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 A KNOT INVARIANT DERIVED FROM GAMMA MOVESTOSHIFUMI TANAKA

Abstract. Any knot can be changed into either the unknot or the trefoil knot by aﬁnite sequence of gamma moves. We deﬁne the gamma index of a knot as the minimum
number of moves among all such sequences. We show that the gamma indices of theknots 41 , 52 , 61 and 63 are equal to one and the gamma index of the connected sum of
the trefoil knot and its mirror image is equal to two.
Key words: Knot; Gamma move.

1. Introduction
A knot is an embedding of a circle in the oriented 3-dimensional sphere S 3 . Two knots are said tobe equivalent if they represent ambient isotopic embeddings.
We deﬁne gamma moves to be changes in a diagram of a knot as in Figure 1. We call two knots
gamma equivalent if there exists a sequence of gamma moves that take us from the one knot to theother, where we can arrange the diagram of the knot any way that we want after each gamma move.
Kauﬀman showed that every knot is gamma equivalent to either the unknot or the trefoil knot [2].

Figure 1
We deﬁne the gamma index of a knot K as the minimal number of gamma moves needed to convert
the knot into either the unknot or the trefoil knot where the minimum is taken over all possible
diagrams of the knot. We denote it by Γ(K).
Theorem 1.1. For any knot K in {41 , 52 , 61 , 63 } as in Figure 2, we have Γ(K) = 1.
Let K̂ be the connected sum of the trefoil knot and its mirror image as in Figure 3. Then let Kn
be the connected sum of n K̂’s.
2
Theorem 1.2. We have n ≤ Γ(Kn ) ≤ n for any positive integer n. In particular, Γ(K2 ) = 2.
312TOSHIFUMI TANAKA

41

52

61

63

Figure 2

the trefoil knot

KFigure 3
2. Proof of Theorem
Lemma 2.1. Each of the following moves is obtained from a single gamma move.

Figure 4
Proof. The result is obtained as shown in Figure 5.

=oror
gamma move

Figure 5       

A KNOT INVARIANT DERIVED FROM GAMMA MOVES3

For a knot K, we deﬁne the Arf invariant Arf (K) [1] [5] of K by Arf (K) = a2 (K) (mod 2), wherea2 is the second coeﬃcient of the Conway polynomial of K. We need the following result.
Proposition 2.2. (Kauﬀman)[2] Two knots are gamma-equivalent if and only if the Arf invariants
are equal.
By the following proposition, we know that there is a knot of arbitrarily large gamma index.
Proposition 2.3. For a knot K, u(K) ≤ 3Γ(K)+1 if Arf (K) = 1 and u(K) ≤ 3Γ(K) if Arf (K) = 0,
where u(K) is the unknotting number of K.
Proof. A gamma move is realized by applying three crossing changes and the unknotting number of
the trefoil knot is equal to one. Thus the inequalities are obtained from Proposition 2.2.
Proof of Theorem 1.1. The theorem is shown by using the deﬁnition of gamma moves and lemma 2.1
as follows. We can show Γ(41 ) = 1 as shown in Figure 6.

==
gamma move

Figure 6
We have Γ(52 ) ≤ 1 as shown in Figure 7.

=gamma move===

Figure 7

4

TOSHIFUMI TANAKA

We have Γ(61 ) ≤ 1 as shown in Figure 8.

=gamma move===Figure 8
We have Γ(63 ) ≤ 1 as shown in Figure 9.

=
gamma move===Figure 9
It is easily seen that each knot K of {41 , 52 , 61 , 63 } is non-trivial and not isotopic to the trefoil knot.
Thus we have Γ(K) ≥ 1. This completes the proof.
Proof of Theorem 1.2. We can show that u(Kn ) = 2n by using the Nakanishi index of a knot [3].
In fact, Nakanishi index is a lower bound of the unknotting number and it is easily seen that theNakanishi index of Kn is equal to 2n. (See [4].) Thus we know that u(Kn ) ≥ 2n. On the one hand,
Kn can be changed into the unknot by applying 2n crossing changes. So we know that u(Kn ) = 2n.
Next, we know that Γ(Kn ) ≤ n and Arf (Kn ) = 0 as shown in Figure 10. Thus by Proposition 2.3,
we know that 2n ≤ 3Γ(Kn ) ≤ 3n.

       

A KNOT INVARIANT DERIVED FROM GAMMA MOVES5==gamma moveFigure 10References
1. L.H. Kauﬀman, The Conway polynomial, Topology 20 (1981), 101–108.
2. L. H. Kauﬀman, The Arf invariant of classical knots. Combinatorial methods in topology and algebraic geometry(Rochester, N.Y., 1982), 101–116, Contemp. Math., 44, Amer. Math. Soc., Providence, RI, 1985.
3. A. Kawauchi, A survey of knot theory, Birkhäuser Verlag, Basel-Boston-Berlin, 1996.
4. Y. Nakanishi, A note on unknotting number. Math. Sem. Notes Kobe Univ. 9 (1981), no. 1, 99–108.
5. R. A. Robertello, An invariant of knot cobordism, Commun. Pure. Appl. Math. 18 (1965), 543–555.
Department of Mathematics, Faculty of Education, Gifu University, Yanagido 1-1, Gifu, 501-1193,
Japan.
E-mail address: tanakat@gifu-u.ac.jp