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2. Preliminaries The symbol  signals either the end or the omission of a proof. Spaces, maps, etc., are smooth (C∞) unless otherwise stated. Manifolds may have boundary and are always oriented; in particular, R, Cn, 2n n 2 2 D :={(z1,...,zn) ∈ C : |z1| + ··· + |zn| ≤ 1}, and S2n−1 :=∂D2n have standard orientations, as does S2 when it is identified with the Riemann sphere P1(C) := C ∪ {∞}. For suitable Q ⊂ M, let NM (Q) denote a closed regular neighborhood of Q in (M,∂M). If Q is a codimension–2 submanifold 2 of M with trivial normal bundle, then a trivialization τ : Q × D → NM (Q) is adapted to a map f : M \ Q → S1 if τ(ξ, 0) = ξ and f(τ(ξ,z)) = z/|z| for ξ ∈ Q, z ∈ D2 \{0}. A surface is a compact 2– no component of which has empty boundary. A Seifert surface is a surface F ⊂ S3. A link L is the boundary of a Seifert surface F , and any F with ∂F = L is a Seifert surface for L. A knot is a connected link. 3 3 Let L ⊂ S be a link. The image in H2(S ,L; Z) of the fundamental class [F ] ∈ H2(F,L; Z) is independent of the choice of a Seifert surface F for L; denote 1 3 3 the corresponding orientation class for L in H (S \ L; Z) =∼ H2(S ,L; Z) by ξL. 3 1 3 1 1 3 Call f : S \ L → S simple if [f] =∼ ξL in π0(Map(S \ L,S )) =∼ H (S \ L; Z). The Morse-Novikov number of L is MN(L) := min{n : there exists a simple Morse map f : S3 \L → S1 with exactly n critical points}. On general principles, MN(L) < ∞. Definitions. A simple Morse map f : S3 \ L → S1 is 2 1. boundary-regular if there is a trivialization τ : L × D → NS3 (L) which is adapted to f; 2. moderate if no critical point of f is a local extremum; 3. self-indexed if the value of f at a critical point is an injective function of the index of the critical point; 4. minimal if no Morse map in ξL has strictly fewer critical points than f.

Proposition 1. Let f : S3 \ L → S1 be a simple Morse map. 1. If f has only finitely many critical points, then f is properly isotopic to a boundary-regular simple Morse map. 2. If f is boundary-regular then f has only finitely many critical points, and for every regular value exp(iθ) ∈ S1, S(f,θ) := L ∪ f −1(exp(iθ)) is a Seifert surface for L. 3. If f is minimal, then f is moderate and has precisely MN(L) critical points. 4. If f (has only finitely many critical points and) is moderate, then up to (proper) isotopy f is moderate and self-indexed. MURASUGISUMSOFMORSEMAPSTOTHECIRCLE 3

5. If f is boundary-regular, moderate, and self-indexed, then either (a) f has no critical points (so MN(L) = 0) and is in fact a fibration, and the Seifert surfaces S(f,θ) are mutually isotopic in S3, or (b) f has 2m > 0 critical points, m each of index 1 and index 2, and the Seifert surfaces S(f,θ) fall into exactly two distinct isotopy classes in S3. Proof. Mostly straightforward (for (3) and (5b), see [16]). In case (5a), L is (as usual) called a fibered link and S(f,θ) is called a fiber surface of f. (Also as usual, any Seifert surface for L isotopic to an S(f,θ) is called a fiber surface for L.) In case (5b), the two isotopy classes of Seifert surfaces S(f,θ) for L (which could well be a fibered link even though f is not a fibration) are distinguished by their Euler characteristics, which differ by 2m. Any of the Seifert surfaces S(f,θ) with first homology group of larger (resp., smaller) rank will be called a large (resp., a small) Seifert surface of the boundary-regular moderate self-indexed simple Morse map f (note: not “of L”). Any fiber surface of a fibration f may be called either large or small. Convention. Henceforth, all Morse maps are boundary-regular and simple.

3. Morse maps from Milnor maps The first explicit Morse maps (in fact, fibrations) for an infinite class of links were given by Milnor’s celebrated Fibration Theorem [10], where they appear as (the instances for functions C2 → C of) what are now called the “Milnor maps” associated to singular points of complex analytic functions Cn → C. For present and future purposes, it is useful to somewhat extend the framework in which Milnor studied these maps. Given a non-constant meromorphic function F : M 99K P1(C) on a complex manifold M, let D(F ) be the (possibly singular) complex which is the closure in M of F −1(0) ∪ F −1(∞). The argument of F is arg(F ) := F/|F |: M \ D(F ) → S1. For M = Cn, the restriction of arg(F ) to rS2n−1 \ D(F ) is the Milnor map of F at radius r. Call the Milnor map of F at radius 1 simply the Milnor map of F . Milnor’s proof of [10, Lemma 4.1], stated for analytic F , applies equally well to meromorphic F . n 2n−1 Lemma 1. If F : C 99K P1(C) is meromorphic, then (z1,...,zn) ∈ rS \D(F ) is a critical point of the Milnor map of F at radius r iff the complex vectors 1 ∂F ∂F (z1,..., zn), (z1,...,zn),..., (z1,...,zn) i F (z1,...,zn) ∂z1 ∂z1 are linearly dependent over R. The next lemma rephrases several more results from [10]. Lemma 2. Let F : C2 → C be analytic (so D(F ) = F −1(0)), with F (0, 0) = 0, and suppose F has no repeated factors in OC2 (so the multiplicity of each irreducible component of D(F ) is 1). 1. There is a set X(F ) of radii r ∈ ]0, ∞[ such that X(F ) is discrete in [0, ∞[ and, if r∈ / X(F ), then D(F ) intersects rS3 transversally, so that L(F, r) := (1/r)(D(F ) ∩ rS3) is a link in S3; L(F, r) and L(F, r′) are isotopic for r, r′ in the same component of ]0, ∞[ \ X(F ). 4 LEE RUDOLPH

