A Geometric Proof of the Flyping Theorem

A GEOMETRIC PROOF OF THE FLYPING THEOREM

THOMAS KINDRED

Abstract. In 1898, Tait asserted several properties of alternating knot diagrams. These assertions became known as Tait’s conjectures and remained open until the discovery of the Jones polynomial in 1985. The new polynomial invariants soon led to proofs of all of Tait’s conjectures, culminating in 1993 with Menasco–Thistlethwaite’s proof of Tait’s flyping conjecture. In 2017, Greene (and independently Howie) answered a longstanding question of Fox by characterizing alternating links geometrically. Greene then used his characterization to give the first geometric proof of part of Tait’s conjectures. We use Greene’s characterization, Menasco’s crossing ball structures, and re-plumbing moves to give the first entirely geometric proof of Menasco–Thistlethwaite’s flyping theorem.

1. Introduction P.G. Tait asserted in 1898 that all reduced alternating diagrams of a given prime non-split link in S3 minimize crossings, have equal writhe, and are related by flype moves (see Figure 1) [11]. Tait’s conjectures remained unproven for almost a century, until the 1985 discovery of the Jones polynomial, which quickly led to proofs of Tait’s conjectures about crossing number and writhe. Tait’s flyping conjecture remained open until 1993, when Menasco–Thistlethwaite gave its first proof [8, 9], which they described as follows: The proof of the Main Theorem stems from an analysis of the [chessboard surfaces] of a link diagram, in which we use geometric techniques [introduced in [7]]... and properties of the Jones and Kauffman polynomials.... Perhaps the most striking use of polynomials is... where we “detect a flype” by using the fact that if just one crossing is switched in a reduced alternating diagram of n crossings, and if the resulting link also admits an alternating diagram, then the crossing number of that link is at most n − 2. Thus, although the proof of the Main Theorem has a strong geometric flavor, it is not entirely geometric; the question 1

2 THOMAS KINDRED

T 2 T1 T2 T1

Figure 1. A flype along an annulus A ⊂ S2. Two link diagrams are flype-related if there is a sequence of flype moves taking one diagram to the other.

remains open as to whether there exist purely geometric proofs of this and other results that have been obtained with the help of new polynomial invariants. We answer part of Menasco–Thistlethwaite’s question by giving the first entirely geometric proof of the flyping theorem. Flyping theorem. All reduced alternating diagrams of a given prime, nonsplit link are flype-related. As a corollary, we obtain a new geometric proof of other parts of Tait’s conjectures, which were first proven independently by Kauff- man [6], Murasugi [10], and Thistlethwaite [12] using the Jones polynomial, and were first proved geometrically by Greene [3]: Theorem 5.28. All reduced alternating diagrams of a given link have the same crossing number and writhe. It does not follow a priori that reduced alternating link diagrams minimize crossings. This and other parts of Menasco–Thistlethwaite’s question remain open; see Problems 2.6–2.8. Like Menasco–Thistlethwaite’s proof, ours stems from an analysis of chessboard surfaces and uses the geometric techniques introduced in [7]. Recent insights of Greene and Howie regarding these chessboard surfaces also play a central role in our analysis [3, 4]. (Those insights answered another longstanding question, this one from Ralph Fox: “What [geometrically] is an alternating knot [or link]?”) The most striking difference between our proof and the original proof in [9] is that we “detect flypes” via re-plumbing moves. Specif- ically, suppose D and D0 are reduced alternating diagrams of a prime nonsplit link L with respective chessboard surfaces B,W and B0,W 0, where B and B0 are positive-definite. With this setup:1 Theorem 4.11. D and D0 are flype-related if and only if B and B0 are related by isotopy and re-plumbing moves, as are W and W 0.

1B and W are essential, since D is reduced alternating and L is nonsplit. A GEOMETRIC PROOF OF THE FLYPING THEOREM 3

Theorem 4.11 is the first of two intermediate results which, together with background from [3], immediately imply the flyping theorem. With the same setup as Theorem 4.11, the second result is: Theorem 5.26. Any essential positive-definite surface spanning L is related to B by isotopy and re-plumbing moves.2 Our proof of the flyping theorem is entirely geometric, not just in the formal sense that it does not use the Jones polynomial, but also in the more genuine sense that it conveys a geometric way of under- standing why the flyping theorem is true. Namely, using Greene’s characterization, we leave the diagrammatic context for a geometric context; in that context, we describe moves that correspond to flypes; finally we prove that these moves are sufficiently robust: • Reduced alternating diagrams D correspond to geometric ob- jects: pairs (B,W ) of essential definite surfaces of opposite sign (this is Greene’s characterization); • Flype moves on D correspond to isotopy and re-plumbing moves on (B,W ) (this is Theorem 4.11); and • The pair (B,W ) is related to any other such pair (B0,W 0) by some sequence of these moves (this is Theorem 5.26). An outline: Section 2 reviews Greene’s characterization and other facts about spanning surfaces, then establishes further properties of definite surfaces, culminating with Lemma 2.14, which describes how a positive-definite surface and a negative-definite surface can intersect. Section 3 addresses generalized plumbing and re-plumbing moves, with a brief interlude about proper isotopy of spanning surfaces through the 4-ball.3 Section 4 sets up crossing ball structures and uses them, together with facts from §3, to prove Theorem 4.11; in particular, Figure 14 shows the type of re-plumbing move associated with flyping. Section 5 uses crossing balls structures and Lemma 2.14 to prove Theorem 5.26, and thus the flyping theorem and Theorem 5.28. Several of the arguments in §§4-5 are adapted from [1, 5]. Thank you to Colin Adams for posing a question about flypes and chessboard surfaces during SMALL 2005 which eventually led to the insight behind Figure 14 and Theorem 4.11. Thank you to Josh Howie and Alex Zupan for helpful discussions. Thank you to Josh Greene for helpful discussions and especially for encouraging me to think about this problem.

2. Definite surfaces

2Likewise for negative-deﬁnite surfaces and W . 3Generalized plumbing is also called Murasugi sum. 4 THOMAS KINDRED

Figure 2. Constructing chessboard surfaces; chessboards near a vertical arc (yellow) at a crossing

2.1. Background. Let L be a link in S3 with a closed regular neigh- 4 borhood νL and projection πL : νL → L. One can define spanning surfaces for L in two ways; in both definitions, F is compact, but not necessarily orientable, and each component of F has nonempty boundary.5 First, a spanning surface for L is an embedded surface F ⊂ S3 with ∂F = L. Alternatively, a spanning surface is a properly embedded surface F in the link exterior S3 \ νL◦ such that ∂F intersects each meridian on ∂νL transversally in one point.6 We use the latter definition. The rank β1(F ) of the first homology group of a spanning surface F counts the number of “holes” in F . When F is connected, β1(F ) = 1 − χ(F ) counts the number of cuts — along disjoint, properly embedded arcs — required to reduce F to a disk. Thus: Observation 2.1. If α is a properly embedded arc in a spanning 0 ◦ 0 surface F and F = F \ να is connected, then β1(F ) = β1(F ) + 1. A spanning surface F is (geometrically) incompressible if any simple closed curve in F that bounds a disk in S3 \\(F ∪νL) also bounds a disk in F .7 A spanning surface F is ∂-incompressible if any properly embedded arc in F that is ∂-parallel in S3 \\(F ∪ νL) is also

4Convention: Given X ⊂ S3, νX denotes a closed regular neighborhood of X and is taken in S3 unless stated otherwise. 5When L is nonsplit and alternating, F is connected, by Corollary 5.2 of [1]. 6 −1 Deﬁnition: A meridian on ∂νL is a circle πL (x) ∩ ∂νL for a point x ∈ L. 7For compact X,Y ⊂ S3, X \\Y denotes the metric closure of X \ Y . If each x ∈ X ∩ Y has a neighborhood ν(x) such that Z ∩ ν(x) is connected or empty for each component Z of X \ Y, then X \\Y is the disjoint union of the closures in S3 of the components of X \ Y . Hence, each component of X \\Y embeds naturally in S3, although X \\Y may not. For a general construction for X \\Y , let {(Uα, φα)} be a maximal atlas for X. About each x ∈ X, choose a suﬃciently tiny chart (Ux, φx) and construct Ux \\Y as above. Denote the components of Ux ∩ (Ux \\Y ) by Uα, α ∈ Ix, and denote S each natural embedding fα : Uα → Ux. Then x∈X {(Uα, φx ◦fα)}α∈Ix is an atlas 3 for X \\Y . Gluing all the maps fα yields a natural map f : X \\Y → X ⊂ S . A GEOMETRIC PROOF OF THE FLYPING THEOREM 5

∂-parallel in F . If F is incompressible and ∂-incompressible, then F is essential. This geometric notion of essentiality is weaker than the algebraic notion of π1-essentiality, which holds F to be essential if inclusion F,→ S3 \\νL induces an injective map on fundamental groups and F is not a mobius band spanning the unknot. Proposition 2.2. If an essential surface F contains a properly embedded arc β which is parallel in S3 \\(F ∪νL) to a properly embedded arc α ⊂ ∂νL \\∂F , then α is parallel in ∂νL to ∂F . Proof. Since F is essential and β is parallel in the link exterior to α ⊂ ∂νL, β is also parallel in F to an arc α0 ⊂ ∂F . Thus, α and α0 are both parallel in S3 \\(F ∪ νL) to β, and so α is parallel through a disk X ⊂ S3 \\(F ∪ νL) to α0 ⊂ ∂F . Since L is nontrivial and nonsplit, X is parallel in S3 \\(F ∪ νL) to a disk X0 ⊂ ∂νL, through 0 which α is parallel to α ⊂ ∂F . Given any diagram D of L, one may a color the complementary regions of D in the projection sphere S2 black and white in chessboard fashion. See Figure 2.8 One may then construct spanning surfaces B and W for L such that B projects into the black regions, W projects into the white, and B and W intersect in vertical arcs which project to the the crossings of D. Call the surfaces B and W the chessboard surfaces from D. A connected alternating diagram is reduced iﬀ both chessboard surfaces are essential. The euler number e(F ) of a spanning surface F is the algebraic self- intersection number of the properly embedded surface in the 4-ball obtained by perturbing F . Alternatively, −e(F ) can be computed by summing the component-wise boundary slopes of F .9 For brevity, we call −e(F ) the slope of F and denote −e(F ) = s(F ). 0 3 0 Let F and F be spanning surfaces for a link L ⊂ S with F t F 0 and ∂F t ∂F on ∂νL. Orient L arbitrarily, and orient ∂F and ∂F 0 so that each is homologous in νL to L. Then the algebraic 0 0 intersection number i(∂F, ∂F )∂νL equals s(F ) − s(F ). Given any spanning surface F for a link L ⊂ S3, Gordon-Litherland constructed a symmetric, bilinear pairing h·, ·i : H1(F ) × H1(F ) → Z as follows [2]. See Figure 3 Let νF be a neighborhood of F in the link exterior S3 \\νL with projection p : νF → F , such that p−1(∂F ) = νF ∩ ∂νL; denote Fe = ∂νF \\∂νL. Given any oriented −1 simple closed curve γ ⊂ F , denote γe = ∂(p (γ)), and orient γe following γ. Let τ : H1(F ) → H1(Fe) be the transfer map characterized

