Crosscap Number Three Alternating Knots

CROSSCAP NUMBER THREE ALTERNATING KNOTS

NOBORU ITO AND YUSUKE TAKIMURA

Abstract. In this paper, we have completed the determination of the set of alternating knots with the crosscap number three explicitly, which implies the lower bound 4 ≤ C(K) of the crosscap number C(K) of an alternating knot K.

1. Introduction In this paper, for every alternating knot K with the crosscap number C(K), we obtain the lower bound 4 ≤ C(K) by fixing the alternating knots with the crosscap number three (Main result 1, Tables 1 and 2, note that the set of alternating knots with C(K) ≤ 2 is known by Clark [4] and by us [12]). For example, we show 11-crossing knots which estimates are improved from the latest KnotInfo [3] as in Table 3. Our main result is given here. Main result 1 (a part of Theorem 2). Let K be a prime alternating knot and C(K) the crosscap number of K. Let X be the set of knot projections in the list of Tables 1 and 2, and let Xknot = {K | K is an alternating knot that is a knot type obtained from an alternating projection P ∈ X }. Then, K is a knot in Xknot if and only if C(K) = 3. We would like to mention that crosscap numbers of knots are discussed in the literature. Clark obtained that for a knot K, C(K) = 1 if and only if K is a 2-cable knot (in particular, for an alternating knot K, K is a (2, p)-torus knot) [4]. Clark also obtained an upper bound C(K) ≤ 2g(K) + 1 by using g(K), which is the orientable genus of K (this inequality holds for every knot K)[4]. For the inequality, Murakami and Yasuhara [15] gave an example C(K) = 2g(K) + 1 by K = 74, and sharp bounds C(K) ≤ ⌊n(K)/2⌋ for the minimum crossing number n(K) of a knot K (note that [5] of Hatcher and Thurston includes the particular case 74, which is discussed explicitly in Hirasawa-Teragaito [6]). Further, Murakami and Yasuhara [15] gave the necessary and sufficient condition for the crosscap number to be additive under the connected sum [15]. Historically, the orientable knot genus has been well studied, and a general algorithm for computations has been already known. For knots with low crossing numbers, effective calculations are made from genus bounds using invariants such as the Alexander polynomial and the Heegaard Floer homology. Although crosscap numbers are harder to compute, crosscap numbers of several families are known by Teragaito (torus knots) [17], Hatcher-Thurston [5] (2-bridge

Date: March 20, 2019. Key words and phrases. crosscap number (non-orientable genus), alternating knot, knot projection, spanning surfaces. MSC 2010: 57M25. 1 2 NOBORU ITO AND YUSUKE TAKIMURA

Table 1. Prime alternating projections with C(K) = 3 (List I, over/under informations are omitted) … …

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Table 2. Prime alternating projections with C(K) = 3 (List II, over/under informations are omitted) … …

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… … 4 NOBORU ITO AND YUSUKE TAKIMURA

Table 3. Sharp lower bounds of crosscap number three alternating knots with 11 crossings. There are 367 prime alternating knots with 11 crossings, and for 193 prime alternating knots for which the crosscap number is listed as unknown. For 82 of these knots, we give sharp lower bounds, and for 59 of these knots, we obtain the exact values of the crosscap numbers.

knot data in KnotInfo this paper knot data in KnotInfo this paper 11a 4 [3, 4] 4 11a 237 [3, 4] 4 11a 7 [3, 4] 4 11a 241 [3, 4] 4 11a 10 [3, 4] 4 11a 245 [3, 4] 4 11a 15 [3, 4] 4 11a 257 [3, 4] 4 11a 22 [3, 4] 4 11a 280 [3, 4] 4 11a 33 [3, 4] 4 11a 291 [3, 4] 4 11a 37 [3, 4] 4 11a 295 [3, 4] 4 11a 39 [3, 4] 4 11a 296 [3, 4] 4 11a 40 [3, 4] 4 11a 299 [3, 4] 4 11a 45 [3, 4] 4 11a 304 [3, 4] 4 11a 46 [3, 4] 4 11a 320 [3, 4] 4 11a 50 [3, 4] 4 11a 324 [3, 4] 4 11a 55 [3, 4] 4 11a 325 [3, 4] 4 11a 57 [3, 4] 4 11a 331 [3, 4] 4 11a 58 [3, 4] 4 11a 345 [3, 4] 4 11a 60 [3, 4] 4 11a 347 [3, 4] 4 11a 61 [3, 4] 4 11a 354 [3, 4] 4 11a 63 [3, 4] 4 11a 366 [3, 4] 4 11a 82 [3, 4] 4 11a 3 [3, 5] [4, 5] 11a 86 [3, 4] 4 11a 12 [3, 5] [4, 5] 11a 88 [3, 4] 4 11a 16 [3, 5] [4, 5] 11a 92 [3, 4] 4 11a 17 [3, 5] [4, 5] 11a 102 [3, 4] 4 11a 20 [3, 5] [4, 5] 11a 103 [3, 4] 4 11a 23 [3, 5] [4, 5] 11a 106 [3, 4] 4 11a 38 [3, 5] [4, 5] 11a 108 [3, 4] 4 11a 42 [3, 5] [4, 5] 11a 118 [3, 4] 4 11a 44 [3, 5] [4, 5] 11a 133 [3, 4] 4 11a 47 [3, 5] [4, 5] 11a 139 [3, 4] 4 11a 48 [3, 5] [4, 5] 11a 143 [3, 4] 4 11a 49 [3, 5] [4, 5] 11a 153 [3, 4] 4 11a 51 [3, 5] [4, 5] 11a 156 [3, 4] 4 11a 56 [3, 5] [4, 5] 11a 165 [3, 4] 4 11a 64 [3, 5] [4, 5] 11a 181 [3, 4] 4 11a 94 [3, 5] [4, 5] 11a 194 [3, 4] 4 11a 105 [3, 5] [4, 5] 11a 199 [3, 4] 4 11a 202 [3, 5] [4, 5] 11a 200 [3, 4] 4 11a 275 [3, 5] [4, 5] 11a 201 [3, 4] 4 11a 318 [3, 5] [4, 5] 11a 219 [3, 4] 4 11a 319 [3, 5] [4, 5] 11a 222 [3, 4] 4 11a 321 [3, 5] [4, 5] 11a 231 [3, 4] 4 11a 353 [3, 5] [4, 5] CROSSCAP NUMBER THREE ALTERNATING KNOTS 5 knots, in theory), Hirasawa-Teragaito (2-bridge knots, explicitly) [6], and Ichihara- Mizushima (many pretzel knots) [7]. Adams and Kindred [1] determine the crosscap number of the alternating knots, in theory. Kalfagianni and Lee [14] improve the eﬃciency of these computations by using the Jones polynomial to establish two sided bounds on crosscap numbers. In this paper, by using methods of Adams-Kindred [1], as a continution of [12], we ﬁx the set of alternating knots with the crosscap number three.