Figure 1. A small and a large Seifert surface of u.

2. If 1 ∈/ X(F ), then the Milnor map of F is simple, and NS3 (L(F, 1)) has a trivialization which is adapted to the Milnor map of F . 3. If 1 < inf X(F ), then the Milnor map of F is a fibration (the Milnor fibration of F ), and in particular L(F, 1) (the link of the singularity of F −1(0) at (0, 0)) is a fibered link. Note that, in case (2), the Milnor map of F may have degenerate critical points, but if it does not, then it is Morse (in the sense of the convention in section 2). In practice, Lemma 1 makes it easy to locate the critical points of a Milnor map and check them for non-degeneracy. 2 Example 1. Given m,n ≥ 0 with m + n = 1 in case mn = 0, let Fm,n : C → m n C : (z, w) 7→ mz + nw ; easily, X(Fm,n) = ∅. The link of the singularity of −1 −1 F0,1 (0) at (0, 0) is an unknot O (of course F0,1 (0) is not in fact singular at (0, 0)); the Milnor fibration of F0,1 = pr2 will be denoted by o. More generally, the link of −1 the singularity of Fm,n(0) at (0, 0) is a torus link O{m,n} of type (m,n), and the Milnor fibration of Fm,n will be denoted by o{m,n}.

2 2 Example 2. If Gε : C → C : (z, w) 7→ 4w − 8εw − 1, then for sufficiently small ε ≥ 0, Uε :=L(Gε, 1) is an unlink of two unknotted components, and if ε> 0, then (by direct calculation using Lemma 1) the Milnor map of Gε is Morse, with two critical points (one each of index 1 and index 2). Let U :=Uε for a sufficiently small ε > 0, and denote by u: S3 \ U → S1 this Morse Milnor map. (The situation is pictured in Fig. 1.)

Proposition 2. MN(U)=2 and u is a minimal Morse map. Proof. This is immediate from the properties of u just asserted, given that U is not 3 fibered (which follows, for instance, from the fact that π2(S \ U) 6= {0}). The non-fibration u, trivial though it be, is an ingredient of fundamental impor- tance in the constructions of section 5.

4. Murasugi sums of Morse maps n 1 For n ∈ N, write Gn := {z ∈ C : z = 1}, and let pn : P1(C) \ G2n → S be the n n argument of the rational function P1(C) → P1(C): z 7→ (1 + z )/(1 − z ), viz., (1 + zn)/|1+ zn| p (z)= for z∈ / G ∪ {∞}, p (∞)= −1. n (1 − zn)/|1 − zn| 2n n MURASUGISUMSOFMORSEMAPSTOTHECIRCLE 5

Figure 2. Q1(π/2) and some level sets of pr2 |Q1(π/2); Q2(π/2) and some level sets of pr2 |Q2(π/2).

1 Define Pn : P1(C) \ G2n × [0, π] → S by Pn(z,θ) = exp(iθ)pn(z). Note that pn(ζz)= pn(z) and Pn(z,θ)= Pn(ζz,θ) for any ζ ∈ Gn. −1 Lemma 3. (1) p1 is a fibration with fiber ] − 1, 1[ = p1 (1). For n > 1, pn has −1 −1 exactly two critical points, 0 ∈ pn (1) and ∞ ∈ pn (−1), at each of which the germ n of pn is smoothly conjugate to the germ of z 7→ Im(z ) at 0. (2) P1 is a fibration −1 with fiber ] − 1, 1[ × [0, π]= P1 (1). For all n, Pn has no critical points, and there 2 is a trivialization τPn : (G2n × [0, π]) × D → NP1(C)×[0,π](G2n × [0, π]) which is adapted to Pn. −1 P C Let Qn(θ) denote the closure of Pn (exp(iθ)) in 1( ) × [0, π]. By inspection 1 (and Lemma 3), for all exp(iθ) ∈ S , Q1(θ) is a 4–gonal 2–disk with smooth interior bounded by the union of G2 × [0, π] and 2 semicircles in P1(C) ×{0, π}; for n> 1 1 and all exp(iθ) ∈ S \{1, −1}, Qn(θ) is a 4n–gonal 2–disk with smooth interior bounded by the union of G2n × [0, π] and 2n circular arcs in P1(C) ×{0, π}. (The cases n =1, 2, θ = π/2, are pictured in Fig. 2.)