8That is, so that regions of the same color meet only at crossing points. 9 If L1,...,Lm are the components of ∂F , then the boundary slope of F along each Li equals the framing of Li in F , given by the linking number lk(Li, Lci), Pm where Lci is a co-oriented pushoﬀ of Li in F . Hence, s(F ) = i=1 lk(Li, Lci). 6 THOMAS KINDRED

Figure 3. An oriented curve γ on F , with γe on Fe.

10 by p∗ ◦ τ = 2 · 1. The Gordon-Litherland pairing

h·, ·i : H1(F ) × H1(F ) → Z is the symmetric, bilinear mapping given by the linking number ha, bi = lk(a, τ(b)).

Any projective homology class g ∈ H1(F )/± has a well-defined self- pairing hg, gi. When F is connected and g is primitive, there is a 1 simple closed curve γ ⊂ F representing g, and 2 hg, gi equals the framing of γ in F . Given an ordered basis B = (a1, . . . , am) on H1(F ), the Goeritz m×m matrix G = (xij) ∈ Z given by xij = hai, aji represents h·, ·i with respect to B.11 The signature of G is called the signature of F and is denoted σ(F ). Gordon-Litherland showed that the quantity σ(F ) − 1 2 s(F ) is independent of F , and in fact equals the Murasugi invariant ξ(L), which is the average signature of L across all orientations. They also showed that σ(F ) is the signature of the 4-manifold obtained by pushing the interior of F into the interior of the 4-ball B4, while fixing ∂F in ∂B4 = S3, and taking the double-branched cover of B4 along this surface. In particular, when L is a knot, ξ(L) is the signature of L and of the 4-manifold obtained as a double- branched cover of B4 along any perturbed Seifert surface. A spanning surface F is positive-definite if hα, αi > 0 for all nonzero α ∈ H1(F ) [3]. Equivalently, F is positive-definite iff σ(F ) = β1(F ). When F is connected, an equivalent condition, more geometric in flavor, is that F is positive-definite iff, for each simple closed curve γ ⊂ F , either: • The framing of γ in F is positive, or • γ bounds an orientable subsurface of F . Greene defines negative-definite surfaces analogously. Observe:

10 In other words, given any primitive g ∈ H1(F ), choose an oriented simple closed curve γ ⊂ int(F ) representing g, and construct γe as above; then, τ(g) = [γe]. 11 Pm Pm That is, any y = i=1 yiai and z = i=1 ziai satisfy T hy, zi = y1 ··· ym G z1 ··· zm . A GEOMETRIC PROOF OF THE FLYPING THEOREM 7

Proposition 2.3. If F+ and F−, respectively, are positive- and negative- deﬁnite spanning surfaces for the same nonsplit link L ⊂ S3, then

s(F+) − s(F−) = 2(β1(F+) + β1(F−)).

Proof. Deﬁniteness implies that β1(F±) = ±σ(F+), and [2] gives s(F±) = 2(σ(F±) − ξ(L)). Thus:

s(F+) − s(F−) = 2(σ(F+) − ξ(L)) − 2(σ(F−) − ξ(L)) = 2(β1(F+) + β1(F−)). Greene characterized alternating links in terms of definite surfaces: Theorem 1.1 of [3]. If B and W are positive- and negative-definite spanning surfaces for a link L in a homology Z/2Z sphere with irreducible complement, then L is an alternating link in S3, and it has an alternating diagram whose chessboards are isotopic to B and W . Moreover, this diagram is reduced if and only if neither B nor W has a homology class with self-pairing ±1.12 Convention 2.4. Isotopies of properly embedded surfaces and arcs are always taken rel boundary.13 Two properly embedded surfaces or arcs are parallel if they have the same boundary and are related by an isotopy which fixes this boundary. The converse of the first sentence of the theorem is also true: Proposition 4.1 of [3]. A connected link diagram is alternating if and only if its chessboard surfaces are definite and of opposite signs. Convention 2.5. If D is a connected alternating link diagram, then its chessboard surfaces B and W are labeled such that B is positive- definite and W is negative-definite. Likewise for chessboard surfaces 0 0 0 B and W (resp. Bi and Wi) from such a diagram D (resp. Di). Greene used Theorem 1.1 of [3] and lattice flows to give a geometric proof of part of Tait’s conjectures: Theorem 1.2 of [3]. Any two reduced alternating diagrams of the same link have the same crossing number and writhe. Alternatively, this theorem follows from the flyping theorem, since flypes preserve crossing number and writhe. Thus, our proof of the flyping theorem will give a new geometric proof of this theorem. Remark. Theorem 1.2 of [3] does not imply, a priori, that a reduced alternating diagram realizes the underlying link’s crossing number. All existing proofs of this fact [6, 10, 12, 15] use the Jones polynomial.

12A diagram D is reduced if each crossing abuts four distinct regions of S2 \D. 13For example, an isotopy of a spanning surface F ⊂ S3 \\νL is a homotopy h : F × I → S3 \\νL for which h(F × {t}) is a spanning surface at each t ∈ I. 8 THOMAS KINDRED

Problem 2.6. Give an entirely geometric proof that any reduced alternating diagram realizes the underlying link’s crossing number. More generally, Thistlethwaite proved that any adequate link diagram minimizes crossings. He then deduced that any reduced alternating tangle diagram minimizes crossings. See Corollary 3.4 of [13] and Definition 2.2 and Theorem 3.1 of [14]. Problem 2.7. Prove Corollary 3.4 of [13] geometrically. Problem 2.8. Give a geometric proof of Theorem 3.1 of [14]. 2.2. Operations on definite surfaces; arcs of intersection. Definite surfaces behave well under boundary-connect sum: Proposition 2.9. If F is a boundary-connect sum of positive-definite surfaces F1 and F2, then F is positive-definite.

Proof. Let Gi be a Goeritz matrix for Fi for i = 1, 2. Then each Gi has β1(Fi) positive eigenvalues (counted with multiplicity). Further, G 0 G = 1 0 G2 is a Goeritz matrix for F with β1(F ) positive eigenvalues. Next, consider subsurfaces of definite surfaces. Greene proved: Lemma 3.3 of [3]. If F is a definite surface and F 0 ⊂ F is a compact subsurface with connected boundary, then F 0 is definite. Here is a related fact: Proposition 2.10. If F 0 is a compact subsurface of a definite surface F , and every component of F \ F 0 intersects ∂F , then every component of F 0 is definite. 0 0 0 Proof. Let F0 be a component of F , let j : F0 → F denote inclusion, 0 and let g ∈ H1(F0) be primitive with j∗(g) = 0 ∈ H1(F ). Choose a 0 00 simple closed curve γ ⊂ F0 representing g. Then γ = ∂F for some 00 0 00 0 orientable F ⊂ F . The fact that γ ⊂ F0 implies that F ⊂ F , or else F 00 would intersect, and therefore contain, some component of 0 00 F \ F , implying contrary to assumption that ∂F 6= γ. Ergo, j∗ is 0 injective, so F is definite. In particular, Proposition 2.10 immediately implies: Lemma 2.11. If α is a system of disjoint properly embedded arcs in a definite surface F , then each component of F \ να◦ is definite. Note also: Proposition 2.12. Let F be a spanning surface for a prime, nonsplit link L ⊂ S3 and α a properly embedded arc in F . If α is not ∂-parallel in S3 \\νL, then the link L0 = ∂(F \ να◦ ) is nonsplit. A GEOMETRIC PROOF OF THE FLYPING THEOREM 9

Figure 4. Adding twists to a spanning surface

Proof. Assume for contradiction that L0 is split. Then there is a sphere Q ⊂ S3 \ L0 which intersects F along α, and perhaps some simple closed curves. The fact that L is prime implies, contrary to 3 assumption, that α is ∂-parallel in S \\νL. Next, consider the operation of adding (half) twists, shown in Fig- ure 4. It works like this. Let F be a spanning surface for a link L, α ⊂ F a properly embedded arc, and m an integer. Let A be an unknotted annulus or mobius band whose core circle has framing m 0 ◦ 0 2 , and let α be a properly embedded arc in A such that A \ να is a disk. Construct F \A in such a way that α and α0 are glued together to form an arc α00. Depending on the sign of m, the sur- 0 ◦ 00 m face F = (F \A) \ να is said to be obtained from F by adding 2 (positive or negative) twists along α. Proposition 2.13. If F 0 is obtained by adding positive twists to a positive-definite surface F , then F 0 is positive-definite.14 Indeed, if G is a positive-definite symmetric matrix and G0 is obtained by increasing a diagonal entry of G, then G0 is also positive- definite. Alternatively, here is a geometric proof: Proof. Let A be an unknotted annulus or mobius band with m half- twists for some m > 0. Then A is also positive-definite, as are F \A 0 and F , by Proposition 2.9 and Lemma 2.11. In the following way, one can compute the slope difference s(F ) − s(F 0) by counting, with signs, the arcs of F ∩ F 0. Given an arc α of F ∩ F 0, let ν∂α be a regular neighborhood of the endpoints ∂α in ∂νL. Then ∂F ∩ ν∂α and ∂F 0 ∩ ν∂α each consist of two properly embedded arcs, one in each disk of ν∂α. The two arcs in each disk intersect transversally in a single point, giving 0 i(∂F, ∂F )ν∂α ∈ {0, ±2}. Note: 0 X 0 s(F ) − s(F ) = i(∂F, ∂F )ν∂α arcs α of F ∩F 0

14Likewise for adding negative twists to a negative-deﬁnite surface. 10 THOMAS KINDRED

−→

Figure 5. A positive-deﬁnite surface F cannot intersect a negative-deﬁnite surface W along a non-∂- parallel arc α with i(∂F, ∂W )ν∂α = −2.

