors uniquely through the quotient group

[F7]

The explicit valid figure-eight drawing has exactly the four side triples and oriented under-arc transitions stated here. Explicit unknot and figure-eight oriented Wirtinger codes

Proof

1.1givenF1F3F4F5algebra

The crossing-free code has one under-arc and no crossing relators, so [F1] gives $⟨ x ∣ ⟩$ . This is the free group on one generator, isomorphic to $(Z, +)$ by exponent sum using [F5]; [F3] identifies it with the unknot group.

1.2givenF1F2F4algebra

The trefoil panels give the crossing triples $(b, c, a)$ , $(c, a, b)$ , and $(a, b, c)$ . Applying [F2] gives the three crossing relations in the statement. The crossing with overarc $a$ gives $c = a b a^{- 1}$ . Substituting this into $a = b c b^{- 1}$ gives $a = b a b a^{- 1} b^{- 1}$ . Right multiplication by $b a$ yields $a b a = b a b$ . Under that braid relation, $c a c^{- 1} = a b a b^{- 1} a^{- 1} = b$ , so the third crossing equation follows from the first two. Thus eliminating $c$ gives exactly $⟨ a, b ∣ a b a = b a b ⟩$ .

1.3givenF1F2F7algebra

By [F7], these local charts belong to the explicit figure-eight code, rather than merely prescribing four abstract crossing relations. Each panel is an oriented local crossing chart. In every panel [F2] gives $x_{o} x_{L} = x_{R} x_{o}$ . The side triples $(o, L, R)$ are $(x_{1}, x_{2}, x_{3})$ , $(x_{4}, x_{2}, x_{1})$ , $(x_{2}, x_{4}, x_{3})$ , and $(x_{3}, x_{4}, x_{1})$ . Substitution, and reversal of the written equality when needed, gives exactly the four equations in the statement. The under-arrow direction does not select a different conjugation rule.

2.1step 1.2F4F6algebra

In $S_{3}$ , the two words $(12) (23) (12)$ and $(23) (12) (23)$ both equal $(13)$ , so the assignment respects the trefoil relation. The transpositions $(12)$ and $(23)$ generate all six permutations $e, (12), (23), (12) (23), (23) (12), (13)$ , hence the homomorphism is surjective by [F4] and [F6]. Since $S_{3}$ is nonabelian, this group is nonabelian.

3.1step 1.3algebra∎

From $R_{3}$ , left multiplication by $x_{1}$ gives $x_{1} x_{3} x_{2} = x_{1} x_{2} x_{4}$ . Applying $R_{1}$ to the right side gives $x_{3} x_{1} x_{4}$ , and $R_{2}$ then gives $x_{3} x_{4} x_{2}$ . Right cancellation of $x_{2}$ yields $x_{1} x_{3} = x_{3} x_{4}$ , which is $R_{4}$ .

Depends on

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Sources

Alessia Cattabriga and Stefano Francaviglia, Knot Theory and Applications: Lecture Notes, §4.2, printed pp. 19–21 (standard reference, not scraped)
Cole Charbonneau, Computing the Wirtinger Presentation of a Knot Group from a Knot Diagram, §4 (standard reference, not scraped)
Stefan Friedl, Knot Theory, §§5.2–5.4, printed pp. 68–74 (standard reference, not scraped)