2. Preliminaries 2.1. Deﬁnitions and notations.

Deﬁnition 1 (knot, knot projection, knot diagram, alternating knot). A knot is a smooth embedding from a circle to R3. Two knots K and K′ are equivalent if there exists a homeomorphism of R3 onto itself which maps K onto K′. Each equivalence class of knots is called a knot type.A knot projection is an image of a generic immersion from a circle into a 2-sphere where every singularity is a transverse double point. In this paper, a transverse double point of a knot projection is simply called a double point, and a knot projection with no double points is called the simple closed curve and is denoted by O in this paper. A knot diagram is a knot projection with over/under information of each double point. Throughout this paper, in general, a knot diagram (a knot projection, resp.) is deﬁned not to be distinct from its mirror image. For a knot diagram, a double point with over/under information is called a crossing. As a special case, for a knot projection, one can arrange crossings in such a way that an under-path and an over-path alternate when we travel along the knot projection. The resulting knot diagram is called an alternating knot diagram. A knot is called an alternating knot if there exists a knot diagram D of the knot such that D is an alternating knot diagram.

Definition 2 (splices, splices of type S− or type RI−, Seifert splice, their inverses). Let P be a knot projection. For every double point d of P , we take a sufficient small open disk N(d) as in Figure 1 (a). Note that ∂cl(N(d)) ∩ P consists of four distinct points. Then, there exists an open disk N ′(d) with two simple arcs connecting four points on ∂cl(N(d)) ∩ P , as shown in Figure 1 (b) or (c) such that N ′(d) ∪ cl(S2 \ N(d)) = S2 and cl(N ′(d)) ∩ cl(S2 \ N(d)) = ∂cl(N(d)). Then, we call the replacement of N(d) with N ′(d) a splice. Note that for P , even if we apply a splice of the double point d in N(d), S2 \ N(d) is fixed. In this paper, we often present the connection by dotted curves as in Figure 2. For a splice preserving the number of components, as shown in a splice from (a-1) to (a-2) in Figure 2, there is a special case such that one of the two dotted curve is a simple arc, as in (b-1) to (b-2) as in Figure 2 and it is called a splice of type RI−. The inverse operation of a single RI− is denoted by RI+. A splice of type RI− is often denoted simply by RI−. The other case is called a splice of type S−.A splice of type S− is often denoted simply by S−. The inverse operation of a single S− is denoted by S+. A splice that appears in Figure 2 (c-1), (c-2) is called a Seifert splice or a splice of type Seifert. Here, dotted arcs as in (c-1) and (c-2) of Figure 2 present the global connection. State circles appearing in the process Seifert’s algorithm are called Seifert circles.

Remark 1. By deﬁnition, every splice is one of three types: S−, RI−, or Seifert. 6 NOBORU ITO AND YUSUKE TAKIMURA

Figure 1. (a) : N(d), (b) : N ′(d), and (c) : another type N ′(d)

Figure 2. A splice from (a-1) to (a-2), a splice of type RI− from (b-1) to (b-2), and a Seifert splice from (c-1) to (c-2). Since the deﬁnition of type Seifert does not need the orientation, thus, in Deﬁnition 2, we ignore the orientation. However, we often need the property of type Seifert whose orientation is preserved, as shown in (c-1) and (c-2).

Remark 2. An operation as shown in Figure 2 (a-1), (a-2) or (b-1), (b-2) is introduced by Calvo [2] (for the full twisted version) and [10] (for the half-twisted version), and in [10], it is called the inverse of a half-twisted splice operation, denoted by A−1. Fact 1 is known, and by definition, it is elementary to show it. Fact 1 (cf. Section 1 of [10]). Let P be a knot projection having exactly n double points. There exist at most 2n distinct sequence of splices of type S− and type RI− from P to the simple closed curve O, where each sequence consists of n splices in total. Definition 3 (u−(P ) and u−(K)). Let P be a knot projection and O the simple closed curve. Let S and RI be as in Definition 2. The nonnegative integer u−(P ) is defined by u−(P ) = min{ the number of S−’s in seq(P ) | seq(P ) : a finite sequence of splices of type S− and of type RI− from P to O }. Here, to define u−(P ), we implicitly use Fact 1. Let Z(K) be the set of knot projections − − obtained from alternating knot diagrams of K and let u (K) = minP ∈Z(K) u (P ). By definition, u−(K) is an alternating knot invariant. − b − b Example 1. Figure 3 gives examples of knot projections with u (11) = 0, u (51) − b = 1, or u (73) = 2 (for the symbol nbi, see [13]). In this paper, letting i be a positive integer, for a knot diagram ni in a table of a well-known book [16], the corresponding knot projection is denoted by Dˆ (for details, see [13]). Since the definition of a connected sum of two knot projections is (slightly) different from that of two knots, we obtain Definition 4 as follows. Definition 4 (a connected sum of two knot projections, a prime knot projection). Let Pi be a knot projection (i = 1, 2). Suppose that the ambient 2-spheres corresponding to P1,P2 are oriented. Let pi be a point on Pi where pi is not a double point (i = 1, 2). Let di be a sufficiently small disk with the center pi (i = 1, 2) CROSSCAP NUMBER THREE ALTERNATING KNOTS 7

− b − b − b Figure 3. u (11) = 0, u (51) = 1, or u (73) = 2.

e satisfying di ∩ Pi consists of an arc which is properly embedded in di. Let di = 2 e e e e cl(S \ di), Pi = Pi ∩ di, and let h : ∂d1 → ∂d2 be an orientation reversing home- e e e e omorphism where h(∂P1) = ∂P2. Then P1 ∪h P2 gives a knot projection in the e e e e oriented 2-sphere d1 ∪h d2. The knot projection P1 ∪h P2 in the oriented 2-sphere is denoted by P1♯(p1, p2, h)P2 and is called a connected sum of the knot projections P1 and P2 at the pair of points p1 and p2 (Figure 4). If a knot projection is not the simple closed curve and is not a connected sum of two knot projections, each of which is not the simple closed curve, it is called a prime knot projection. In this paper, a connected sum of knot projections is often simply denoted by P1♯P2 when no confusion is likely to arise.