Lemma 2.14. If F and W respectively are positive- and negative- deﬁnite surfaces spanning a prime nonsplit link L ⊂ S3, and α is a non-∂-parallel arc of F ∩ W , then i(∂F, ∂W )ν∂α = 2.

Proof. Assume for contradiction that i(∂F, ∂W )ν∂α 6= 2. 0 ◦ First, consider the case i(∂F, ∂W )ν∂α = −2. Let F = F \ να, and let W 0 be the surface obtained by adding one (negative) half- twist to W along α. Then F 0 and W 0 span the same link L0, with 0 0 β1(F ) = β1(F ) − 1 and β1(W ) = β1(W ). See Figure 5. Lemma 2.11 implies that F 0 is positive-deﬁnite, and Proposition 2.13 implies that W 0 is negative-deﬁnite. Hence, L0 is alternating, by Theorem 1.1 of [3]. Further, L0 is nonsplit by Proposition 2.12. Therefore, by Corollary 5.2 of [1], F 0 and W 0 are both connected. Thus, by Observation 2.1 and Proposition 2.3:

s(F 0) − s(W 0) = s(F ) − s(W ) + 2

= 2(β1(F ) + β1(W )) + 2 0 0 > 2(β1(F ) + β1(W )).

This contradicts Proposition 2.3. Now, consider the case i(∂F, ∂W )ν∂α = 0. The argument here is identical to the ﬁrst case, except that one constructs W 0 by cutting ◦ 0 out να, rather than adding twists, giving β1(F ) = β1(F ) − 1 and 0 β1(W ) = β1(W ) − 1, hence:

s(F 0) − s(W 0) = s(F ) − s(W )

= 2(β1(F ) + β1(W )) 0 0 > 2(β1(F ) + β1(W )).

Again, this contradicts Proposition 2.3. A GEOMETRIC PROOF OF THE FLYPING THEOREM 11

= ∗ along

Figure 6. A plumbing cap and its shadow for a spanning surface, and the associated de-plumbing.

Figure 7. Re-plumbing a spanning surface replaces a plumbing shadow with its cap.

3. Generalized plumbing 3.1. Basic deﬁnitions. Let F be a spanning surface for a nonsplit link L (not necessarily alternating). A plumbing cap for F is a properly embedded disk V ⊂ S3 \\(F ∪νL) with the following properties: • The natural map f : S3 \\(F ∪ νL) → S3 restricts to an embedding f : V → S3, and f(∂V ) bounds a disk Ub ⊂ F ∪νL. 3 • Denoting the 3-balls of S \\(Ub ∪ V ) by Y1,Y2, neither subsurface Fi = F ∩ Yi is a disk. The disk U = Ub ∩ F is the shadow of V . If the ﬁrst property holds but the second fails, we call V a fake plumbing cap for F ; we still call U the shadow of V . The decomposition F = F1 ∪ F2 is a plumbing decomposition or de-plumbing of F along U and V , denoted F = F1 ∗ F2. See Figure 6. The reverse operation, in which one glues F1 and F2 along U to produce F , is called generalized plumbing or Murasugi sum. If V is a plumbing cap for F with shadow U, then one can construct another spanning surface F 0 = (F \ U) ∪ V ; we call the operation of changing F to F 0 re-plumbing F along U and V . See Figure 7. Call the analogous operation along a fake plumbing cap a fake re-plumbing. This is an isotopy move. Two spanning surfaces are plumb-related if there is sequence of re-plumbing and isotopy moves taking one surface to the other. Proposition 3.1. If V is a plumbing cap or fake plumbing cap for a spanning surface F , and U is the shadow of V , then F is plumb- related to (F \\U) ∪ V . Proof. If V is a fake plumbing cap, then F is isotopic to (F \\U)∪V . Otherwise, a re-plumbing move takes F to (F \\U) ∪ V . 12 THOMAS KINDRED

3.2. Re-plumbing in S3 and proper isotopy through B4. As an aside, it is interesting to note that plumb-related surfaces in S3 are properly isotopic in B4. More precisely, let L be a link in S3, and for 3 i = 1, 2, let Fi ⊂ S be an embedded surface with ∂Fi = L, and let 0 4 Fi be the properly embedded surface in B obtained by perturbing 3 int(Fi), while ﬁxing ∂Fi = L ⊂ S . With this setup:

Proposition 3.2. Assume F1 and F2 are plumb-related. Then: 0 0 4 • F1 and F2 are related by an ambient isotopy of B which fixes S3 ⊃ L. • There is an isomorphism φ : H1(F1) → H1(F2) satisfying hα, βi = hφ(α), φ(β)i for all α, β ∈ H (F ). F1 F2 1 1 • If F1 is definite, then F2 is definite of the same sign. • In particular, if F1 is a chessboard surface from a reduced alternating diagram, then so is F2. Proof. The first statement follows from the observation that any re- plumbing move can be realized as an isotopy through B4 in which one fixes the entire surface except the plumbing shadow and pushes the plumbing shadow through B4 to the plumbing cap. The second then follows from Theorem 3 of [2], which states that the pairing on Fi corresponds to the intersection form on the 2-fold branched cover 4 0 0 0 of B with branch set Fi , since F1 and F2 give the same branched cover. The last two statements follow immediately, using [3]. 3.3. Acceptable and irreducible plumbing caps. Definition 3.3. A plumbing cap V is acceptable if both: • No arc of ∂V ∩ ∂νL is parallel in ∂νL to ∂F . • No arc of ∂V ∩ F is parallel in F \\U to ∂F . Any, possibly fake, plumbing cap V for F can be adjusted near any arcs described in Definition 3.3 so as to remove those arcs one at a time. Doing so for a plumbing cap eventually yields an acceptable plumbing cap V 0. With this setup: Observation 3.4. The surfaces obtained by re-plumbing F along V and V 0 are isotopic. In the fake case, the same procedure gives an F -parallel disk, so there is no such thing as an acceptable fake plumbing cap: Observation 3.5. No fake plumbing cap satisfies both properties from Definition 3.3. The first bulleted property in Definition 3.3 implies that if V is an acceptable plumbing cap with shadow U then (∂U ∪ ∂V ) ∩ ∂νL is a system of circles on ∂νL, each isotopic to a meridian. Consider one such circle α ∪ β, where α is an arc of ∂V ∩ ∂νL and β an arc of A GEOMETRIC PROOF OF THE FLYPING THEOREM 13

Figure 8. Given an acceptable plumbing cap V for F , each arc of ∂V ∩ F joins arcs of V ∩ ∂νL with opposite signs, here −1 left and +1 right.

∂U ∩ ∂νL. Since β ⊂ ∂F is transverse to each meridian on ∂νL, we may assign opposite local orientations to β and πL(β) ⊂ L. Extend these local orientations to global orientations on L. Convention 3.6. With the setup above, sign(α) = lk(L, α ∪ β); α is called positive if sign(α) = +1 and negative if sign(α) = −1. Proposition 3.7. For any acceptable plumbing cap V , the arcs of ∂V ∩ ∂νL alternate in sign around ∂V . Proof. Consider any arc c of ∂V ∩ F . Orient c so that U ∩ νc lies to the left of c when viewed from V ∩ νc. Then the initial point of c has sign +1 and the terminal point has sign −1. See Figure 8. A plumbing cap V is reducible if there is a properly embedded disk X ⊂ S3 \\(νL ∪ F ∪ V ) such that ∂X = α ∪ β for arcs α ⊂ U, β ⊂ V , and both disks V1,V2 obtained by surgering V along X are (non-fake) plumbing caps for F ; X is a reducing disk. If no such X exists, then V is irreducible. See Figure 9. Proposition 3.8. If V is an acceptable, irreducible plumbing cap for F with shadow U, and if ε is a properly embedded arc in U with neither endpoint on ∂νL which is parallel in S3 \\(F ∪ νL ∪ V ) to an arc ε0 ⊂ V , then ε is parallel in U to ∂V . Proof. Let W be a properly embedded disk in S3 \\(F ∪νL∪V ) with 0 ∂W = ε ∪ ε . Surger V along W to obtain two disks V1,V2; since V is irreducible, we may assume wlog that V1 is a fake plumbing cap. Denote the shadow of each Vi by Ui. If ∂V1 ⊂ F , then ε is parallel through U1 to ∂F . Assume otherwise. Then, because V1 is fake, Observation 3.5 implies that either some arc of ∂V1 ∩ ∂νL is parallel in ∂νL to ∂F , or some arc ρ of ∂V1 ∩ F is parallel in F \\U1 to ∂F . Yet, V is acceptable, and ∂V contains all arcs of ∂V1 ∩ ∂νL and all but one arc of ∂V1 ∩ F . Therefore, the one arc ρ of ∂V1 ∩ F which ∂V does not contain must cut oﬀ a disk 14 THOMAS KINDRED

Figure 9. A reducing disk for a reducible plumbing cap; the arc ρ in the proof of Proposition 3.8 and the neighborhood Y in the proof of Lemma 3.10. from F \\U. That disk, however, must contain U2, implying that V2 is also a fake plumbing cap. This implies, contrary to assumption, that V was a fake plumbing cap as well.

