Lecture 10

Reviewing end of Lecture 9.

Lemma 1. Let Γ be a finite angled graph, whose angle girth is ≥ 2π, i.e. |σ|_{∠} ≥ 2π for each immersed closed path σ. Then there exists a constant c_{2} ≤ 0 such that |γ|_{∠} ≥ π − c_{2} for arbitrary immersed path γ → Γ.

Proof. (figure) Divide γ into closed paths and two “tails”, which give some bounded negative contribution.

Exercise. Write the proof.

Lemma 2. If X satisfies negative weight test and X is compact, then ∃K = K(X) such that Area (D) ≤ K|∂_{P} D| for any near-immersed D → X.

Proof. Without loss of generality assume D is non-singular. Pull back angles to D. Apply combinatorial Gauss-Bonet theorem.

2π = 2πχ(D) = ∑_{f} κ(f) + ∑_{v∈∂D} κ(v) + ∑_{v∈IntD} κ(v)

(figure: D)

Let c_{1},c_{2} be such that κ(f) ≤ −c_{1} < 0, c_{2} as in lemma 1 applied to any link(x), x ∈ X^{0}. |γ|_{∠} ≥ π − c_{2}, so κ(v) = π − |γ|_{∠} ≤ c_{2}.

Now we have

2π ≤ c_{1}Area (D) + c_{2}|∂_{P} D| + 0
c_{1}Area (D) ≤ c_{2}|∂_{P} D| − 2π
Area (D) ≤ \frac{c_{2}}{c_{1}} |∂_{P} D|
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Consider ⟨a_{1}, a_{2}, …, a_{g}|a_{1}^{2}a_{2}^{2} ⋯ a_{g}^{2}⟩, whose standard complex is genus 2g nonorientable surface. It is (2g, 2g)-complex, and corresponding disk diagram is a 2g-gon.

g = 2 (4, 4)-complex — nonpositive weight test, g ≥ 3 (2g, 2g)-complex — negative weight test. Exercise. Let 0 < g < h and Σ_{g} denote genus g surface. There is no injection

π_{1}Σ_{g} ↪ π_{1}Σ_{h} (both surfaces orientable or both non-orientable).

Dehn Complex of a Link.

Example. (Max Dehn) Naturally occuring (4, 4)-complexes. (figure: knot) Knot is embedding S^{1} ↪ ℝ^{3}, or S^{1} ↪ S^{3}. (Smooth or piecewise linear.) Link is embedding ⨆ S^{1} ↪ ℝ^{3}. π_{1}(S^{3} \ L) is a nice presentation of a link. (figure: knot over a plane) Imagine link L covering above or below a “projection

plane” ℝ^{2} ⊂ ℝ^{3}.

On the plane, triple crossings are not allowed (see figure) (figure: allowed crossing, forbidden crossing) Since valency of each vertex is 4, projection has a checkerboard coloring. Dehn complex D is determined by the following:

two 0-cells α, β above and below ℝ^{2}; 1-cells correspond to regions in projection; 2-cells correspond to crossing points in projection. They come in two flavors, depending on coloring of crossing point neighborhood (see figure). 2-cells are attached to four 1-cells each as shown (see figure).

(figure: two types of crossing)

(figure: attaching a 2-cell)

(figure: Dehn complex)

Theorem 1. D ↪ (ℝ^{3} \ L) induces a π_{1}-isomorphism π_{1}D → π_{1}(ℝ^{3} \ L).

Definition 1. Link projection is alternating, if each circle alternatively travels above and below at crossing.

(figure: alternating link projection)

Definition 2. Link projection is prime, if every circle C in ℝ^{2}, which intersects the projection in two regular points, has a simple arc on one side.

(figure: prime link projection)

A projection is not prime iff it is a “connected sum” of two projections.

Theorem 2. If projection is prime and alternating, then Dehn complex D is (4, 4)- complex.

Proof. Each 2-cell has ≥ 2 sides. We need to check that link(α), link(β) have girth ≥ 4. Examine link(α) (link(β) similar).

(figure)

In fact, link(α) inbeds in the projection picture. Checkerboard coloring gives bipartile structure to link(α). Therefore, there are no sycles of length 1, 3. Then girth ≥ 4 if no bigons. Two types of bigons:

(figure) — not alternating

(figure) — not prime

Exercise. Think through logical details.

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4-valent graph → make alternating projection → Dehn (4, 4)-complex.

Quadratic isoperimetric function for (4, 4)-complex. Let D be a disc diagram, whose 2-cells are squares, 0-cells have links of girth ≥ 4. D has a system of “dual curves”, each curve passing through opposite sides of squares (see figure).

(figure: disc diagram with dual curves)

Lemma 3. No dual curve is a circle. (figure)

Lemma 4. No dual curve self-intersects. (figure)

Lemma 5. No two dual curves intersect in a bigon. (figure)

Lemma 6. No two three curves intersect in a triangle. (figure)

Proof. Subdivide D along dual curves. Consider a subdiagram E bounded by dual curves. Note: E is a (4, 4)-complex. Apply combinatorial Gauss-Bonet theorem to E:

\begin{aligned}2πχ(E) & = & ∑_{v∈∂E} κ(v) & + & ∑_{v∈IntE} κ(v) & + & ∑_{f} κ(f) \\ 2π ≤ & = & ∑_{v∈∂E} κ(v) & + & (≤ 0) & + & 0 \end{aligned}

Conclusion: E has at least four \frac{π}{2} corners. Finally we show that Area (D) ≤ (|∂_{P} D|)^{2} (without loss of generality D is non-

singular). Each 2-cell in D gives distinct dual curves (by lemma 4). By lemma 3, these curves hit the boundary. By lemma 5, the map {2-cells} → {distinct pairs of dual curves} is injective.

(figure: square and corresponding dual curves) This gives inequality:

Area (D) ≤ \binom{\frac{∂_{p}D}{2}}{2} ≤ |∂_{P} D|^{2}
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Exercise. Find an interesting (thick) (3, 6)-complex.