Figure 4. A connected sum P1♯(p1, p2, h)P2 of two knot projections P1 and P2

− − − Fact 2 ([12]). For knot projections P1 and P2, u (P1♯P2) = u (P1) + u (P2).

Definition 5 (the connected sum of two knots). Let Ki be a knot (i = 1, 2) and Di a knot diagram of Ki. Let Pi be a knot projection obtained from Di by ignoring over/under information for every crossing. A connected sum D1♯(p1, p2, h)D2 is defined as a connected sum P1♯(p1, p2, h)P2 in Definition 4. Then, a knot having a knot diagram D1♯(p1, p2, h)D2 is called a connected sum of K1 and K2. Because it is well-known that a connected sum of K1 and K2 does not depend on (p1, p2, h), the connected sum is denoted by K1♯K2. Definition 6 (crosscap number). The crosscap number C(K) of a knot K is defined by C(K) = min{ 1 − χ(Σ) | a non-orientable surface Σ with ∂Σ = K}, where χ(Σ) is the Euler characteristic of Σ. Traditionally, we define that K is the unknot if and only if C(K) = 0.

Definition 7 (state surface, cf. [1]). Let P be a knot projection and DP a knot diagram by adding any over/under information to each double point of P . Let K(DP ) be the knot type where DP is a representative of the knot type. In this 8 NOBORU ITO AND YUSUKE TAKIMURA paper, by using the identification S2 = R2 ∪ {∞}, we often consider a knot pro- 2 jection P (knot diagram DP , resp.) on R . For a knot projection, by applying a splice to each double point, we have an arrangement of disjoint circles on R2. The resulting arrangement of circles on R2 are called a state, and these circles in a state are called state circles. For a state σ, every circle is filled with disks, and the nested disks stacked in some order. Then the surface is given by attaching half-twisted bands across the crossings of DP to obtain a surface Σσ spanning the knot K(DP ). For each crossing, the twisting is fixed by the type of the crossing. The surface generated by this algorithm is called a state surface.

Notation 1 (a state surface Σσ). For a knot projection P with n double points, { | − −}n splices σi σi = S or RI i=1 of the double points of P imply the sate surface { | − −}n Σσ, where σi is deﬁned by Figure 5 and let σ = σi σi = S or RI i=1.

− − Figure 5. If a double point corresponds to S (RI , resp.), σi is a splice of type S− (a Seifert splice, resp.).

Definition 8 (Seifert’s algorithm). For a knot, we orient it. Then, for every crossing of a knot diagram of the knot, if we choose the splice from (c-1) to (c-2) as in Figure 2, then the state surface given by Definition 7 is orientable (cf. Fact 3). The resulting surface does not depend on the orientation of the knot. Traditionally, for a given knot, this process from a knot diagram to a surface bounded by the knot is called Seifert’s algorithm. The state obtained by Seifert’s algorithm is called a Seifert state, and its state surface is called a Seifert state surface. Fact 3 (a well-known fact). Let n be a positive integer. For 2n states from an n crossing knot diagram, all except the Seifert state give non-orientable state surfaces. For every alternating knot K, there exists a Seifert state surface whose genus is g(K) by the algorithm in Definition 8. 2.2. Known facts. In Sections 4 and 5, we use Facts 4–6.

Fact 4 ([12]). Let P be a knot projection and DP a knot diagram by adding any over/under information to each double point of P . Let K(DP ) be the knot type having a knot diagram DP and C(K(DP )) the crosscap number of K(DP ). Then, − C(K(DP )) ≤ u (P ).

In particular, by letting χ be the Euler characteristic, Σ0 a non-orientable surface with the maximal χ spanning K(DP ), and Σu a state surface realizing the splices − − of u (P ), χ(Σ0) = χ(Σu) if and only if C(K(DP )) = u (P ). CROSSCAP NUMBER THREE ALTERNATING KNOTS 9

Notation 2. Let P be a knot projection and DP a knot diagram. A particular state surface introduced in Fact 4 is denoted by Σu, which is a state surface corresponding − to a sequence of splices that realized u (P ) (Remark 3). In other words, a Σσ − (Notation 1) realizing u (P ) is denoted by Σu. Remark 3 (cf. [12]). For the latter statement in the above, the outline of the proof as follows. Let n(P ) be the number of double points and |Sσ| the number of the circles in the state σ realizing u−(P ). Then,

χ(Σ0) = χ(Σu)

⇔1 − C(K(DP )) = |Sσ| − n(P ) − ⇔C(K(DP )) = u (P ).

1 Following [1], a genus is defined to be the orientable genus of a knot or 2 of the crosscap number. Adams and Kindred [1] showed that the orientable genus and the crosscap number are given by a state surface (Definition 7). We obtain Definition 9, and we review Fact 5.

Deﬁnition 9 (n-gon). Recall that a knot diagram is a knot projection with over/under information (Deﬁnition 1). Let D be a knot diagram and ∂F the boundary of the closure of a connected component F in S2 \ D. Suppose that for a positive integer n, when the components of D that lie on ∂F are removed, the reminder consists of n connected components, each of which is homeomorphic to an open interval. Then, ∂F is called an n-gon.

Fact 5 (Adams-Kindred, Theorem 3.3 of [1]). For every alternating knot diagram, the following algorithm (1)–(3) always generates a minimal genus state surface.

Minimal genus algorithm. Let DP be an alternating knot diagram.

(1) Find the smallest m for which DP contains an m-gon. (2) If m ≤ 2, then we apply the splice(s) to the crossing(s) so that the m- gon becomes a state circle. If m > 2, then m = 3 by a simple Euler characteristic argument on the knot projection (see, e.g., [14, Lemma 3.1] or [11, Lemma 2]). Then, choose a triangle of DP . From here, the process has two branches: For one branch, we apply splices to the crossings on this triangle’s boundary so that the triangle becomes a state circle. For the other branch, we apply splices to the crossings the opposite way. (3) Repeat Steps (1) and (2) until each branch reaches a state. Of all resulting state surfaces, choose the one with the smallest genus. Note that following [14], an expression of Fact 5 is obtained.