1 Lemma 4. For all exp(iθ) ∈ S , the restriction pr2 |Q1(θ): Q1(θ) → [0, π] has no critical points. For n > 1 and exp(iθ) ∈ S1 \{1, −1}, there is exactly one critical point of pr2 |Qn(θ): Qn(θ) → [0, π] (to wit, (0,θ) for 0 <θ<π and (∞,θ − π) for π<θ< 2π), at which the germ of pr2 |Qn(θ) is smoothly conjugate to the germ of z 7→ Im(zn) at 0.

By further inspection, for 0 <θ< 2π, θ 6= π, Qn(θ) is piecewise-smoothly isotopic (by an isotopy fixing ∂Qn(θ) ∪ (Qn ∩ P1(C) ×{θ −⌊θ/π⌋π}) pointwise) to ′ ′ a piecewise-smooth 4n–gonal 2–disk Qn(θ) which is so situated that both Qn(θ) ∩ P C ′ P C ( 1( )×[0,θ −⌊θ/π⌋π]) and Qn(θ)∩( 1( )×[θ −⌊θ/π⌋π, π]) are piecewise-smooth ′ P C 4n–gonal 2–disks, while Qn(θ) ∩ ( 1( ) ×{θ −⌊θ/π⌋π}) is a (smooth) 2–disk in P1(C)×{θ −⌊θ/π⌋π} naturally endowed with the structure of a 2n–gon. (The cases ′ ′ n = 1, 2, θ = π/2, are pictured in Fig. 3. Versions of Q2(π/2) and Q2(3π/2) in 2 NS3 (S ) =∼ P1(C) × [0, π] are pictured in Fig. 4.) Definitions. Let L ⊂ S3 be a link, f : S3 \ L → S1 a Morse map, exp(iθ) ∈ S1 a regular value of f, and ψ ⊂ S(f,θ) an n–star on S(f,θ) (in the sense of [19]: that is, ψ is the union of n arcs αs, pairwise disjoint except for a common endpoint ∗ψ ∈ Int S(f,θ), such that ∂S(f,θ) ∩ αs = ∂αs \ {∗ψ} for each s). A regular neighborhood NS3 (ψ) is f–good provided that NS3 (ψ) ∩ S(f,θ) is a regular 6 LEE RUDOLPH

′ ′ Figure 3. Q1(π/2) and Q2(π/2).

′ ′ Figure 4. Q2(π/2) and Q2(3π/2) (the latter with viewports), as 2 they appear in in NS3 (S ) =∼ C × [0, π]. b neighborhood NS(f,θ)(ψ) (and thus an n–patch on S(f,θ) in the sense of [19]: that is, a 2–disk naturally endowed with the structure of a 2n–gon whose edges are alternately boundary arcs and proper arcs in S(f,θ)) and there is a diffeomorphism h: (P1(C), G2n) → (∂NS3 (ψ),L ∩ ∂NS3 (ψ)) such that (f ◦ h)|(P1(C) \ G2n)= pn.

Lemma 5. Every neighborhood of ψ in S3 contains an f–good regular neighbor- hood. Proof. First suppose that L = O and f = o (as in Example 1). Stereographic projection σ : S3 \{(0, −i)} → C × R : (z, w) 7→ (z, Re(w))/(1 + Im(w)) maps O to 1 2 −1 S ×{0} and S(o,π/2) to D ×{0}. The radius ψ1 :=σ ([0, 1] ×{0}) is a 1–star in S(o,π/2), and the preimage σ−1(B) of an appropriate ellipsoidal 3–disk B ⊂ C × R (say, with one focus at (0, 0), center at (1 − ε, 0) for sufficiently small ε > 0, and minor axes much shorter than the major axis), is an o–good regular neighborhood NS3 (ψ1). The n–sheeted cyclic branched covering (C × R) ∪ {∞} → (C × R) ∪ {∞} : (ζ,t) 7→ (ζn,t), ∞ 7→ ∞, branched along ({0}× R) ∪ {∞}, can be modified in a neighborhood of ∞ so as to −1 3 3 induce via σ a smooth n–sheeted cyclic branched covering cn : S → S , branched 1 3 −1 −1 along A := {0}× S ⊂ S , with cn (S(o,π/2)) = S(o,π/2); then ψn :=cn (ψ1) is −1 an n–star in S(o,π/2), and cn (NS3 (ψ1)) is (if minimal care has been taken) an o– good regular neighborhood NS3 (ψn). (The cases n =1, 2, 3 are pictured in Fig. 5.) MURASUGISUMSOFMORSEMAPSTOTHECIRCLE 7

Figure 5. Some level sets of o|(∂NS3 (ψn) \ O) for n =1, 2, 3.

Figure 6. A “side view” (indicating some level sets of f) of the part of the non–f–good neighborhood M near an endpoint of ψ.