Proposition 3.9. If X is a reducing disk for an acceptable plumbing cap V , then the disks V1,V2 obtained by surgering V along X both 15 satisfy |Vi ∩ F | < |V ∩ F |. Moreover, the surface obtained by re- plumbing along V is isotopic to the surface obtained by re-plumbing along V1 and then along V2. Proof. Let ∂X = α ∪ β, where α ⊂ U and β ⊂ V . If necessary, perturb X so that neither point of ∂α = ∂β lies on ∂F . Then |V1 ∩ F | + |V2 ∩ F | = |V ∩ F |. Moreover, both |Vi ∩ F | > 0, since α is not parallel in U to ∂F . This conﬁrms the ﬁrst statement Next, let Ui denote the shadow of Vi for i = 1, 2, and let Y be a bicollared neighborhood of X in S3 \\(F ∪νL∪V ), so that ∂Y ∩F = να is a regular neighborhood in U of α and ∂Y ∩ V = νβ is a regular neighborhood in V of β. Note that ∂Y \ ν◦ (α ∪ β) consists of two disks, X1 ⊂ V1 and X2 ⊂ V2. See Figure 9. Re-plumb F along V1 and then V2 by replacing U1 with V1 and then U2 with V2; next, push the disk X1 ∪ να ∪ X2 through Y to νβ. These two moves together replace U1 ∪ U2 ∪ να = U with

((V1 ∪ V2) \\(X1 ∪ X2)) ∪ νβ = V, the same as re-plumbing F along V . Together, Observation 3.4 and Proposition 3.9 immediately imply: Lemma 3.10. If F and F 0 are plumb-related, then there is a sequence of re-plumbing and isotopy moves taking F to F 0 in which each re-plumbing move follows an acceptable irreducible plumbing cap.

15Here and throughout, | − | counts the connected components of −. A GEOMETRIC PROOF OF THE FLYPING THEOREM 15

Figure 10. A link near a crossing ball with S+ and S−.

4. Crossing balls and plumbing caps Section 4 uses the crossing ball structures introduced in [7] to study plumbing caps for chessboards from alternating link diagrams. 4.1. Crossing ball setup. Let D be a reduced diagram (not necessarily alternating) of a prime nonsplit link L with crossings c1, . . . , cn, and let νS2 be a neighborhood of S2 with projection π : νS2 → S2.16 ◦ 2 Insert disjoint closed crossing balls Ct into νS , each centered at the Fn respective crossing point ct. Denote C = t=1 Ct. Perturb each Ct 2 near its equator ∂Ct ∩ S so that, for the balls Y+,Y− comprising 3 2 S \\(S ∪ C), each π|∂Y± is a diﬀeomorphism. Construct a (smooth) embedding of L in (S2 \ int(C)) ∪ ∂C by perturbing the arcs of D ∩ C following the over-under information at the crossings, while ﬁxing D∩S2\int(C). Call the arcs of L∩S2 edges, and those of L ∩ ∂C ∩ ∂Y± overpasses and underpasses, respectively. Near each crossing, this looks like Figure 10, center. Let νL be a closed regular neighborhood of L in νS◦ 2 with projection map πL : νL → L. For each edge e ⊂ L, call the cylinder −1 E = πL (e) ∩ ∂νL an edge of ∂νL; call the rectangles E ∩ Y± the top and bottom of the edge, respectively. For each over/underpass e± of −1 L, call E± = πL (e±)∩∂νL an over/underpass of ∂νL. Call E+ ∩Y+ and E+\\Y+ the top and bottom of the overpass, respectively, and call E− ∩ Y− and E− \\Y− the bottom and top of the underpass, respec- −1 tively. Assume that the meridia comprising πL (L ∩ ∂C ∩ S+ ∩ S−) −1 also comprise ∂νL∩π (∂C ∩S+ ∩S−). Then these meridia cut ∂νL into its edges, overpasses, and underpasses. 3 2 Denote the two balls of S \\(S ∪ C ∪ νL) by H±, so that each ◦ H± = Y± \ νL. Also denote ∂H± = S±. See Figure 10. −1 ◦ For each crossing ct, denote the vertical arc π (ct) ∩ Ct \ νL by S 2 ◦ vt, and denote v = t vt. For each t, ∂Ct ∩ S \ νL consists of four arcs on the equator of ∂C. Two of these arcs lie in black regions of S2 \ D, two in white. For the two arcs α, β in black regions, a core circle in α ∪ β ∪ (∂νL ∩ Ct) bounds a disk Bt ⊂ Ct such that π(Bt) is 2 disjoint from the white regions of S \ D and intersects D only at ct.

16The assumption that L is prime implies that L is nontrivial. 16 THOMAS KINDRED

This disk Bt contains the vertical arc vt and is called the standard positive crossing band at Ct. The arcs in the white regions likewise give rise to a standard negative crossing band Wt in Ct; note that ◦ Bt ∩Wt = vt. Any properly embedded disk in Ct \νL which contains ◦ vt and is isotopic in Ct \νL to Bt (resp. Wt) is called a positive (resp. negative) crossing band. See Figure 2. Denote the union of the black (resp. white) regions of S2 \ D by Bb (resp. Wc). Then [ (1) B = Bb \ int(C ∪ νL) ∪ Bt t and [ (2) W = Wc \ int(C ∪ νL) ∪ Wt t are the black and white chessboard surfaces from D.17 3 Denote the two balls of S \\(B ∪ W ∪ νL) by Hd±, such that each Hd± ⊃ H±; also denote ∂Hd± = Sc±. 4.2. Plumbing cap setup. Keeping the setup from §4.1, assume that D is alternating and V is an acceptable plumbing cap for B. Deﬁnition 4.1. V is in standard position if: • V is transverse in S3 to B, W, ∂C, v, and ∂V ∩ ∂νL is transverse in ∂νL to each meridian on ∂νL; • no arc of ∂V ∩ B \\∂C is parallel in B to ∂C; 18 • ∂V ∩ ∂νL ∩ C = ∅; and • W intersects V only in arcs, hence cuts V into disks. Let us check that it is possible to position V in this way. The ﬁrst three conditions are straightforward. Achieve the fourth condition by removing any simple closed curves (“circles”) of V ∩ W one at a time through the following procedure. Choose a circle γ of V ∩W which bounds a disk V 0 of V \\W . Since W is incompressible, γ also bounds a disk W 0 ⊂ W . Choose a circle 0 0 00 0 0 γ ⊂ W ∩V which bounds a disk W of W \\V . (If int(W )∩V = ∅, then γ0 = γ.) The circle γ0 also bounds a disk V 00 ⊂ V . The sphere

17Note that B ∩ W = v. Also: 2 S+ ∩ S− = S \ int(C ∪ νL) = (B ∪ W ) \ int(C), and 2 ◦ S+ ∪ S− = (S \ int(C ∪ νL)) ∪(∂C \ νL) ∪ (∂νL \ int(C)). | {z } =(B∪W )\int(C)

18That is, ∂V is disjoint from the top of each overpass and bottom of each underpass of ∂νL. Hence, for any overpass that ∂V intersects, ∂V traverses the entire top of the overpass; likewise for underpasses of ∂νL. A GEOMETRIC PROOF OF THE FLYPING THEOREM 17

Figure 11. Each arc of ∂V ∩ ∂νL disjoint from W is negative and traverses exactly one over/underpass.

V 00 ∪ W 00 bounds a ball Y in the link exterior, since L is nonsplit. Push V 00 through Y past W 00, removing γ0 from V ∩ W . Repeat this process until W intersects V only in arcs, and thus cuts V into disks. Observation 4.2. If V is in standard position, then all components of V ∩ Hd± are disks, and all components of Hd± \\V are balls.

Observation 4.3. If V is in standard position, then each arc of ∂V ∩ ∂νL which is disjoint from W is a negative arc that traverses exactly one overpass or underpass (along the top or bottom, respectively). This follows from Convention 3.6 and the fact that D is alternating; see Figure 11. Observation 4.3 implies, in particular: Observation 4.4. Any positive arc of ∂V ∩ ∂νL lies entirely on a single edge of ∂νL and contains an endpoint of an arc of V ∩ W .

Proposition 4.5. V ∩W 6= ∅, and each arc α of ∂V \\W intersects at most two disks of B \\W .

Proof. Observation 4.4 and Proposition 3.7 imply that V ∩ W 6= ∅. Consider an arc α of ∂V \\W . Proposition 3.7 and Observation 4.4 imply that α intersects at most one negative arc of ∂V ∩ ∂νL. Also, each arc of α \\∂νL lies in a single disk of B \\W . The result now follows from Observations 4.3 and 4.4.

Next, consider an outermost disk V0 ⊂ V \\W . Its boundary consists of an arc α ⊂ ∂V \\W and an arc β ⊂ V ∩ W . Let να be a regular neighborhood of α in ∂V with να ∩ ∂B ⊂ α.

Lemma 4.6. If such V0 ⊂ Hd+, then V0 appears as in Figure 12. In particular, the endpoints of να lie in opposite disks of Sc+ \\∂V0.

Proof. Let ∂V0 = α ∪ β as above. Since V is in standard position, the endpoints of α cannot lie on the same vertical arc or edge of ∂νL. This and the fact that D is prime and reduced imply that α must traverse an overpass. Hence, by Proposition 4.5, α intersects 18 THOMAS KINDRED

Figure 12. The possible conﬁgurations for V near an innermost circle of V ∩ Sc+. exactly two disks of B \\W , and its conﬁguration depends only on its endpoints, each of which lies either on v or on a positive arc of ∂V ∩ ∂νL. Figure 12 shows the three possibilities.

With V0 as in Lemma 4.6: Observation 4.7. There is exactly one arc δ of ∂V \\W whose endpoints lie in opposite disks of Sc+ \\∂V0. This arc δ traverses the underpass at the same crossing where ∂V0 traverses the overpass. 4.3. Plumb-equivalence and ﬂype-equivalence. Maintain the setup from §§4.1-4.2. In particular, D is a reduced alternating diagram of a prime nonsplit link L; B and W are the chessboard surfaces from D; and a plumbing cap V for B is in standard form. Say that V is apparent in D if |V ∩ W | = 1. Lemma 4.6 implies that any apparent plumbing cap for B appears as in Figure 13, left. Proposition 4.8. If V is apparent, then V is irreducible. Proof. If V were reducible, then it would be possible to surger V to give two plumbing caps V1,V2 with both |∂Vi∩∂νL| < |∂V ∩∂νL| = 4. Since both |∂Vi ∩ ∂νL| must be even, they must both equal two. But then V1, V2 would each decompose F as a boundary connect sum. This is impossible because F is essential and L is prime. Theorem 4.9. If V is an acceptable, irreducible plumbing cap for B in standard position, then V is apparent in D.