Fact 6 (Theorem 3.3 of [14]). Let K be an alternating knot, C(K) the crosscap number of K, and g(K) the orientable genus of K. Let Σ be a state surface with the maximal Euler characteristic obtained from the minimal genus algorithm as in Fact 5. Then, (1) If there exists Σ that is a non-orientable, then C(K) = 1 − χ(Σ). (2) If every Σ is orientable, then C(K) = 2 − χ(Σ) and C(K) = 2g(K) + 1. 10 NOBORU ITO AND YUSUKE TAKIMURA

3. Alternating knots with crosscap numbers ≤ 2 and knot projections with u−(P ) ≤ 2 Deﬁnition 10 (set ⟨S⟩). Let P and P ′ be knot projections. We say that P ∼ P ′ if P and P ′ are transformed each other by a ﬁnite sequence of operations of types RI. It is easy to see that ∼ gives an equivalence relation. Let S be a given set of knot projections. Let ⟨S⟩ = {P | P ∼ Q (∃Q ∈ S)}. Notation 3. Let l, m, n, p, q, and r be positive integers. Let P be a knot projection, T the set of (2, 2l − 1)-torus knot projections (l ≥ 2), R the set of (2m, 2n − 1)-rational knot projections (m ≥ 1, n ≥ 2), and P the set of (2p, 2q − 1, 2r − 1)-pretzel knot projections (p, q, r ≥ 1) as in Figure 6. Let ⟨T ⟩♯⟨T ⟩ = {P1♯P2 | P1,P2 ∈ ⟨T ⟩}. Let ⟨T ⟩♯⟨R⟩ = {P1♯P2 | P1 ∈ ⟨T ⟩,P2 ∈ ⟨R⟩}. Let ⟨T ⟩♯⟨P⟩ = {P1♯P2 | P1 ∈ ⟨T ⟩,P2 ∈ ⟨P⟩}. Let ⟨T ⟩♯⟨T ⟩♯⟨T ⟩ = {P1♯(P2♯P3) | P1,P2,P3 ∈ ⟨T ⟩}.

Notation 4. Let l, m, n, p, q, and r be positive integers. Let Tknot be the set of (2, 2l − 1)-torus knots (l ≥ 2), Rknot be the set of (2m, 2n − 1)-rational knots (m ≥ 1, n ≥ 2), and Pknot the set of (2p, 2q − 1, 2r − 1)-pretzel knots (p, q, r ≥ 1), ′ ′

as shown in Figure 7. Let Tknot♯Tknot = {L♯L | L, L ∈ Tknot}. … … … … … …

Figure 6. A (2, 2l − 1)-torus knot projection (l ≥ 2), a (2m, 2n − 1)-rational knot projection (m ≥ 1, n ≥ 2), and a (2p, 2q−1, 2r−1)-

pretzel knot projection (p, q, r ≥ 1) … … … … … …

Figure 7. A (2, 2l − 1)-torus knot (l ≥ 2), a (2m, 2n − 1)-rational knot (m ≥ 1, n ≥ 2), and a (2p, 2q−1, 2r−1)-pretzel knot (p, q, r ≥ 1)

In [12], we obtain Fact 7. CROSSCAP NUMBER THREE ALTERNATING KNOTS 11

Fact 7 (Proposition 1 and Theorem 3 in [12]). Let Tknot, Rknot, Pknot, and Tknot♯Tknot be as in Notation 4. Let K be an alternating knot and C(K) the crosscap number of K. Then, the following conditions are mutually equivalent.

(A) K ∈ Tknot. (B) C(K) = 1. (C) u−(K) = 1. Further, the following conditions are mutually equivalent.

(A) K ∈ Rknot ∪ Pknot ∪ Tknot♯Tknot. (B) C(K) = 2. (C) u−(K) = 2. To obtain Fact 7, we use the preliminary result (Fact 8), also given by [12]. Fact 8 (Theorem 1 in [12]). Let ⟨T ⟩, ⟨R⟩, ⟨P⟩, and ⟨T ⟩♯⟨T ⟩ be as Notation 3. Let P be a knot projection. (1) u−(P ) = 1 if and only if P ∈ ⟨T ⟩. (2) u−(P ) = 2 if and only if P ∈ ⟨R⟩ ∪ ⟨P⟩ ∪ ⟨T ⟩♯⟨T ⟩. In the next section, we discuss on alternating knots with the crosscap number 3 and knot projections with u−(P ) = 3.

4. Alternating knots with crosscap number 3 Notation 5. The symbol odd (odd3, even, even4, resp.) indicates that the number of double points is odd (odd (≥ 3), even, even (≥ 4), resp.). We show that three typical local cases that use the symbol “odd”, as shown in Figure 8. For example, the symbol “odd” indicates the number of labels x is odd, as shown in Figure 8. 1−1 Also we deﬁne the symbols (odd) or (even). For example, for Pαγ in Tables 1 and 2, there exist exactly four variables, each of which is “odd” or “even”, with no round brackets. Then, there exists the other possibility obtained by replacing “even” with “odd” at two places that are marked by round brackets and “odd” is replaced with “even” in the place that is marked by the round bracket (Figure 9). The symbols in the other knot projections are deﬁned in similar way.

Notation 6. Let P be a knot projection, and K(DP ) the knot type as in Deﬁni- alt tion 7. If K(DP ) is an alternating knot, we denote K(DP ) by K (P ). Tables 1 and 2 give a list of prime knot projections by using Notation 5. Let X be the set of alt knot projections in the list of Tables 1 and 2, and let Xknot = {K | K = K (P ) of P ∈ X }. Here, we introduce Theorem 1 to ﬁx the set of crosscap three alternating knots. Theorem 1. Let P be a knot projection. Let ⟨T ⟩♯⟨R⟩, ⟨T ⟩♯⟨P⟩, and ⟨T ⟩♯⟨T ⟩♯⟨T ⟩ be in Notation 3 and let X be the set as in Notation 6. • Suppose that P is a prime knot projection. Then, u−(P ) = 3 if and only if P is a knot projection in X . • Suppose that P is a non-prime knot projection. Then, u−(P ) = 3 if and only if P is a knot projection in ⟨T ⟩♯⟨R⟩, ⟨T ⟩♯⟨P⟩, or ⟨T ⟩♯⟨T ⟩♯⟨T ⟩. The proof of Theorem 1 is given in Section 6.

12 NOBORU ITO AND YUSUKE TAKIMURA

… … …

…

… … … … …

Figure 8. The symbol odd (odd3, even, even4, resp.) indicates the number of labels x is odd (odd (≥ 3), even, even (≥ 4), resp.). … … … … … …

… … … … … …

Figure 9. 1−1 The two possibilities of Pαγ

Theorem 2. Let K be a prime alternating knot, C(K) the crosscap number of K. − Let u (K) be as in Deﬁnition 3 and let Xknot the set as in Notation 6. Then, the following conditions are mutually equivalent.