3 Clearly any neighborhood of ψn in S contains an o–good regular neighborhood of the type just constructed. The general case follows immediately, upon observing that, for any link L, Morse map f : S3 \ L → S1, and regular value exp(iθ) of f, if ψ ⊂ S(f,θ) is an n–star on S(f,θ), then there is a diffeomorphism h: M → h(M) from a (non–f–good) 3 3 neighborhood M of ψ in S to a neighborhood h(M) of ψn in S , and a diffeo- morphism k : (S1, exp(iθ)) → (S1,i), such that h(M ∩ S(f,θ)) = h(M) ∩ S(o,π/2) and k ◦ f|(M \ L)= o ◦ h|(M \ L). (A “side view” of part of one such M is pictured in Fig. 6.)

3 3 1 Construction. For s = 0, 1, let Ls ⊂ Ss be a link and fs : Ss \ Ls → S a 3 Morse map with critical points crit(fs) ⊂ Ss \ Ls. Fix n > 0 and exp(iΘ) ∈ 1 S \ (f0(crit(f0)) ∪ f1(crit(f1)) ∪{0, π}). Let ψn,s ⊂ S(fs, Θ) be an n–star with fs– 3 3 3 good regular neighborhood Ns :=NSs (ψn,s). Let Es :=Ss \Int Ns; let ds : Es → D be a diffeomorphism. Let hs : (P1(C), G2n) → (∂Ns,L ∩ ∂Ns) be a diffeomorphism s with (fs ◦ hs)|(P1(C) \ G2n) = pn. For a fixed ζ ∈ Gn, let (z,s) ≡ζ hs(ζ z,s) (z ∈ P1(C),s =0, 1). The identification space Σ(N0,N1, ζ) := (E0 ⊔ P1(C) × [0, π] ⊔ E1)/≡ζ has a natural piecewise-smooth structure, imposed on it by the identification map Π, with respect to which the 1–submanifold

L(N0,N1, ζ):=(L0 ∩ E0 ⊔ G2n × [0, π] ⊔ L1 ∩ E1)/≡ζ and the map

f(N0,N1, ζ):=((f0 ⊔ Pn ⊔ f1)/≡ζ )|(Σ(N0,N1, ζ) \ L(N0,N1, ζ)) are smooth where this is meaningful (i.e., in the complement of the identification locus Π(P1(C) ×{0, π}), along which Σ(N0,N1, ζ) is itself a priori only piecewise- smooth). It is not difficult to give Σ(N0,N1, ζ) a smooth structure everywhere, in which L(N0,N1, ζ) and f(N0,N1, ζ) are everywhere smooth. The smoothing 8 LEE RUDOLPH can be done very naturally off Π((G2n ∪ {0, ∞}) × {0, π}), using the fact that by construction (see Lemmas 3 and 4) the map f(N0,N1, ζ) and the function 2 2 (kd0k − 1) ⊔ pr2 ⊔(π +1 −kd1k )/≡ζ are piecewise-smoothly transverse (in an evident sense) on Π((P1(C) \ (G2n ∪{0, ∞})) ×{0, π}). A somewhat less natural, but not difficult, construction smooths Σ(N0,N1, ζ) on Π({0, ∞}×{0, π}) in such a way as to make f(N0,N1, ζ) smooth there also (in the process, possibly forcing 2 2 (kd0k −1)⊔pr2 ⊔(π+1−kd1k )/≡ζ to be not smooth at those points). Nor is there is any difficulty in smoothing Σ(N0,N1, ζ) on Π(G2n ×{0, π}). Further details will be suppressed. When Σ(N0,N1, ζ), with the smooth structure just constructed, is identified with 3 S , L(N0,N1, ζ) (resp., f(N0,N1, ζ)) will be called a 2n–gonal Murasugi sum of L0 and L1 (resp., of f0 and f1) and denoted by L0 * L1 (resp., f0 * f1) (N0,N1,ζ) (N0,N1,ζ) or simply by L0 * L1 (resp., f0 * f1). (A 2–gonal Murasugi sum of links is simply a connected sum, and a 2–gonal Murasugi sum of Morse maps f0 and f1 may be denoted f0=k f1. A more detailed description of the construction in the case of connected sums, which goes more smoothly than the general case, is given in [16].)

3 3 1 Proposition 3. If Ls ⊂ Ss is a link and fs : Ss \ Ls → S is a Morse map (s =0, 1), then

1. any Murasugi sum L0 * L1 is a link, and (N0,N1,ζ) 2. any Murasugi sum f0 * f1 is a Morse map and its critical points (N0,N1,ζ) crit(f0 * f1) = crit(f0) ∪ crit(f1) have indices and critical values that (N0,N1,ζ) are inherited unchanged from those of f1 and f2; in particular, (a) if f0 and f1 are moderate, then f0 * f1 is moderate, and (N0,N1,ζ) (b) if f0 and f1 are self-indexed and have the same critical value for each index, then f0 * f1 is self-indexed. (N0,N1,ζ)