Proof. Proposition 4.5 implies that V ∩ W 6= ∅. Assume for contradiction that |V ∩ W | > 1. Then there is a disk V1 ⊂ V whose boundary consists of two arcs, α ⊂ ∂V and β ⊂ V ∩ W , and whose interior intersects W in a nonempty collection of arcs β1, . . . , βk, such that each βi is parallel through a disk Xi ⊂ V1 to an arc αi ⊂ α. Assume wlog that the disks Xi all lie in Hd−. Denote γ = α ∪ β = ∂V1, and for each i = 1, . . . , k, denote the circle αi ∪ βi = ∂Xi by γi. Then γ1, . . . , γk are disjoint simple closed curves in Sc+. Relabel if necessary so that γ1 bounds a disk Y ⊂ Sc+ A GEOMETRIC PROOF OF THE FLYPING THEOREM 19

Figure 13. Every plumbing cap for W is either iso- topically apparent (left) or reducible (right).

19 0 which is disjoint from β∪γ2 ∪· · ·∪γk. Let δ, δ be the arcs of ∂V \\W 0 that each share an endpoint with α1. The other endpoints of δ, δ 20 must both lie in the disk Sc+ \\Y . Lemma 4.6 implies wlog that the endpoints of δ lie in opposite disks of Sc+ \ ∂γ1. Hence, Observation 4.7 implies that δ traverses the underpass at the same crossing where γ1 traverses the overpass. Moreover, ∂V is disjoint from the vertical arc at this crossing, or else an arc of ∂V \\W would have both endpoints on that vertical arc. Therefore, there is a properly embedded arc ε in the shadow U of V which is disjoint from W and has one endpoint on δ ∩ int(B) and the other on δ0 ∩ int(B), as in Figure 13, right. Further, since δ and 0 δ both lie on the boundary of the disk V1 \\(X1 ∪· · ·∪Xk) ⊂ V \W , that disk contains an arc ε0 with the same endpoints as ε. The arcs ε 0 and ε lie on the boundary of the same ball of Hd−.Thus, ε is parallel in S3 \\(F ∪νL∪V ) to the arc ε0 ⊂ V , and so Proposition 3.8 implies that ε is parallel in U to one of the two arcs of ∂V \\∂ε. The arc of ∂V \\∂ε which contains α1 is not a subset of ∂U, and so ε must be parallel in U to the other arc of ∂V \\∂ε. But then that arc is disjoint from W and so |V ∩ W | = 1, contrary to assumption. Given an apparent plumbing cap V for B (or W ), there is a corresponding flype move, as shown in Figure 14. Namely, the flype move proceeds along an annular neighborhood of a circle γ ⊂ S2 comprised of the arc V ∩ W together with an arc in U ∪ νL. (The resulting link diagram might be equivalent to D.) Conversely, if D → D0 is a flype move along an annulus A ⊂ S2, then there is an apparent plumbing cap V for B (or W ) with V ∩ W ⊂ A (resp. V ∩ B ⊂ A).

19 This is possible because at least two circles among γ1, . . . , γk bound disks of Sk Sc+ \\ i=1 γi, and most one of these disks intersects β. 20 This is because these endpoints lie on β ∪ γ2 ∪ · · · ∪ γk ⊂ Sc+ \\Y . 20 THOMAS KINDRED

T T 2 2 2 T

T T 1 1

Figure 14. A ﬂype move corresponds to an isotopy of one chessboard surface (here, W ) and a re- plumbing of the other.

Proposition 4.10. Let V be an apparent plumbing cap for B, D → D0 the ﬂype move corresponding to V , B0 and W 0 the chessboard surfaces from D0, and B00 the surface obtained by re-plumbing B along V . Then B0 and B00 are isotopic, as are W 0 and W .21

Proof. Figure 14 demonstrates the isotopies. Theorem 4.11. Let D and D0 be reduced alternating diagrams of a prime, nonsplit link L with respective chessboard surfaces B,W and B0,W 0. Then D and D0 are flype-related if and only if B and B0 are plumb-related as are W and W 0. Greene’s characterization in [3] implies that another reduced alternating diagram D00 of L has chessboard surfaces which are isotopic to B0 and W . We will see in the proof of Theorem 4.11 that D00 0 (which will be denoted Dm) is related to D (resp. D ) by a sequence of flypes each of which corresponds to an isotopy of W (resp. B0). Proof. If D and D0 are flype-related, then B and B0 are plumb- related as are W and W 0, by Proposition 4.10. Assume conversely that B and B0 are plumb-related, as are W and W 0. Then, by Lemma 3.10, there are two sequences of re-plumbing 0 0 moves B = B0 → · · · → Bm = B and W = W0 → · · · → Wn = W , such that each re-plumbing move Bi → Bi+1 (resp. Wi → Wi+1) 0 follows an acceptable, irreducible plumbing cap Vi (resp. Vi ). Let D0 = D. Put the plumbing cap V0 in standard position with respect to D. Theorem 4.9 implies that V0 is now apparent. Let D0 → D1 be flype move corresponding to V0. By Proposition 4.10, the negative-definite chessboard surface from D1 is isotopic to W , and the positive-definite chessboard surface from D1 is isotopic to B1. Repeat in this manner. Ultimately, this gives a sequence of flypes from D = D0 to a diagram Dm of L whose positive-definite chessboard surface is isotopic to B0 and whose negative-definite chessboard surface is isotopic to W . To complete the proof, construct a sequence 0 0 0 of flypes Dm = D0 → · · · → Dn = D in the same manner, but now

21An analogous statement holds for apparent plumbing caps for W . A GEOMETRIC PROOF OF THE FLYPING THEOREM 21

Figure 15. Positive (left) and negative (right) crossing bands in a surface F in fair (or good) position. with the ﬂypes coming from the (apparent) acceptable, irreducible 0 plumbing caps Vi for Wi. 5. Spanning surfaces in the crossing ball setting

Adopt the setup from 4.1 of D, B, W , C, νL, H±, S±, etc. Now, also let F be an incompressible spanning surface for a prime nonsplit link L. Keep this setup throughout §5. The opening of each subsection of §5 will declare any additional hypotheses for that subsection. Starting in §5.4, these hypotheses will become increasingly specific regarding F and L. Namely: • Starting in §5.4, F is essential. • Starting in §5.5, L and D are alternating, hence B is positive- definite, and W is negative-definite. • Starting in §5.6, F is definite. 5.1. Fair position for F . Definition 5.1. F is in fair position if: • F is transverse in S3 to B, W , ∂C, and v; • ∂F is transverse in ∂νL to each meridian on ∂νL; • whenever Ct ∩ ∂F 6= ∅, Ct contains a crossing band in F ; • each crossing band in F is disjoint from S+; and • S+ ∪ S− cuts F into disks. (Later, we will define a slightly more restrictive good position for F .) It is certainly possible to position F so that the first four conditions hold. One way to achieve the fifth condition is by minimizing 22 |F \ (S+ ∪ S−)|, subject to the first four conditions.

22 If some component X of F \\(S+ ∪ S−) is not a disk, choose a component of ∂X and push it into int(X). The resulting circle γ is 0-framed in F , bounds 3 no disk in X, and does bound a disk in S \ (S+ ∪ S− ∪ νL). Among all such disks, choose one Z t F with minimal |Z ∩ F |. Since γ is 0-framed in F , Z ∩ F is comprised of circles. Choose an innermost disk Z0 ⊂ Z \\F , and denote ∂Z0 = γ0. The incompressibility of F implies that γ0 bounds a disk X0 in F . 3 The 2-sphere X0 ∪ Z0 bounds a ball Y ⊂ S \\νL whose interior is disjoint from 22 THOMAS KINDRED

Figure 16. The three types of bigon moves; the mid- dle one also appears in Figure 17.

The fourth condition implies that every crossing band in F appears as in Figure 15. This creates an asymmetry between F ∩ S− versus F ∩S+ which will be strategically useful. The idea is that by pushing F ∩ (S+ ∪ S−) into S− as much as possible, the remaining circles of F ∩ S+ will be more constrained; ultimately, these added constraints will enable the key re-plumbing move in the proof of Theorem 5.26. Proposition 5.2. If F is in fair position, then all components of C \\F and H± \\F are balls, and all components of F ∩ S+ ∩ S−, F ∩ ∂C ∩ S±, and ∂F ∩ S± are arcs. Proof. By assumption, F intersects each crossing ball in disks, hence it cuts these crossing balls into balls. Likewise for H±. All components of ∂C ∩ S+, ∂C ∩ S−, and S+ ∩ S− are disks. If F intersected one of these disks in a circle, γ, then, since all components of F \\(S+ ∪ S− ∪ νL) are disks, γ would bound disks of F in both components of F \\(S+ ∪S− ∪νL) whose boundaries contain γ. This contradicts the assumption that F is connected. The assumption that D is nontrivial and nonsplit implies that each component of ∂F ∩ S± is an arc. 5.2. Bigon Moves.

Deﬁnition 5.3. Let F be in fair position and α ⊂ S± an arc that: • intersects F precisely on its endpoints, which lie on the same circle γ of F ∩ S±, but not on the same arc of F ∩ S+ ∩ S−; • is disjoint from over/underpasses; and • intersects S+ ∩ S− in exactly one component. Then α is parallel through a properly embedded (“bigon”) disk Z ⊂ H± \\F to an arc β ⊂ F ∩ H±, and one can push F ∩ νβ through νZ past α, as in Figures 16 and 17. This is called a bigon move.

Proposition 5.4. Given an arc α ⊂ S± as in Deﬁnition 5.3, the endpoints of α do not lie on the same circle of F ∩ S∓.