(A) K is a knot in Xknot. (B) C(K) = 3. (C) u−(K) = 3.

5. Proof of Theorem 2 To obtain the proof of Theorem 2, we prepare the following technical lemmas (Lemma 1–Lemma 4).

Lemma 1. Let P be a knot projection and Kalt(P ) be as in Notation 6. If P ∈ X , then C(Kalt(P )) = u−(P ) = 3. CROSSCAP NUMBER THREE ALTERNATING KNOTS 13

Proof. Suppose that P ∈ X . Letting X ′ = X\{P | Dalt(P ) : a (2m, 2n)-rational alternating knot diagram (m, n ≥ 2), or a (2p − 1, 2q − 1, 2r − 1)-pretzel alternating knot diagram (p, q ≥ 2, r ≥ 1) }, we have two cases. Remark 4. Note that {P | Dalt(P ) : a (2m, 2n)-rational alternating knot diagram (m, n ≥ 2), or a (2p − 1, 2q − 1, 2r − 1)-pretzel alternating knot diagram (p, q ≥ 2, ≥ } 1−2 1−4 2−1 2−4 3 r 1) belong to Pαβ , P , Pαβ , P , and Pγδ (see Figure 10). Any other sets cannot obtain such rational and pretzel knots. … … … … … … … … …

… … … … …

Figure 10. 1−2 1−4 2−1 The set consisting of types Pαβ , P , Pαβ , 2−4 3 P , and Pγδ contain (2m, 2n)-rational alternating knot diagrams (m, n ≥ 2) and a (2p − 1, 2q − 1, 2r − 1)-pretzel alternating knot diagrams (p, q ≥ 2, r ≥ 1) in special cases.

• Case 1 (P ∈ X ′). Note that, by the minimal genus algorithm of Fact 5, alt there exists a surface Σ0 that spans K (P ) and has the maximal Euler characteristic χ(Σ0). Note also that a state surface Σu obtained from the − computation of u (P ) coincides with an element Σ0 having the maximal Euler characteristic χ(Σ0) in the set of state surfaces obtained by the minimal genus algorithm of Fact 5. Then, the equality χ(Σ0) = χ(Σu) holds. By Fact 4, the equality implies C(Kalt(P )) = u−(P ). Further, by the latter part of Theorem 1, u−(P ) = 3. • Case 2 (P ∈ X \ X ′): P ∈ {P | Dalt(P ) : a (2m, 2n)-rational alternating knot diagram (m, n ≥ 2), or a (2p − 1, 2q − 1, 2r − 1)-pretzel alternating knot diagram (p, q ≥ 2, r ≥ 1) }. By the latter part of Theorem 1 together with Facts 8 and 4, C(Kalt(P )) ≤ u−(P ) = 3. This fact together with Fact 7 implies that 3 ≤ C(Kalt(P )) ≤ 3, which implies C(Kalt(P )) = 3. □ Lemma 2. Let P be a knot projection that is f(S1) of an immersion. Then, for any two double points d, d′ of P , the conﬁguration of four points f −1({d, d′}) on S1 is one of two types (a) and (b) in Figure 12 (upper line). In other words, any two double points d and d′ are represented by Figure 12 (a) or (b)(lower line) where the dotted curves indicate the connections of the double points. Proof. Every knot projection is a 1-component curve, and thus, the possibilities of connections of the two double points are shown in Figure 12. □ Lemma 3. Suppose that there exist two double points as in Figure 12 (a). Then, after a Seifert splice at one of the two double point, any splice at the other double point yields another knot projection.

14 NOBORU ITO AND YUSUKE TAKIMURA … … … …

Figure 11. A rational knot in the left half is a pretzel knot in the right half (m, n ≥ 2). The pretzel knot is often denoted by a (2m − 1, 2n + 1, −1)-pretzel knot.

Figure 12. (a) : the leftmost knot projection and (b) : the two knot projections in the right half. Two double points and their connections. Dotted arcs indicate the connections of double points.

Proof. By Figure 13, it is elementary to show the claim. □

Notation 7. Let P be a knot projection with exactly n(P ) double points. For a { }n(P ) sequence Opi i=1 :

Op1 Op2 Op3 Opn(P ) P = P0 → P1 → P2 → · · · → Pn(P ) − − giving the state s, Opi is of type Seifert, S , RI , or an operation as in Figure 15. The operation as shown in Figure 15 joining two components into one component − − is denoted by Sjoin. An orientation presentation of S is given by Figure 14. We denote a Seifert splice splitting one component (joining two components, resp.) into two components (one component, resp.) by Tsplit (Tjoin, resp.).

− − Proposition 1. Let S , Sjoin, Tsplit, or Tjoin as in Notation 7. Let P be a prime { }n(P ) knot projection with exactly n(P ) double points. Let Opi i=1 be the set of splices obtained from the minimal genus algorithm of Fact 5. Let Pi be a union of some knot projections. CROSSCAP NUMBER THREE ALTERNATING KNOTS 15

Figure 13. Two successive Seifert splices on the two double points (upper arrow) and one Seifert splice and the other splice on the two double points (lower arrow)

Figure 14. An oriented presentation of S− (left half) corresponding to the unoriented version with orientations (right half)

Figure 15. − The operation Sjoin joining two components into one component. Dotted curves indicate connections of curves.

{ }n(P ) { }n(P ) Suppose that we arrange splices Opi i=1 and this sequence Opi i=1 is

Op1 Op2 Op3 Opn(P ) P = P0 → P1 → P2 → · · · → Pn(P ), − − which includes exactly one S and exactly one Tjoin or Sjoin. { ′ }n(P ) { }n(P ) Then there exists a sequence Opi i=1 consisting of splices in Opi i=1 of the ′ − ′ − ′ − − state such that Op1 = S , Op2 = S , Op3 = S , which are followed by m 3 Seifert splices, each of which is Tsplit. In order to show Proposition 1, we prepare Definitions 11 and 12. Definition 11 (chord diagram (cf. [9])). We define a chord diagram as a configu- ration of n pair(s) of points on a circle up to ambient isotopy and reflection of the circle. Traditionally, two points of each pair are connected by a straight arc. The arc is called a chord. 16 NOBORU ITO AND YUSUKE TAKIMURA