With Θ chosen as in the construction, the Seifert surface S(f0 * f1, Θ) (N0,N1,ζ) is isotopic (up to smoothing) to Π((S(f0, Θ) ∩ E0) ⊔ Qn(Θ) ⊔ (S(f1, Θ) ∩ E1)), and thus piecewise-smoothly isotopic to

′ Π(S(f0, Θ) ∩ E0 ⊔ Qn(Θ) ⊔ S(f1, Θ) ∩ E1)= ′ P C Π(S(f0, Θ) ∩ E0 ⊔ Qn(Θ) ∩ 1( ) × [0, Θ −⌊Θ/π⌋π]) ∪ ′ P C ′ ′ Π(Qn(Θ) ∩ 1( ) × [Θ −⌊Θ/π⌋π, π] ⊔ S(f1, Θ) ∩ E1)=:S0 ∪ S1 ′ where Ss is piecewise-smoothly isotopic to S(fs, Θ) by an isotopy carrying the 2– ′ ′ disk S0 ∩ S1 (with its naturally structure of 2n–gon, noted after Lemma 4) to the n–patch Ns ∩ S(fs, Θ). Thus, for a suitable diffeomorphism H : N0 ∩ S(f0, Θ) → N1 ∩ S(f1, Θ) (determined by ζ, up to isotopy), S(f0 * f1, Θ) is a 2n– (N0,N1,ζ) gonal Murasugi sum S(f0, Θ) *H S(f1, Θ) as described in [19] (see also the primary sources [12, 23, 1]).

On the level of Seifert surfaces, a 2–gonal Murasugi sum S0 *H S1 is the same as a boundary-connected sum S0 S1. After boundary-connected sum, the most commonly encountered case of Murasugi sum is 4–gonal, and the most familiar and probably most useful 4–gonal Murasugi sums are annulus plumbings, as described in [20] (see also the primary sources [22, 2], as well as [3] and [21]). Specifically, for any knot K, let A(K,n) denote any Seifert surface diffeomorphic to an annulus such that K ⊂ ∂A(K,n) and the Seifert matrix of A(K,n) is [n]. MURASUGISUMSOFMORSEMAPSTOTHECIRCLE 9

1 Tα (0) (1) (0) hv hs α hu α

*α A(O, 0) *α A(O, 0)

=∼

1 1 Figure 7. α on S and Tα(S); S *α A(O, 0) and Tα(S) *α A(O, 0).

(For example, A(O, −1) is a positive Hopf annulus, isotopic to a fiber surface of o{2, 2}, as in Example 1; the mirror image A(O, 1) of A(O, −1) is a negative Hopf annulus; and A(O, 0) is isotopic to a large Seifert surface of u as in Example 2.) Let γ(K) ⊂ A(K,n) be an arc from K to ∂A(K,n) \ K, so that NA(K,n)(γ(K)) =: C1 is a 2-patch (though γ(K), as given, is not a 2-star). Let S be a Seifert surface S, α ⊂ S a proper arc, C0 ⊂ S a 2–patch with α ⊂ ∂C0 (respecting orientation). Let H : (C0, α) → (C1, C1 ∩ K) be a diffeomorphism. Each of γ(K), C0, and H is unique up to isotopy, so the 4–gonal Murasugi sum S *H A(K,n) depends (up to isotopy) only on α, and there is no abuse in denoting it by S *α A(K,n). When S = A(K′,n′) is also an annulus, it is slightly abusive—but handy—to write simply A(K′,n′) * A(K,n), with the understanding that α = γ(K′). Example 3. Given any proper arc α on a surface S, there is a (highly non-unique!) (0, 1)–handle decomposition

(0) (1) (†) S = [ hx ∪ [ hz x∈X z∈Z

(1) for which α is an attaching arc of some 1–handle hs . The subsurface

(0) (1) S ≀ α:= [ hx ∪ [ hz s=6 x∈X z∈Z is independent, up to isotopy on S, of the handle decomposition (†), and can be thought of as “S cut open along α”. If S is a Seifert surface, then for every n ∈ Z n 3 there is a reimbedding Tα : S → S (unique up to isotopy) which is the identity (1) on (S ≀ α) ∪ κ, where κ ⊂ hs is a core arc, and which puts n counterclockwise (1) full twists in hs . For any n, the annulus-plumbed Seifert surfaces S *α A(O, 0) n and Tα (S) *α A(O, 0) are isotopic, and the boundary of each is isotopic to ∂(S ≀ α). (The situation is pictured in Fig. 7, for n = 1.) In particular, for any knot K and n integer n, A(K,n)= Tγ(K)(A(K, 0)), and ∂(A(K,n) * A(O, 0)) is an unknot. 10 LEE RUDOLPH

Figure 8. Piecewise-smooth Seifert surfaces for U (two disks; an annulus) isotopic to those in Figure 1.

Figure 9. Piecewise-smooth Seifert surfaces for U * U =∼ O (a disk; a punctured torus, with viewport) isotopic to a small Seifert surface and a large Seifert surface of u * u.

