       THE REGION INDEX AND THE RASMUSSEN INVARIANT OF A KNOT
KANA FURUTA AND TOSHIFUMI TANAKAAbstract. For a knot in the 3-sphere S 3 , there exists a diagram such that we have the unknot bychanging all crossings on the boundary of some region of the diagram. The minimal number of thecrossing changes over all such diagrams is called the region index of a knot. In this paper, we showthat there exists inﬁnite families of knots which have the region indices equal to two and three byusing the Rasmussen invariant.

 unknotting  number; the 
Key words: Knots;
theRasmusseninvariant.

1. Introduction
3Let D be a diagram of a knot in S and let D+ be the four-valent graph which is obtained fromD by replacing each crossing with a vertex. We call the closure of each connected component ofS 2 − D+ a region of D. A region crossing change at a region R of D is the local deformation on Dby the changing all the crossings on the boundary of R [8]. For any knot K, it is known that there
exists a diagram D such that D can be transformed into a diagram of the unknot by performing asingle region crossing change to some region R of D [1]. The region index of a knot K, denoted by
Reg(K), is deﬁned to be the minimal number of crossings on the boundary of an unknotting regionof an unknotting region diagram of K over all such diagrams. The region index has been introduced
by A. Kawauchi, K. Kishimoto and A. Shimizu. The unknotting number of a knot K, denoted byu(K), is the minimal number of crossing changes needed to create the unknot, the minimum being
taken over all possible sets of changes in all diagrams of K [4][5]. The unknotting number is less thanor equal to the region index. In particular, we can easily see that the region index of any knot withunknotting number one is larger then or equal to two by the deﬁnition. In general, it is not easy tocalculate the region index. In this paper, we obtain the following results.
Theorem 1.1. There exists an inﬁnite family of knots {Km } in S 3 such that Reg(Km ) = 2.
Theorem 1.2. There exists an inﬁnite family of knots {Kn } in S 3 such that Reg(Kn ) = 3.
In Sections 2, we shall give proofs of Theorems 1.1 and 1.2. In Section 3, we shall show examples
of calculation of the region index.
Acknowledgements. The second author is partially supported by the Ministry of Education, Science,Sports and Culture, Grant-in-Aid for Scientiﬁc Research(C), 2016-2018ʢ16K05145).
2. Proofs of Theorems 1.1 and 1.2
Proof of Theorem 1.1. Let m be a positive integer. We consider the knot Km which has a diagram
as in Figure 1. We can calculate that the Goeritz invariant of Km is equal to (4m + 1) [2][3]. In
particular, it is non-trivial. Thus 2 ≤ Reg(Km ) by the deﬁnition of the region index. On the otherhand, if we perform a region crossing change at the region R as indicated in Figure 1, then we have adiagram of the unknot. So we know that Reg(Km ) ≤ 2. Thus we have Reg(Km ) = 2. This completes
the proof.
Proof of Theorem 1.2. Let n be a positive odd integer. We consider the knot Kn which has a diagram
as in Figure 2. Then we know that Kn is a positive knot and the Rasmussen invariant is equal tothree by Theorem 4 in [7]. Thus by Theorem 1 in [7], we know that 3 ≤ g∗ (Kn ) ≤ u(Kn ) where1    2 

KANA FURUTA AND TOSHIFUMI TANAKAg∗ (Kn ) is the smooth slice genus of Kn . On the other hand, if we perform a region crossing change
at the region R as indicated in Figure 2, then we have a diagram of the unknot. So we know thatReg(Kn ) ≤ 3. Thus we have Reg(Kn ) = 3. We can also calculate that the Goeritz invariant of Kn isequal to (4n − 1) [2][3]. This completes the proof.

RmFigure 1. The knot Km

Rn : odd

Figure 2. The knot Kn

3. Examples
Example 3.1. Reg(819 ) = 3.
The knot 819 is the knot which has a diagram as in Figure 3. If we perform a region crossing change
at R as indicated in Figure 3, then we have the unknot. So we know that Reg(819 ) ≤ 3. On the otherhand, the unknotting number is known to be three [4]. Thus we know that Reg(819 ) = 3.
Example 3.2. Reg(938 ) = 3.

       THE REGION INDEX AND THE RASMUSSEN INVARIANT OF A KNOT

3

RFigure 3. The knot 819
The knot 938 is the knot which has a diagram as in Figure 4. By performing a region crossingchange at R, then we have the unknot. So we know that Reg(938 ) ≤ 3. On the other hand, Owens
showed that u(938 ) = 3 [6]. Thus we know that Reg(938 ) = 3.

RFigure 4. The knot 938
Example 3.3. Reg(949 ) = 3.
The knot 949 is the knot as in Figure 5. If we perform a region crossing change at R, then we havethe unknot. So we know that Reg(949 ) ≤ 3. On the other hand, Stoimenow showed that u(949 ) = 3[9]. Thus we know that Reg(949 ) = 3.
RFigure 5. The knot 949
References
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2. K. Ikeda, T. Ikeda, T. Kawakami, H. Sugimoto, T. Sugiura, J. Yagi, S. Yamanaka, Goeritz invariants of two-bridgelinks and torus links, Kochi J. Math. 5 (2010), 163–172.
3. T. Ikeda, H. Sugimoto, Goeritz invariants of pretzel links, J. Geom. 104 (2013), no. 1, 127–136.
4. A. Kawauchi, A survey of knot theory, Translated and revised from the 1990 Japanese original by the author,
Birkhauser Verlag, Basel, 1996.

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KANA FURUTA AND TOSHIFUMI TANAKA

5. W. B. R. Lickorish, An introduction to knot theory, Graduate Texts in Mathematics, 175. Springer-Verlag, NewYork, 1997.
6. B. Owens, Unknotting information from Heegaard Floer homology, Adv. Math. 217 (2008), no. 5, 2353–2376.
7. J. A. Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 (2010), no. 2, 419–447.
8. A. Shimizu, Region crossing change is an unknotting operation, J. Math. Soc. Japan 66 (2014), no. 3, 693–708.
9. A. Stoimenow, Polynomial values, the linking form and unknotting numbers, Math. Res. Lett. 11 (2004), no. 5-6,755–769.
Department of Mathematics, Faculty of Education, Gifu University, Yanagido 1-1, Gifu, 501-1193, Japan.
Department of Mathematics, Faculty of Education, Gifu University, Yanagido 1-1, Gifu, 501-1193, Japan.
E-mail address: tanakat@gifu-u.ac.jp