Definition 12 (a chord diagram CDP of a knot projection P ). Let P be a knot projection. Then, there is a generic immersion g : S1 → S2 such that g(S1) = P . We define a chord diagram of P , for example, Figure 16, as follows. Let k be the number of the double points of P . We fix a base point, which is not a double point

Figure 16. A chord diagram CDP of a knot projection P . on P , and choose an orientation of P . Starting from the base point, proceed along P by the orientation of P . Assign 1 to the first double point that we encounter. Then, we assign 2 to the next double point that we encounter provided it is not the first double point. Suppose that we have already assigned 1, 2, . . . , p. Then, we assign p + 1 to the next double point that we encounter if it has not been assigned yet. Following the same procedure, we finally assign all the double points. Here, note that g−1(double point assigned i) consists of two points on S1 and we shall assign i to them. The chord diagram represented by g−1(double point assigned 1), g−1(double point assigned 2), . . . , g−1(double point assigned k) on S1 is denoted by CDP . This CDP is called a chord diagram of the knot projection P . (Proof of Proposition 1.) Proof. Suppose that we give an orientation to the knot projection P and each orientation of Pi is obtained by Figure 2 (Figure 14, Figure 15, resp.) for every − − splice of type Seifert (S , Sjoin, resp.). Note that every Seifert splice does not { }n(P ) change a given orientation. If the given Opi i=1 is presented by − ··· − ··· S Tsplit TsplitSjoin , then there exists Tsplit splitting one component into two components, which will − be joined by the Sjoin (such Tsplit may not be unique) where the two double points − corresponding to Tsplit and Sjoin are represented by the left figure of Figure 13. It is easy to see that the two splices correspond to the lower arrow of Figure 13, and − thus, by exchanging the order of the two splices, these Tsplit and Sjoin are replaced − ′ − ′ − ′ − by two S ’s. Then we set Op1 = Op1 = S , Op2 = S , and Op3 = S . Thus, in the above cases, the statement holds. Therefore, next, we consider the other case, which implies that every Opi (2 ≤ i ≤ n(P )) is a splice of type Seifert. Then the { }n(P ) sequence Opi i=1 is presented as − S Tsplit ··· TsplitTjoin ··· .

For the Tjoin that we ﬁrstly encounter, there exists an operation Tsplit splitting one component into two components, which will be joined by Tjoin, i.e., this Tsplit should be followed by this Tjoin (such Tsplit may not be unique). Then we can retake Op1 − = S , Op2 = Tsplit, and Op3 = Tjoin. CROSSCAP NUMBER THREE ALTERNATING KNOTS 17

In this case, for the pair TsplitTjoin appears in Figure 13 (upper arrow). Then − we have Figure 17 that shows the three double points corresponding to S , Tsplit, + and Tjoin in the prime knot projection P are realized by applying S to the knot projection as shown in type (a) of Figure 12. There are two cases. − • Case 1: For S , Tsplit, and Tjoin, first we consider six cases as in Figure 18, ′ ≤ ≤ it is elementary to see that we can take Opi (1 i 3) where every ′ − ′ − ′ − Op1 = S , Op2 = S , and Op3 = S , which implies that the statement of Proposition 1 holds. • Case 2: The other case is presented by the left figure of Figure 19. The corresponding CDP is shown as the right figure of Figure 19. By the assumption, P is prime, then there exist chords c1, c2, . . . , ck intersecting − the chord corresponding to the splice S as shown in CDP of the right figure of Figure 19 (Here, this is the case that the part (A) intersects the part (B). The other cases consist the pair (A), (C) and the pair (A), (C’). Since the argument of each case is the same as the pair (A), (B), the other cases are omitted and we consider the pair (A), (B) only). Note that k must be an even positive integer (see [8, Figure 10.20]). After we apply − Op1 (= S ) to P , CDP and CDP1 are presented by Figure 20 since now Op2 = Tsplit, Op3 = Tjoin, and every Opi (4 ≤ i ≤ n(P )) is Tsplit.

Figure 17. An application of a single S+ to each knot projection (in six cases) implies a knot diagram (three types).

Since chords will be parallel by Op1 (Figure 20), c1, c2, . . . , ck are bound- aries of bigons, which are presented by Figure 21. Note that a state by the minimal genus algorithm of Fact 5 applying a splice to a double point so that a bigon becomes a state circle. Note also that the algorithm of Fact 5 always generates a minimal genus state surface bounding a given alternating knot. Therefore, we may ﬁx splices implying that each bigon becomes { }n(P ) − a state circle. Then Opi i=1 includes at least two S ’s, as shown in Figure 21, which contradicts the assumption. □

Proposition 1 implies Lemma 4. 18 NOBORU ITO AND YUSUKE TAKIMURA

Figure 18. For every knot diagram of three types (1)–(3) obtained from Figure 17, splices on the three double points that are of type S−.

Figure 19. Case 2. Three splices Opi (1 ≤ i ≤ 3) consisting of − S , Tsplit, and Tjoin of a prime knot projection P (left) and its chord diagram. The dotted part (A) ((B), (C), (C’), resp.) of P corresponds to (A) ((B), (C), (C’), resp.) of CDP .

Lemma 4. Let P be a prime knot projection, n(P ) the number of double points, and s a state of P obtained from the minimal genus algorithm of Fact 5 where the number of circles in s is n(P ) − 2. Let Opi be a splice (1 ≤ i ≤ n(P )). ≥ { }n(P ) Suppose that n(P ) 3 and there exists a sequence Opi i=1 :

Op1 Op2 Op3 Opn(P ) P = P0 → P1 → P2 → · · · → Pn(P ) = s − where exactly one Opi is S . { }n(P ) { ′ }n(P ) − Then, by modifying Opi i=1 , we can retake Opi i=1 consisting of Opi = S (1 ≤ i ≤ 3) followed by n(P ) − 3 Seifert splices. CROSSCAP NUMBER THREE ALTERNATING KNOTS 19

Figure 20. Even (e.g., four, in the ﬁgure) chords that intersect − the chord labeled by S (= Op1), which corresponds to Figure 19. If we apply Op1, then we have a chord diagram as shown in the right ﬁgure.

Figure 21. A knot projection with odd (e.g., three, in this figure) bigons consisting of even (e.g., four, in this figure) double points that correspond to non-labeled even (e.g., four, in this figure) chords as in Figure 20.