F . The minimality of |Z ∩ F | implies that X0 ∩ (S+ ∪ S−) 6= ∅; also, X0 must be disjoint from all crossing bands and must contain any saddle disks that it intersects. Therefore, it is possible to push F near X0 through Y past Z0, thus decreasing |F \ (S+ ∪ S−)|, while ensuring that the resulting positioning satisﬁes the ﬁrst four conditions. This contradicts the minimality of |F \ (S+ ∪ S−)|. A GEOMETRIC PROOF OF THE FLYPING THEOREM 23

Figure 17. One of three types of bigon moves.

Proof. If either endpoint of α is on ∂νL, instead of in S+ ∩ S−, there is nothing to prove. Assume that α ⊂ S+ ∩ S−, and assume for contradiction that γ±, respectively, are circles of F ∩ S±, both of which contain both endpoints of α. Let X±, respectively, denote the disks of F ∩ H± with ∂X± = γ±. Choose arcs β± ⊂ X± with the same endpoints as α, and let δ± = α ∪ β±. The circles δ± bound properly embedded disks Z± ⊂ H± \\F . Gluing Z+ and Z− along α produces a disk Z with Z ∩ F = ∂Z = β+ ∪ β−. Since F is incompressible, β+ ∪ β− must bound a disk X ⊂ F . The boundary of this disk X is the curve β+ ∪β−, which intersects S+∩S− only in two points. Therefore, X intersects S+∩S− in a single arc, and this arc has the same endpoints as α. Contradiction. Lemma 5.5. Performing a bigon move on a surface in fair position produces a surface in fair position. Proof. Consider a bigon move along an arc α. Assume wlog that the endpoints of α lie on the same circle γ+ of F ∩ S+. Then, by 0 Proposition 5.4, the endpoints of α lie on distinct circles γ−, γ− of F ∩ S−. Let X+ be the disk of F ∩ H+ bounded by γ+, and let X− 0 0 and X− be the disks of F ∩ H− bounded by γ− and γ−, respectively. 0 Since X− 6= X−, the bigon move along α must involve pushing an arc β+ ⊂ X+ past α. This has the effect of splitting X+ into two 0 distinct disks of F ∩H+, while merging X− and X− into a single disk of F ∩ H−. The rest of F is unaffected. Thus, this move preserves the fact that S+ ∪ S− cuts F into disks. Hence, it produces a surface in fair position. 5.3. Good position for F . Definition 5.6. F is in good position if: • F is in fair position, • no arc of ∂F ∩ S± has both endpoints in the same arc of S+ ∩ S− ∩ ∂νL, and • no arc of F ∩ ∂C ∩ S± has both endpoints on the same arc of ∂C ∩ S+ ∩ S−. Given a surface F in fair position, it is possible to use the moves in Figures 18 and 19 to remove any arcs described in Definition 5.6, all 24 THOMAS KINDRED

Figure 18. Removing arcs of ∂F ∩ S± with both endpoints on the same arc of S+ ∩ S− ∩ ∂νL.

Figure 19. Removing arcs of F ∩∂C ∩S± with both endpoints on the same arc of ∂C ∩ S+ ∩ S−. while preserving the conditions of fair position. That is, any surface in fair position admits local adjustments that put it in good position. Proposition 5.7. If F is in good position, then each component of F ∩ C is either a crossing band or a saddle disk as in Figures 15, 20.

Proof. Consider a crossing ball Ct where F does not have a crossing band. Each component γ of F ∩ Ct is a circle because F does not have a crossing band at Ct, and so, by assumption, ∂F ∩ ∂Ct = ∅. Also, by assumption, each such circle γ bounds a disk of F ∩ Ct. Moreover, Proposition 5.2 implies that S2 cuts each such γ into arcs; by assumption, the endpoints of each arc of γ \\S2 are on distinct arcs of ∂Ct ∩ S+ ∩ S−. Since each disk of ∂Ct ∩ S± contains only two arcs of of ∂Ct ∩ S+ ∩ S−, the result follows.

5.4. Essential spanning surfaces. Assume in §5.4 that F is an essential surface in good position. Deﬁne the complexity of F to be (3) ||F || = |v \\F |. A GEOMETRIC PROOF OF THE FLYPING THEOREM 25

Figure 20. When F is in good position, F ∩ C is comprised of crossing bands and saddle disks.

Figure 21. Left: If an arc β of ∂F ∩ S− is disjoint from underpasses, then its endpoints lie on distinct circles of F ∩ S+. Right: In particular, no arc of F ∩ S+ ∩ S− is parallel in S+ ∩ S− to ∂νL.

Since F is in good position, Proposition 5.7 implies that any disk R of S+ ∩ S− and any crossing ball Ct incident to R satisfy: ( 0 F has a crossing band at Ct (4) |vt \\F | = |∂R ∩ ∂Ct ∩ F | otherwise.

Proposition 5.8. For each arc β of ∂F ∩ S± that traverses no over/underpasses, the endpoints of β lie on distinct circles of F ∩S∓. Proof. Assume for contradiction that the endpoints of such an arc β lie on the same circle γ of F ∩ S∓, and let X be the disk of F ∩ H∓ bounded by γ. Then X contains a properly embedded arc α with the same endpoints as β. The fact that β traverses no overpasses implies 3 0 that α is parallel in S \\νL to an arc α on ∂νL ∩ S∓ (drawn yellow in Figure 21, left). Since F is in good position, the shared endpoints 0 of these arcs must lie on distinct disks of S+ ∩ S−. Therefore, α ∪ β is a meridian on ∂νL; hence, α0 is not parallel in ∂νL to ∂F . This contradicts Proposition 2.2.

Proposition 5.9. No arc of F ∩ S+ ∩ S− is parallel in S+ ∩ S− to ∂νL. 26 THOMAS KINDRED

Figure 22. No arc of ∂F ∩ S+ ∩ S− has both endpoints on the same crossing ball.

Proof. If such an arc exists, then choose one, α which is outermost in S+ ∩ S−. Up to mirror symmetry, α must appear as in Figure 21, right. This is impossible, by Proposition 5.8.

Proposition 5.10. If |v\\F | is minimal, then no arc of ∂F ∩S+∩S− has both endpoints on the same crossing ball.

Proof. If both endpoints of an arc of ∂F ∩ S+ ∩ S− lie on the same crossing ball, then there is an outermost such arc α in S+ ∩S−. Both endpoints of α lie on saddle disks, by 5.7, so α abuts two saddle disks. The sequence of isotopy moves shown in Figure 22 decreases |v \\F |, so it was not minimal.

Proposition 5.11. If |v \\F | is minimized and a circle γ of F ∩ S± traverses the over/underpass at a crossing ball Ct, then γ ∩∂Ct = ∅.

Proof. Assume for contradiction that γ traverses the overpass at Ct and γ ∩ ∂Ct 6= ∅. Then there is a disk R of S+ ∩ S− containing a point x of γ ∩ ∂Ct ∩ S+ ∩ S−. Also, as shown in Figure 23, there is an arc α ⊂ S+ disjoint from overpasses with α ∩ S+ ∩ S− ⊂ R and ∂α ⊂ γ; Proposition 5.10 implies that such may position α with α ∩ F = ∂α.23 Following Figure 23, perform a bigon move along α, and then perform an additional isotopy move, which removes a saddle disk and thus decreases |v \\F |, contrary to assumption. In particular, Proposition 5.11 implies the following two facts:

Proposition 5.12. If |v \\F | is minimized, no arc α of F ∩S+ ∩S− has one endpoint on C and the other on an incident edge of ∂νL. Proof. If such an arc exists, choose one, α, which is outermost in S+ ∩ S−. Let x denote the endpoint of α on a crossing ball Ct, and assume wlog that the edge of ∂νL containing the other endpoint y of α abuts the overpass at Ct. Consider the arc β of ∂F on this edge between y and the overpass at Ct.

23The proof of Proposition 5.14 employs a similar argument and provides greater detail on this point. A GEOMETRIC PROOF OF THE FLYPING THEOREM 27

Figure 23. If |v \\F | is minimized and γ ⊂ F ∩ S+ traverses the overpass at Ct, then γ ∩ Ct = ∅.

If β∩S+∩S− = ∅, the result follows immediately from Proposition 5.11. Otherwise, by Proposition 5.9 and the fact that α is outermost, β intersects S+ ∩S− in exactly one point. In this case, there is now a bigon move which causes the circle of F ∩ S+ traversing the overpass 24 at Ct to intersect ∂Ct. This contradicts Proposition 5.11. Proposition 5.13. If |v \\F | is minimized, then for each crossing ball Ct where F does not have a crossing band, the arcs of F ∩∂Ct∩S± all lie on distinct circles of F ∩ S± from one another and from the arc of ∂F ∩ S± ∩ ∂νL which traverses the over/underpass at Ct.

Proof. Otherwise, a bigon move near Ct gives an immediate contradiction to Proposition 5.10 or Proposition 5.12. Proposition 5.14. If |v \\F | is minimized and a circle γ of F ∩ S± traverses the over/underpass of ∂νL at a crossing Ct, then γ is disjoint from both edges of ∂νL incident to the under/overpass at Ct.

Proof. Assume for contradiction that a circle γ of F ∩ S+ traverses the overpass of ∂νL at Ct and intersects an edge of ∂νL incident to the underpass at Ct. Then this edge of ∂νL contains an endpoint of an arc of γ ∩ S+ ∩ S−. Denote the disk of S+ ∩ S− containing this endpoint by R. As shown in Figure 24, there is an arc α ⊂ S+ disjoint from overpasses with α ∩ S+ ∩ S− ⊂ R and ∂α ⊂ γ. Choose α such that α t F and all points of int(α) ∩ F lie on distinct arcs of F ∩ S+. We claim that α is properly embedded in S+ \\F , i.e. α∩F = ∂α. Suppose otherwise. Then there is an arc α0 of α \\F sharing neither endpoint with α, such that both endpoints of α0 lie on the same circle 0 of F ∩S+. Perform a bigon move along α . Consider the resulting arc

24 To see the bigon move, let ρ be the arc of F ∩ S+ ∩ S− with an endpoint on 0 int(β); let α be the arc of F ∩ ∂Ct ∩ S− sharing an endpoint with α; and let σ be 0 the arc of F ∩ S+ ∩ S−, other than α that shares an endpoint with α . The bigon move is along an arc in S+ ∩ S− with one endpoint on ρ and the other on σ. 28 THOMAS KINDRED

Figure 24. No circle of F ∩ S+ intersects both the top of the overpass at Ct and an edge of ∂νL incident to the underpass at Ct.