Of course S(f0=k f1,θ) is always a connected sum S(f0,θ)=k S(f1,θ). As the fol- lowing example shows, for n > 1, if exp(iθ) and Θ lie in different components of 1 S \ crit(f0 * f1), then S(f0 * f1,θ) need not be a Murasugi sum of S(f0,θ) and S(f1,θ). Example 4. As pictured in Fig. 1 (redrawn piecewise-smoothly in Fig. 8), and noted in Example 3, the large Seifert surface of u is A(O, 0). Of course the small Seifert surface of u is two disks. By the construction and Prop. 3, there is a (moderate, self-indexed) 4–gonal Murasugi sum u * u with 4 critical points whose large Seifert surface A(O, 0) * A(O, 0) is a punctured torus bounded by an unknot; the small Seifert surface of u * u is a disk, which is not a Murasugi sum of two pairs of disks. (The situation is pictured in Fig. 9.) By Example 3, there is also a (moderate, self-indexed) 4–gonal Murasugi sum u * o{2, 2} with just 2 critical points (since o{2, 2} is a fibration) with the same large and small Seifert surfaces as u * u. Example 4 also shows that, if no restriction is placed on the Seifert surfaces along which a Murasugi sum L0 * L1 is formed, then it can easily happen that MN(L0 * L1) > MN(L0)+ MN(L1). Much worse is true: according to Hirasawa [4], any knot K bounds a Seifert surface F such that F = F0 * F1 is a Murasugi sum and K0 := ∂F0, K1 := ∂F1 are unknots; since [16] for every m there is a knot K with MN(K) > m, MN(K0 * K1) − (MN(K0)+ MN(K1)) can be arbitrarily large. However, Prop. 3 does lead immediately to the following generalization of the inequality (∗) stated in section 1. 3 1 Corollary 1. If fs : S \ Ls → S is a minimal Morse map, and f0 * f1 s (N0,N1,ζ) is any Murasugi sum, then MN(L0 * L1) ≤ MN(L0)+ MN(L1). (N0,N1,ζ) MURASUGISUMSOFMORSEMAPSTOTHECIRCLE 11

5. Morse maps and free Seifert surfaces 3 For any X, let h1(X) := rank H1(X; Z). A Seifert surface F ⊂ S is called 3 free iff F is connected and π1(S \ F ) is a free group; alternatively, F is free iff 3 1 2 1 2 S \ Int NS3 (F ) is a handlebody Hg =∼ (S × D )1 ··· (S × D )g (necessarily of genus g = h1(F )). Call the Morse map f free if, for every regular value exp(iθ) of f, the Seifert surface S(f,θ) is free.

Lemma 6. A Murasugi sum F0 * F1 of Seifert surfaces F0 and F1 is free if and only if F0 and F1 are free. 3 Proof. As observed by (for instance) Stallings [23], π1(S \ F0 * F2) is the free 3 3 3 product of π1(S \ F0) and π1(S \ F1). In particular, π1(S \ F0 * F1) is free if and 3 3 only if π1(S \ F0) and π1(S \ F1) are free. It is well known (and obvious) that, if f is a fibration, then f is free. The following proposition is Lemma 4.2 of [16]. Proposition 4. A large Seifert surface of a moderate self-indexed Morse map is free.

Corollary 2. If f is a moderate self-indexed Morse map (in particular, if f is a self-indexed minimal Morse map) then f is free if and only if a small Seifert surface of f is free. Example 6, below, gives a knot K and a non-free self-indexed minimal Morse map f : S3 \ K → S1. 3 3 Recall that a genus g handlebody Hg ⊂ S is Heegard if S \ Int Hg is also a handlebody. By a theorem of Waldhausen, any two Heegard handlebodies of genus g in S3 are isotopic, so it is obvious (by considering standard examples) that, if 3 Hg ⊂ S is a Heegard handlebody and ∆ ⊂ ∂Hg is a 2–disk, then F = ∂Hg \ Int ∆ is a free Seifert surface for the unknot ∂∆. The converse is also true and easily proved.

Lemma 7. A free Seifert surface F for an unknot is of the form F = ∂Hg \ Int ∆ 3 for some Heegard handlebody Hg ⊂ S and 2–disk ∆ ⊂ ∂Hg.

Lemma 8. If S is a free Seifert surface for the unknot of genus g, then there is a 3 1 moderate self-indexed Morse map f : S \ ∂S → S with exactly h1(S)=2g critical points, such that S is a large Seifert surface of f and a small Seifert surface of f is a disk. Proof. This follows immediately from Lemma 7, using a suitable modification of the usual dictionary between Morse functions (to R) and cobordisms. Alternatively, and more in the spirit of this paper, note that (by Prop. 3 and Example 3) a g–fold connected sum (u * o{2, 2})=k ··· =k (u * o{2, 2}) is such a map.

Proposition 5. If F is a free Seifert surface, then F is the small Seifert surface of f for a free moderate self-indexed Morse map f : S3 \ ∂F → S1 which has exactly 2h1(F ) critical points.

Proof. Since F is connected, there exist pairwise disjoint proper arcs αs ⊂ F ,

1 ≤ i ≤ h1(F ), such that (··· ((F ≀ α1) ≀ α2) ··· ) ≀ αh1(F ) is a 2-disk. If T denotes 12 LEE RUDOLPH

Figure 10. S and (T 1 (S) A(O , 1)) A(O′ , −1) have αs *αs s *γ(Os) s isotopic boundaries.