Proof. Note that if the splices are Seifert splices of type Tsplit (Notation 7) to obtain a state of a knot projection with exactly n(P ) double points, the number of circles in s is n(P ) + 1. Now, since the state s of P satisﬁes that the number of circles − − in s is n(P ) − 2 and exactly one Opi is S , the splices should consist of one S , − − n(P ) 2 Tsplit, and the single splice that is Tjoin or Sjoin. Hence, the assumption of Proposition 1 is satisﬁed. □

(Proof of Theorem 2.) (Proof of (A) ⇒ (B).) For a knot projection P , let Kalt(P ) be the knot type as in Notation 6. Then,

Lemma 1 ⟨X ⟩ ⊂ {P | C(Kalt(P )) = u−(P ) = 3} Theorem 1 ⊂ ⟨X ⟩. Then, Claim 1. ⟨X ⟩ = {P | C(Kalt(P )) = u−(P ) = 3}. 20 NOBORU ITO AND YUSUKE TAKIMURA

Note that a knot projection P uniquely determines an alternating knot diagram up to reflection. By Claim 1, the left-hand side ⟨X ⟩ determines {Kalt(P ) | P ∈ alt ⟨X ⟩}, which equals Xknot. On the other hand, the right-hand side {P | C(K (P )) = u−(P ) = 3} determines {Kalt(P ) | C(Kalt(P )) = u−(P ) = 3}, which is equal to {K : an alternating knot | C(K) = 3, u−(K) = 3} (for the definition of u−(K), see Definition 3). Therefore,

Claim 2.

− Xknot = {K : an alternating knot | C(K) = 3, u (K) = 3}.

Claim 2 implies that (A) ⇔ (B). ((B) ⇒ (A).) Suppose that C(K) = 3 and K is an alternating knot. We consider Case 1 and Case 2 corresponding to (1) and (2) of Fact 6, respectively. • Case 1: there exist a knot diagram Dalt(K), a state s of Dalt(K) such that a non- alt orientable state surface Σ0 obtained from D (K) satisﬁes that C(K) = 1−χ(Σ0), i.e., χ(Σ0) = −2. Here, note that the state s is obtained by the algorithm of Fact 5. Let P be a knot projection obtained from Dalt(K) by ignoring over/under information. Let n(P ) be the number of double points of P . The state s is obtained from P by n(P ) splices. In the following, we ﬁnd the state s in the 2n(P ) candidates. Since χ(Σ0) = −2, note that the splices consist of n(P )−3 Seifert splices producing n(P ) − 2-component curves and three S−’s (Case 1-(i)), or the splices consist of n(P ) − 2 Seifert splices producing n(P ) − 1-component curves, a splice decreasing − − components (i.e., Tjoin or Sjoin of Notation 7) and a single S (Case 1-(ii)). Here, note that if there exist four splices of type S− in the n(P ) splices, then the n(P ) splices do not realize Σ0 because χ(Σ0) = 1 − C(K). Thus, there are no other possibilities. Note also that by Lemma 4, if P is prime, then every possibility of Case 1-(ii) returns to Case 1-(i). Then, we interpret the n(P ) splices as a sequence of the n(P ) splices, and we may suppose that there exists a sequence such that

Op1 Op2 Op3 Opn(P ) P = P0 → P1 → P2 → · · · → Pn(P ) = s

− where Opi = S (1 ≤ i ≤ 3) and Opi = Tsplit (4 ≤ i ≤ n(P )). By Lemma 2, for any Opµ and Opν (4 ≤ µ < ν ≤ n(P )), each of which is Tsplit, the two double points corresponding to Opµ and Opν are represented as in Figure 12 (b) since Tsplit must increase components. Thus, noting that the state s has one to one correspondence with the n(P ) splices (Deﬁnition 7), it is easy to choose σ4, σ5, . . . , σn(P ) like Σu (see Notation 1 and Notation 2) corresponding to RI−’s applied successively to P (= P0) to obtain s. Here, note that there exists at least one 1-gon in each Pi (1 ≤ i ≤ n(P )) since every pair consisting of two double points is represented as in Figure 12 (b). As a result, u−(K) ≤ 3. Here, since K is not the unknot, and is not − in Tknot ∪ Rknot ∪ Pknot ∪ Tknot♯Tknot, we have 3 ≤ C(K) ≤ u (K)(∵ Facts 7 and 4). Thus, u−(K) = 3. • Case 2: For an orientable genus g(K), C(K) = 2g(K) + 1. If C(K) = 3, then g(K) = 1. Then, K is a (2m′, 2n′)-rational knot (m′, n′ ≥ 1) or (2p′ − 1, 2q′ − 1, 2r′ − 1)-pretzel knot (p′, q′, r′ ≥ 1). Note that, in each case, for K, there exists an alternating knot diagram D(K) implying projection P such that u−(P ) = 3. This fact together with the assumption C(K) = 3 implies 3 = C(K) ≤ u−(K) ≤ 3 (∵ Fact 4). CROSSCAP NUMBER THREE ALTERNATING KNOTS 21

((A) ⇔ (C).) Since Claim 2 immediately implies that (A) ⇒ (C), it is suﬃcient to − − show that (C) ⇒ (A). Recall that u (K) = minP ∈Z(K) u (P ) (Deﬁnition 3). u−(K) = 3 ⇒∃P ∈ Z(K) such that P ∈ ⟨X ⟩ (∵ Theorem 1)

⇒K ∈ Xknot. □ Example 2. Theorem 2 depends crucially on the primeness. For example, let- b b alt b b ting 74♯31 be the knot projection as in Figure 22, C(K (74♯31)) = 3 by using − b b − b − b [15, Proposition 4.3]. Note also that u (74♯31) = u (74) + u (31) = 3 + 1 = 4 since u−(P ) is additive under a connected sum of knot projections (Fact 2). Its

generalization is given by knot projections in the right ﬁgure of Figure 22.

… …

alt b b − b b − b − b Figure 22. C(K (74♯31)) = 3 and u (74♯31) = u (74) + u (31) = 4 (left). The same results as this hold for general cases (right).

6. Proof of Theorem 1 Recall Notation 3 and Fact 8. We represent R (P, resp.) by Case 1 (Case 2, resp.) as in Figure 23. The set ⟨T ⟩♯⟨T ⟩ will be treated in the last of this section. Move 1 satisﬁes Fact 9. Move 1. For any pair of simple arcs lying on the boundary of a common region, each of the two local replacements as in Figure 24 by applying operations of type RI+ i − 1 times followed by an operation of type S+.