β that lies in the disk of R \\(γ ∪ α) incident to Ct. Each endpoint of β lies either on the arc ∂Ct ∩ R or on the arc of ∂νL ∩ R incident to the underpass at Ct. This is impossible, by Propositions 5.10, 5.9, and 5.12. Therefore, α is properly embedded in S+ \\F . Following Figure 24, perform a bigon move along α, and then adjust F to have a crossing band at Ct. This decreases |v \\F |, contrary to assumption. 5.5. Alternating links. Assume again in §5.5 that F is an essential surface in good position. Assume also that its complexity ||F || = |v \\F | is minimized, and that D is a reduced alternating diagram (of a prime nonsplit link L) with chessboard surfaces B and W .25 Much of the work in §5.5 will address an innermost circle γ of F ∩ S+, in the following setup. Let T+ be an innermost disk of S+ \\F , and denote ∂T+ = γ. Let X be the disk that γ bounds in 0 26 F , and let T+ = S+ \\T+. Denote −1 T− = S− ∩ π (π(T+)) . Before addressing γ, we note two more general consequences of the assumption that D is alternating: Proposition 5.15. F ∩ B and F ∩ W contain no ∂-parallel arcs.

Proof. Assume for contradiction that an arc α of F ∩B is ∂-parallel in S3 \\νL. Then, since D is reduced, B is essential, and so α cuts oﬀ a disk X from B with ∂X = α∪α0 for some arc α0 ⊂ ∂B; good position implies that X ∩ W 6= ∅. Let Z be an outermost disk of X \\W , so that ∂Z = β ∪ β0, where β ⊂ X ∩ W ⊂ v and β0 ⊂ ∂X = α ∪ α0. If β0 ⊂ α, then α ⊂ F ∩ B is parallel in B to v. Then, pushing F near α through Z past β reduces |v \\F |, contrary to assumption.

25Recall Convention 2.5. 26Recall that π denotes the projection map νS2 → S2. A GEOMETRIC PROOF OF THE FLYPING THEOREM 29

Figure 25. The situation at the end of the proof of Proposition 5.15, with α,α 0,X, and β.

Figure 26. How F appears near each edge of ∂νL in §5.5 (up to mirror symmetry) and §§5.6-5.7 (no reﬂection allowed).

If β0 ⊂ α0, then α0 ⊂ ∂B0 is parallel in B to v. This is impossible, because D is reduced. Otherwise, β0 contains one of the shared endpoints x of α, α0; as shown in Figure 25, it is possible to push α through Z past β ⊂ v, while sliding x along α0 ⊂ ∂B, in a way the decreases |v \\F |, contrary to assumption. Because D is alternating and F is in good position: Observation 5.16. Near each edge of ∂νL, F appears as in Figure 26 (up to mirror symmetry). Note in particular that an edge of ∂νL contains an odd number of points of F ∩ S+ ∩ S− if and only if ∂F traverses the top of the overpass incident to that edge. We now turn our attention to the innermost circle γ of F ∩ S+. Proposition 5.17. Any arc of γ ∩ ∂C appears as in Figure 27. 30 THOMAS KINDRED

Figure 27. The vicinity of an arc of γ ∩∂C. Inward- pointing arrows along the innermost circle γ of F ∩S+ identify black and white regions of S+ ∩ S−.

Proof. Suppose γ intersects a crossing ball Ct. Then Proposition 5.13 implies that γ must intersect Ct in one arc only and must not traverse the overpass at Ct. Proposition 5.12 further implies that γ is disjoint from the edge E of ∂νL which abuts both the underpass at Ct and 2 the disk of ∂Ct \(S ∪νL) incident to γ. Hence, E ∩F ∩S+ ∩S− = ∅, and so Observation 5.16 implies that F must have a crossing band at the opposite end of this edge.

Proposition 5.18. No arc of γ ∩ S+ ∩ S− has both endpoints on C. Proof. This follows immediately from Proposition 5.17 and the fact that D is alternating.

Say that γ encloses a crossing ball Ct if π(T+) ⊃ π(Ct). Lemma 5.19. F has a crossing band in every crossing ball which γ encloses, and γ encloses at least one crossing ball.27

Proof. If γ encloses a crossing ball Ct, then ∂F does not traverse the overpass at Ct, since γ is an innermost circle of F ∩ S+. Initial position implies that F has a crossing band at Ct. Assume, contrary to the second claim, that γ encloses no crossing balls. Then γ cannot intersect a crossing ball, due to Proposition 5.17. Further, γ cannot traverse an overpass, because of Proposition 5.14 and Observation 5.16. Therefore, T+ is disjoint from all crossing balls and overpasses. Hence, π(γ) bounds a disk in S2 \ C. This implies, contrary to assumption, that some arc of γ ∩ S+ ∩ S− is parallel in S+ ∩ S− to ∂νL.

27This gives an alternate proof of the crossing band lemma from [1]: Crossing band lemma from [1]. Given a reduced alternating diagram D of a prime nonsplit link L and an essential spanning surface F for L, it is possible to isotope F so that it has a crossing band at some crossing in D. A GEOMETRIC PROOF OF THE FLYPING THEOREM 31

5.6. Definite surfaces. Assume again in §5.6 that F is an essential surface in good position, ||F || = |v \\F | is minimized, and D is a reduced alternating diagram of L with chessboard surfaces B and W . Also maintain the setup from §5.5 of the innermost circle γ ⊂ F ∩S+ 0 and the associated disks T+,T+ ⊂ S+ and T− ⊂ S−. Now assume also that F is positive-definite. Much of the work in §5.6 will address the possibilities for arcs δ of F ∩ T−. First, note the following more general consequences of the assumption that F is positive-definite: Lemma 5.20. Every arc α of F ∩ W satisfies

i(∂F, ∂W )ν∂α = 2. In particular, every crossing band in F is positive. Proof. Proposition 5.15 implies that α is not ∂-parallel, and so Lemma 2.14 gives i(∂F, ∂W )ν∂α = 2. Observation 5.16 and Lemma 5.20 immediately imply: Observation 5.21. ∂F intersects each edge of ∂νL as in Figure 26.

Now consider F ∩ T−. Each component of F ∩ int(T−) is disjoint from S+ (since γ is an innermost circle of F ∩ S+) and thus consists alternately of arcs on ∂νL ∩ S− and arcs on ∂C ∩ S− which abut crossing bands in F . This and the fact (due to Lemma 5.20) that every crossing band in F is positive imply that every circle of F ∩ int(T−) encloses some disk of B \\C in T− and is parallel in (∂νL ∪ ∂C) ∩ S− to the boundary of this disk. Likewise, for each arc δ of F ∩int(T−), some disk B0 of B ∩T− \\C abuts all components of ∂νL ∩ S− and ∂C ∩ S− that intersect δ.

Convention 5.22. Orient each arc δ of F ∩ int(T−) so that this nearby disk B0 lies to the right of δ, when viewed from H+. Denote the initial point of δ by x and the terminal point by y.

Proposition 5.23. Let δ be an arc of F ∩int(T−), following Conven- tion 5.22. Then x either lies on an underpass of ∂νL or is a point in ∂νL∩B\C with i(∂F, ∂B)νx = −1. Also, y ∈ ∂νL∩S+∩S−; if y ∈ B, then i(∂F, ∂B)νy = +1; otherwise, y ∈ W with i(∂F, ∂W )νy = −1.

Proof. Each endpoint of δ lies either on γ ∩ ∂νL ∩ S+ ∩ S− or on an underpass of ∂νL in π−1(π(γ)). Lemma 5.20 implies that any endpoint z of δ on ∂νL ∩ W satisﬁes i(∂F, ∂W )νz = −1. Thus, every endpoint of δ has one of the types described in the proposition. Moreover, because D is alternating and B0 lies to the left of δ, each type of endpoint can occur only at x or only at y as stated. Although there are three possibilities for x and two for y, there are actually just three possibilities for the pair (x, y): 32 THOMAS KINDRED

Figure 28. Endpoints of an arc δ of F ∩ int(T−), oriented as in Convention 5.22.

Proposition 5.24. Let δ be an arc of F ∩int(T−), as in Convention 5.22. Then x ∈ B if and only if y ∈ B. Proof. We argue by contradiction. Suppose first that x ∈ B and y∈ / B. Let γx ⊂ γ be the arc of F ∩ S+ ∩ S− that has x as an endpoint, and let γy ⊂ γ be the arc of F ∩ S+ ∩ ∂νL that has y as 0 0 an endpoint. Let x ∈ int(γx) and y ∈ int(γy). See Figure 29, left. Perform a bigon move along the arc α.28 Then consider the arc of ∂F ∩ S+ that contains y. Both endpoints of this arc lie on the circle of F ∩ S− that contains δ, contradicting Proposition 5.8. Suppose now that x∈ / B and y ∈ B. Let γy ⊂ γ be the arc of 0 F ∩ S+ ∩ S− that has y as an endpoint, and let y ∈ int(γy). The initial point x lies on the underpass of ∂νL at a crossing Ct, and 00 00 00 00 π(x) = π(x ) for some x ∈ γ. Either x ∈ ∂νL or x ∈ ∂Ct. 00 Suppose first that x ∈ ∂νL, as in Figure 29, center. Let γx be the arc of γ on the edge of ∂νL that abuts B0 and the overpass 0 containing x, and let x ∈ int(γx). As in the first case, it is now possible to perform a bigon move along an arc α with endpoints 0 0 x and y . Do so. The circle of F ∩ S− containing δ traverses the underpass at Ct and now has an endpoint on the edge of ∂νL incident to the overpass at Ct. This contradicts Proposition 5.14. In the remaining case, a similar argument leads to a contradiction, using Proposition 5.11. See Figure 29, right.