1 1 1 the (commutative) composition Tα ◦ Tα ◦···◦ Tαh F , and F1 :=T (F ), then F1 is 1 2 1( ) also free and (··· ((F1 ≀ α1) ≀ α2) ··· ) ≀ αh1(F ) = (··· ((F ≀ α1) ≀ α2) ··· ) ≀ αh1(F ). The iterated Murasugi sum F :=(··· (F A(O , 0)) A(O , 0) ··· ) A(O , 0) 2 1 *α 1 *α 2 *αh F h1(F ) 1 2 1( )

(where each Os is an unknot) is a Seifert surface of genus g(F2)=2h1(F ), and (as in Example 3) ∂F2 is an unknot. By Lemma 6, F2 is free. By Lemma 7, F2 = 3 ∂Hh1(F )\Int∆ for some Heegard handlebody Hh1(F ) ⊂ S and 2–disk ∆ ⊂ ∂Hh1(F ). 3 1 By Prop. 8, there is a moderate self-indexed Morse map f2 : S \ ∂F2 → S with exactly 2g(F2)=4h1(F ) critical points such that F2 is a large Seifert surface of f. But now, F :=(··· (F A(O′ , −1)) A(O′ , −1) ··· ) A(O′ , −1) 3 2 *γ O 1 *γ O 2 *γ Oh F h1 (F ) ( 1) ( 2) ( 1( )) ′ (where each Ot is an unknot) has boundary isotopic to ∂F . (The situation in the neighborhood of a single αs is pictured in Fig. 10.) Since the fibrations o{2, 2} contribute no new critical points,

f3 :=(··· (f2 * o{2, 2}) * o{2, 2}··· ) * o{2, 2} is also moderate and self-indexed with exactly 4h1(F ) critical points. If the isotopy 3 1 from ∂F2 to ∂F carries f2 to f : S \ ∂F → S , then f has the properties required of it.

The free genus of a knot K is gf (K) := min{g(F ): K = ∂F , F is a free Seifert surface}. More generally, for any link L let hf (L) := min{h1(F ): L = ∂F , F is a free Seifert surface}; so, for instance, hf (K)=2gf (K) if L = K is a knot, and hf (L) = 1 iff L = ∂A(O,n) for some n ∈ Z, n 6= 0.

Corollary 3. For any link L, MN(L) ≤ 2hf (L). In particular, for any knot K, MN(K) ≤ 4gf (K).

Let g(K), as usual, denote the genus of a knot K. Of course gf (K) ≥ g(K) for every knot K; there are knots K with g(K) = 1 and gf (K) arbitrarily large [11, 8]. For many knots K (satisfying a suitable condition on the Alexander polynomial ∆K (t)), it follows from [16] that MN(K) ≥ 2g(K). I know no example of a knot K for which it can be shown that MN(K) > 2g(K) (the techniques of [16], as they stand, cannot be used to do this). Some examples follow in which MN(K)=2g(K). Lemma 9. MN(∂A(O,n))=2 for all n 6= ±1. Proof. This generalization of Prop. 2 is implicit in Example 3. MURASUGISUMSOFMORSEMAPSTOTHECIRCLE 13

Figure 11. A proper arc α on S(o{2, 3},θ0); a free surface S(o{2, 3},θ0) *α A(O, 0) isotopic to a large Seifert surface S(o{2, 3} * u,θ0).

Figure 12. A non-free knotted annulus A(O{2, 3}, −1) isotopic to a small Seifert surface S(o{2, 3} * u,θ1); a non-free punctured torus A(O{2, 3}, −1) * A(O, −1) which is a large Seifert surface of a minimal Morse function (o{2, 3} * u) * o{2, 2} for the doubled knot D(O{2, 3}, −1, +).

Example 5. A twist knot is, by definition, the boundary of a Seifert surface of genus 1 of the form A(O,n) * A(O, ∓1). By Cor. 1, Lem. 9, and the fiberedness of A(O, ∓1), it follows that, if K is a twist knot, then MN(K) = 2 unless K is fibered, and in every case K has a free minimal Morse function.

Example 6. Let S be a surface of genus 1 with connected boundary. If α ⊂ S is a non-separating (i.e., boundary-incompressible) proper arc, then S ≀ α is an annulus; conversely, if c ⊂ S is a non-separating simple closed curve, then there is a non- separating proper arc α ⊂ S such that S ≀ α = NS(c). By judicious choices of α on the (genus 1) fiber surface F of one of the non-trivial fibered twist knots (that is, either a positive or negative O{2, ±3} = ∂(A(O, ∓1) * A(O, ∓1) or the figure-8 knot ∂(A(O, 1) * A(O, −1)), one may find annuli A(K,n) ⊂ F for knots K of infinitely many isotopy types. Any such annulus A(K,n) is the small surface, and ±1 ′ (Tα (F ) *α A(O, 0)) *γ(O) A(O , ∓1) is the large surface, of a moderate Morse map f : S3 \∂A(K,n) → S1 with exactly 2 critical points. Such a Morse map is minimal and not free. By plumbing on another copy of A(O, ∓1) we obtain infinitely many doubled knots D(K, n, ∓1) := ∂(A(K,n) * A(O, ∓1)) with MN(D(K, n, ∓1)) = 2, each of which has a non-free minimal Morse map. 14 LEE RUDOLPH