Fact 9 ([12]). Let P be a knot projection and O the simple closed curve. Let ⟨{O}⟩ be as in Definition 10. The following conditions are equivalent. (A) P satisfies u−(P ) = n. (B) There exists Q ∈ ⟨{O}⟩ such that P is obtained from Q by applying Move 1 successively n times. In the following, by using Fact 9, for each region (i) (1 ≤ i ≤ 4) of Case 1 and (j) (1 ≤ j ≤ 5) of Case 2 to which Move 1 to be applied, we find a knot projection P with u−(P ) = 3 from a knot projection P ′ with u−(P ′) = 2 in case by case computations. We explain symbols which we use. The symbol (i)’ ((j)’, resp.) shows that for the case of (i)’ ((j)’, resp.), the same argument as that of (i) ((j), resp.) works. 22 NOBORU ITO AND YUSUKE TAKIMURA … …

… … …

Figure 23. Case 1 and Case 2

Figure 24. Two local replacements

Only some cases i = 1, 2 and j = 1, 3, 5 exist. Further, in the following table, the term “arcs” represents two arcs to which Move 1 to be applied. Suppose that Pxy (Pzw, resp.) is a knot projection obtained by applying Move 1 to arcs x and y (z and w, resp.). If Pxy is a mirror image of Pzw, then we denote “xy (zw)” to indicate arcs applied by Move 1 (for example, “αγ (βγ)” appears in Case 1, the region (1)).

Case 1 (Figure 25), the region (1): …

Figure 25. α, β, and γ in the region (1) of Case 1 CROSSCAP NUMBER THREE ALTERNATING KNOTS 23

arcs the resulting knot projection 1-1 αβ Pαβ 1-1 αγ (βγ ) Pαγ 1-1 γγ Pγγ Case 1 (Figure 26), the region (2): …

Figure 26. α, β, and γ in the region (2) of Case 1

arcs the resulting knot projection 1-2 αβ Pαβ 1-2 αγ (βγ) Pαγ 1-2 γγ Pγγ Case 1, each region (3): For any pair consisting of two arcs bounding a region (3), we have P 1-3.

Case 1, each region (4): For any pair consisting of two arcs bounding a region (4), we have P 1-4.

Case 2 (Figure 27), the region (1):

Figure 27. α, β, and γ in the region (1) of Case 2

arcs the resulting knot projection 2-1 αβ Pαβ 2-1 αγ (βγ ) Pαγ Case 2 (Figure 28), the region (2): 24 NOBORU ITO AND YUSUKE TAKIMURA

… …

Figure 28. α, β, and γ in the region (2) of Case 2

arcs the resulting knot projection 2-2 αβ Pαβ 2-2 αγ (βγ ) Pαγ 2-2 αα (ββ ) Pαα Case 2 (Figure 29), the region (3): …

…

Figure 29. α, β, γ, and γ in the region (3) of Case 2

arcs the resulting knot projection 2-3 αβ Pαβ 2-3 αγ Pαγ 2-3 αδ Pαδ 2-3 βγ Pβγ 2-3 βδ Pβδ 2-3 γδ Pγδ 2-3 ββ Pββ 2-3 δδ Pδδ Case 2, each region (4): For any pair consisting of two arcs bounding a region (4), we have P 2-4.

Case 2, each region (5): For any pair consisting of two arcs bounding a region (5), we have P 2-5. CROSSCAP NUMBER THREE ALTERNATING KNOTS 25 … … … … … … … … … …

Figure 30. Cases 3–6

In the following, we discuss on the other cases. For each knot projection of Cases 3–6, there exist the curve C or C′, each of which decomposes S2 into two disjoint open disks, as shown in Figure 30. Therefore, if we apply an operation of Move 1 to a knot projection in one of the simple closed curves, the resulting knot projection is not prime. Then, we decompose S2 by C,C′ and have two distinct 2 disks D1,D2, and the annulus (= S \ (D1 ∪ D2)). Then, we apply an operation of Move 1 to two arcs which are belong to diﬀerent disks D1,D2 for each case. The resulting knot projections are listed in the following.

arcs the resulting knot projection 3 Case 3 αβ Pαβ 3 Case 3 γδ Pγδ 4 Case 4 αβ Pαβ 5 Case 5 αβ Pαβ 5 5 Case 5 γδ Pγδ (= Pαβ) 6 Case 6 αβ Pαβ We check the cases and it completes the proof. □ Acknowledgement The authors would like to thank Professor Charles Livingston for updating the table of KnotInfo [3] by a question of N. I. The authors would like to thank Professor Tsuyoshi Kobayashi and Professor Makoto Ozawa for their comments. References

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[5] A. Hatcher and W. Thurston, Incompressible surfaces in 2-bridge knot complements, Invent. Math. 79 (1985), 225–246. [6] M. Hirasawa and M. Teragaito, Crosscap numbers of 2-bridge knots, Topology 45 (2006), 513–530. [7] K. Ichihara and S. Mizushima, Crosscap numbers of pretzel knots, Topology Appl. 157 (2010), 193–201. [8] N. Ito, Knot projections, CRC Press, Bosca Raton, FL, 2016. [9] N. Ito, Based chord diagrams of spherical curves, Kodai Math. J. 41 (2018), 375–396. [10] N. Ito and A. Shimizu, The half-twisted splice operation on reduced knot projections, J. Knot Theory Ramiﬁcations 21 (2012), 1250112, 10 pp. [11] N. Ito and Y. Takimura, Triple chords and strong (1, 2) homotopy, J. Math. Soc. Japan 68 (2016), 637–651. [12] N. Ito and Y. Takimura, Crosscap number and knot projections, Internat. J. Math., 29 (2018), 1850084, 21 pp. [13] N. Ito and Y. Takimura, The tabulation of prime knot projections with their mirror images up to eight double points, Topology Proc., 53 (2019) pp. 177–199. [14] E. Kalfagianni and C. R. S. Lee, Crosscap numbers and the Jones polynomial. Adv. Math. 286 (2016), 308–337. [15] H. Murakami and A. Yasuhara, Crosscap number of a knot, Paciﬁc J. Math. 171 (1995), 261–273. [16] D. Rolfsen, Knots and links, Mathematical Lecture Series, Publish or Perish, Inc., Berkeley, Calif., 1976. [17] M. Teregaito, Crosscap numbers of torus knots, Topology Appl. 138 (2004), 219–238.

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo, 153-8914, Japan Email address: [email protected] Gakushuin Boys’ Junior High School, 1-5-1 Mejiro, Toshima-ku, Tokyo, 171-0031, Japan Email address: [email protected]