Proposition 5.25. Let δ be an arc of F ∩ int(T−), following Con- vention 5.22. If x, y ∈ B, then δ is parallel in T− to ∂T−.

Proof. In this case, x, y ∈ γ ∩ B0. Thus, Proposition 5.4 implies that x and y lie on the same arc γ0 of γ0 ∩ S+ ∩ S−. Hence, δ ∪ γ0 is a circle in T− \\F , so δ is parallel in T− to γ0 ⊂ γ ∩ S− ⊂ ∂T−. 5.7. Proof of Theorem 5.26 and the ﬂyping theorem. Let L be a prime nonsplit link, and let D be a reduced alternating diagram of

28 0 0 Note that x ∈ ∂B0, and y is on the same edge of ∂νL that contains y. This edge abuts B0, so there is a properly embedded arc α ⊂ S+ \\F with endpoints 0 0 x , y ∈ γ which is disjoint from overpasses and intersects S+ ∩ S− only in B0. Since y0 ∈ ∂νL, it is possible to perform a bigon move along α. A GEOMETRIC PROOF OF THE FLYPING THEOREM 33

y yx y x 0 0 0 0 0 y x y x y x x0

Figure 29. These bigon moves show that the initial point x of δ lies in B iﬀ the terminal point y does.

L with chessboard surfaces B and W , which are respectively positive- and negative-definite. Theorem 5.26. Any essential positive-definite surface F spanning L is plumb-related to B. As an aside, note that Proposition 3.2 and Theorem 5.26 imply: Corollary 5.27. Any two essential positive-definite surfaces spanning L are related by proper isotopy through B4. Proof of Theorem 5.26. While fixing B and W , transform F by isotopy so that it is in good position and ||F || = |v \\F | is minimal. If F ∩ H+ = ∅, then F traverses the top of no overpass, hence has a crossing band at every crossing. Lemma 5.20 implies that each crossing band is positive, and so F is isotopic to B. Assume instead that F ∩ H+ 6= ∅, and let T+ be an innermost disk of S+ \ F , set up as at the beginning of §5.5, with T− = S− ∩ −1 π (π(T+)) and γ = ∂T+ = ∂X for X ⊂ F ∩ H+. Propositions 5.23, 5.24, and 5.25 imply that each arc δ of F ∩ int(T−) has one one the three types (I, II, or III from left to right) shown in Figure 30. Note:

(5) At least one arc δ of F ∩ int(T−) is of type II or III.

Choose a point z in a region of W ∩ T+ abutting γ. While fixing ◦ 2 −1 F ∩ (S+ ∪ S− ∪ C), adjust F ∩ H± so that F ⊂ νS \ π (z) and 29 π restricts to an injection on each disk of F ∩ H±. Then, while fixing F \\X, push X through the point at infinity in H+ and into −1 H+ ∩ π (T+) such that π : X → π(T+) bijectively.

29 This may involve pushing some disks of F ∩ H+ past the “point at inﬁnity.” 34 THOMAS KINDRED

Figure 30. Arcs of δ ∩ int(T−), and the arcs used in the proof of Theorem 5.26 to transform γ into γ0.

For each arc δ of F ∩int(T−) of type II or III, construct a properly −1 embedded arc in T+ \ (B ∪ π (C)), as shown in Figure 30. Denote the union of these arcs by ρ, and denote the disk of T+ \\ρ containing 0 −1 0 0 0 0 z by T+. Also denote S− ∩ π (π(T+)) = T− and ∂T+ = γ . Take an annular neighborhood A of π(γ0) in S2, such that A inter- 0 sects only the crossing balls that π(γ ) intersects, ∂A ∩ C = ∅, and 0 each arc of F ∩S+ ∩S− that intersects A lies on γ or has an endpoint on a saddle disk at a crossing where γ0 abuts a saddle disk. Denote 0 2 ∂A = γ0 ∪ γ1, such that γ0 ⊂ π(T+). Denote S \\A = Z0 t Z1, −1 where each ∂Zi = γi. Denote π (Z0) = Y0. 0 Denote the arcs of γ ∩ W by ω1, . . . , ωm. Each ωi has a dual arc 30 αi ⊂ A ∩ W \ C. Denote the rectangles of A \\(α1 ∪ · · · αm) by A1,...,Am such that each ∂Ai ⊃ αi ∪ αi+1, taking indices modulo −1 m. Denote each prism π (Ai) by Pi. Each prism Pi intersects F in one of the three ways indicated in the 31 left column of Figure 31. For each i, let Fi denote the component of F ∩ Pi which abuts γ. Each Fi is a disk. Observe that Fi and Fj intersect in an arc when i ≡ j ± 1 (mod m) and are disjoint when i 6≡ j, j ± 1 (mod m). Therefore, F1 ∪ · · · ∪ Fm is either an annulus or a mobius band; denote it by FA. Moreover, the disk X ∩ Y0 attaches to FA along its boundary. Therefore, FA is an annulus, and the following subsurface of F is a disk:

(6) U = (X ∩ Y0) ∪ FA. There is a properly embedded disk V ⊂ νS◦ 2 \\(F ∪ νL) which intersects Y0 in a disk (in H−) and intersects each prism Pi as indicated in the right column of Figure 31.32 Note that ∂V ∩F = ∂U ∩F and (∂V ∩ ∂νL) ∪ (∂U ∩ ∂νL) is a system of meridia and inessential circles on ∂νL. Thus V is a, possibly fake, plumbing cap for F , and

30 The arc αi has one endpoint on γ0 and one on γ1, with |ωi ∩ αi| = 1. 31 The green arcs top-left describe a disk in Pi \ νL, which is parallel through −1 a ball Y ⊂ Pi to π (γ1); F intersects Pi as shown and in an arbitrary number of additional disks in Y , each containing a saddle disk. 32A caveat analogous to the one in Note 31 applies here as well. A GEOMETRIC PROOF OF THE FLYPING THEOREM 35

Figure 31. The re-plumbing in the proof of Theorem 5.26.

U is its shadow. Whether or not V is fake, Proposition 3.1 implies that F is plumb-related to (F \\U) ∪ V . Figure 31 shows the move 33 F → (F \\U) ∪ V within each prism Pi. Whereas the disk U intersects C in some number of saddle disks and no crossing bands, the disk V intersects C in no saddle disks and some number of crossing bands. Moreover, (5) implies that either U ∩ C 6= ∅ or V contains a crossing band. Hence, the move F → 34 F1 = (F \\U) ∪ V decreases the complexity of F : ||F1|| < ||F ||. Isotope F1 into good position so as to minimize its complexity. If ||F1|| > 0, then the same argument gives another re-plumbing move F1 → F2 with ||F2|| < ||F1||. Repeat, performing isotopy and re- plumbing moves F → F1 → · · · → Fr until ||Fr|| = 0. Then Fr is disjoint from H+, hence isotopic to B.

33Alternatively, one can visualize the re-plumbing move as follows. Reﬂect the disk X0 across the projection sphere. The resulting surface has arcs of self- intersection, but each of these arcs has one endpoint on ∂νL and the other in its interior, on an arc of γ0 ∩ W . “Unzip” the surface along these arcs. 34This implies that V was not a fake plumbing cap. 36 THOMAS KINDRED

Flyping theorem. Any two reduced alternating diagrams D,D0 of the same prime, nonsplit link L are flype-related. Proof. Let B,W and B0,W 0 be the respective chessboard surfaces of D and D0, where B and B0 are positive-definite. Then B and B0 are plumb-related, by Theorem 5.26, as are W and W 0. Therefore, D 0 and D are flype-related, by Theorem 4.11. Since crossing number and writhe are invariant under flypes and additive under diagrammatic connect sum and disjoint union, we obtain a new geometric proof of more parts of Tait’s conjectures [3, 6, 10, 12, 15]: Theorem 5.28. All reduced alternating diagrams of a given link have the same crossing number and writhe. Problems 2.6–2.8 remain open. Another corollary is worth noting: Corollary 5.29. If D and D0 are reduced alternating diagrams of the same link, and if B,W and B0,W 0 are their respective chessboard surfaces, then s(B) = s(B0) and s(W ) = s(W 0).

References [1] C. Adams, T. Kindred, A classification of spanning surfaces for alternating links, Alg. Geom. Topol. 13 (2013), no. 5, 2967-3007. [2] C. McA. Gordon, R.A. Litherland, On the signature of a link, Invent. Math. 47 (1978), no. 1, 53-69. [3] J. Greene, Alternating links and definite surfaces, with an appendix by A. Juhasz, M Lackenby, Duke Math. J. 166 (2017), no. 11, 2133-2151. [4] J. Howie, A characterisation of alternating knot exteriors, Geom. Topol. 21 (2017), no. 4, 2353-2371. [5] T. Kindred, Alternating links have representativity 2, Alg. Geom. Topol. 18 (2018), no. 6, 3339-3362. [6] L.H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), no. 3, 395-407. [7] W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984), no. 1, 37-44. [8] W. Menasco, M. Thistlethwaite, The Tait flyping conjecture, Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 2, 403-412. [9] W. Menasco, M. Thistlethwaite, The classification of alternating links, Ann. of Math. (2) 138 (1993), no. 1, 113-171. [10] K. Murasugi, Jones polynomials and classical conjectures in knot theory, Topology 26 (1987), no. 2, 187-194. [11] P.G. Tait, On Knots I, II, and III, Scientific papers 1 (1898), 273-347. [12] M.B. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology 26 (1987), no. 3, 297-309. [13] M.B. Thistlethwaite, On the Kauffman polynomial of an adequate link, In- vent. Math. 93 (1988), no. 2, 285-296. [14] M.B. Thistlethwaite, On the algebraic part of an alternating link, Pacific J. Math. 151 (1991), no. 2, 317-333. [15] V.G. Turaev, A simple proof of the Murasugi and Kauffman theorems on alternating links, Enseign. Math. (2) 33 (1987), no. 3-4, 203-225. A GEOMETRIC PROOF OF THE FLYPING THEOREM 37

Department of Mathematics, University of Nebraska, Lincoln, Ne- braska 68588-0130, USA E-mail address: [email protected] URL: www.thomaskindred.com