Historical note. Neuwirth [14] was apparently the first (1960) to consider, in effect, the notion of “free Seifert surface”: in a footnote, he called a Seifert surface F 3 “algebraically knotted” if π1(S \ F ) is not free. Neuwirth’s language was adapted by Murasugi [13], Lyon [9], and others. By 1976, Problem 1.20 (attributed to Giffen and Siebenmann) in Kirby’s problem list [5] uses the phrase “free Seifert surface”, points out that F is free iff S3 \ F is an open handlebody, and apparently introduces the terminology “free genus”. A number of later authors (e.g., Moriah [11], Livingston [8], M. Kobayashi and T. Kobayashi [6]) have studied free Seifert surfaces.

6. The inhomogeneity and Morse-Novikov number of a closed braid

Let σ1,...,σn−1 be the standard generators of the n–string braid group Bn. A b εb(s) braidword of length k in Bn is a k–tuple = (b(1),...,b(k)) such that b(s)= σib(s) for some (ib,εb): {1,...,k}→{1,...,n − 1}×{−1, 1}. The braid of b is β(b):= [ b(1) ··· β(k). The closed braid of b is β(b):=β(b), where the closure of a braid Bn is (as usual) a certain link in S3. A braidwordb b for which ib is surjective is strict. 3 If b is not strict, then β(b) is a split link (i.e., π2(S \ β(b) 6= {0}). b b Definition. Let b be a strict braidword in Bn. For i ∈ {1,...,n − 1} and ε ∈ − {−1, 1}, let v(i, ε, b):=card((ib,εb) 1(i,ε)). The inhomogeneity of b is n−1 I(b):= X min{v(i, −1, b), v(i, 1, b)} i=1 and the inhomogeneity of β ∈ Bn is I(β) := min{I(b): β = β(b), b is strict}. A braid with inhomogeneity 0 is homogeneous. The following result generalizes Theorem 2 of [23], which states that the closure of a homogeneous braid is a fibered link.

Proposition 6. For all n, for all β ∈ Bn, MN(β) ≤ 2I(β). b Proof. There is nothing to prove if n = 1. Let n = 2. Given a strict braidword b = εb(1) εb(k) (σ1 ,...,σ1 ) in B2, application of Seifert’s algorithm to the corresponding closed braid diagram for β(b) produces a Seifert surface S(b) for β(b) naturally b (0) (0) k b (1) equipped with a (0, 1)–handle decomposition S = (h1 ∪ h2 ) ∪ St=1 ht such that (0) (0) R3 3 (1) (0) h1 and h2 are 2–disks embedded in parallel planes in ⊂ S and ht joins h1 (0) to h2 with a single half-twist of sign εb(t). In case b is homogeneous, S(b) is (as is well known) a fiber surface. If b is not homogeneous, then (up to a cyclic permutation of b, which changes neither β(b) nor the isotopy type of S(b)) there exists t < k such that εb(t)=1= −εb(t + 1), and S(b) is isotopic to an annulus plumbing S(b′) * A(O, 0), where b′ := (b(1),...,b(t − 1),b(t + 2),...,b(k)). (The situation is pictured in Fig. 13.) By Prop. 2, Cor. 1, and induction on I(β), the proposition is true for n = 2. Finally, as observed by Stallings [23], for all n ≥ 2, application of Seifert’s algorithm to the braidword diagram of the closed braid of a strict braidword in Bn produces a Seifert surface which is an iterated Murasugi sum of n − 1 Seifert surfaces of strict braidwords in B2. By Cor. 1 and induction, the proposition is true for all n. Just like Theorem 2 of [23], which can be generalized to closed T-homogeneous braids for any espalier T on vertices {1,...,n} (see [20]), Prop. 6 can be generalized MURASUGISUMSOFMORSEMAPSTOTHECIRCLE 15

−1 Figure 13. S(σ1, σ1, σ1 , σ1) is isotopic by handle-sliding to S(σ1, σ1, σ1) * A(O, 0). in terms of a suitably defined notion of the “T-inhomogeneity” IT (β) of a braid β ∈ Bn. Similarly, the explicit construction (by lifting the fibration o through a suitable branched cover S3 → S3) of a fibration f : S3 \ β → S1, given in [17] for a closed homogeneous braid β and easily generalized (as mentionedb in [18]) to closed T-homogeneous braids, actuallyb gives for any T-bandword b (in particular for an ordinary bandword) an explicit self-indexed moderate Morse map f : S3\β(b) → S1 with 2IT(b) critical points. b

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Department of Mathematics, Clark University, Worcester MA 01610 USA E-mail address: [email protected]