       Positive knots and Rasmussen’s invariant
Toshifumi Tanaka
Department of Mathematics Education, Faculty of Education, Gifu University, Yanagido 1-1, Gifu, 501-1193
Email: tanakat@gifu-u.ac.jp

ABSTRACTWe deﬁne the positive (resp. negative) index of a knot as the minimal number of crossing changes
which are needed to make the knot into a positive (resp. negative) knot. We give lower bounds for theinvariants of knots by using the Rasmussen invariant. We detect the invariants for a torus knot. We
also compute the invariant for most knots with crossing number less than 8.
Keywords: positive knot; Rasmussen invariant; unknotting number

1. Introduction
A link is smoothly embedded circles in the 3-sphere S3 . A knot is a link with one connected component. We
assume that every link is oriented. A diagram of a link is a generic projection of a link to the 2-sphere in S3
with signed double points, called positive (or negative) crossings as in Figure 1.

orFigure 1
We say that a knot is positive if it admits a diagram with all positive crossings. We deﬁne an invariant uP(resp. uN ) of a link as the minimal number of crossing changes which are needed to make the knot into apositive (resp. negative) knot. We call the number the positive (resp. negative) index of a link. Let u(K) bethe unknotting number of a knot K. In [4], Rasmussen has introduced an eﬀective concordance invariant s(K)of a knot K which gives a lower bound for the unknotting number. Let K! be the mirror image of a knot Kwith reversed orientation. In this paper, we show the following.
Theorem 1.1. Let K be a knot in S3 .
s(K)≤ uP (K) ≤ u(K);(1) −2s(K)≤ uN (K) ≤ u(K);(2)2(3) uP (K) = uN (K!).
Corollary 1.2. Let T (p, q) be the torus knot of type (p, q).
(1) uP (T (p, q)) = 0;
(p − 1)(q − 1)(2) uN (T (p, q)) =.
2
We say that a knot is slice if it bounds a smooth disc in the 4-ball.



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Proposition 1.3. Let K be a knot with unknotting number one. If K is slice, then both of the positive index
and the negative index equal one.
Corollary 1.4. There exists a knot K such that s(K) = 0 and uP (K) = uN (K) = 1.
This paper is organized as follows. In Section 2, we will recall properties of Rasmussen invariant and prove ourresults. In Section 3, we will give a result of computations of the positive and the negative indices for knots
with crossing number less than 8.
2. Rasmussen invariant
In a paper of Khovanov published in 2000 [2], a cohomology has been constructed from a diagram of a knot
or a link via (1+1)-TQFT. Lee modiﬁed Khovanov’s TQFT and simpliﬁed the cohomology [3], and Rasmussen
has deﬁned an eﬀective concordance invariant s(K) of a knot K from Lee’s cohomology [4]. (We call theinvariant the Rasmussen invariant.) Rasmussen has detected the unknotting number of a torus knot by using
the invariant and proved Milnor’s conjecture combinatorially.
The slice genus of a knot K in S 3 is the least integer g such that K is the boundary of a connected, orientable
2-manifold S with genus g smoothly embedded in the 4-ball B 4 bounded by S 3 . The slice genus of K is denotedby g4 (K).
Two links are concordant if there is a smooth embedding(nS 1 ) × [0, 1] → S 3 × [0, 1]
which restricts to the given links(nS 1 ) × {i} → S 3 × {i}where i = 0, 1. The set of concordance classes of knots forms an abelian group under connected sum [1]. The
group is called the knot concordance group.
Main properties of Rasmussen invariant are summarized as follows [4].
Theorem 2.1. If K is a knot in S 3 , then we have the following.
(1) s induces a homomorphism from the knot concordance group to Z;(2) |s(K)| ≤ 2g4 (K), where g4 (K) is the slice genus of K;(3) If K is alternating, then s(K) = σ(K), where σ(K) is the classical knot signature of K;(4) If K1 is obtained from K2 by performing a single crossing change, then |s(K1 ) − s(K2 )| ≤ 2.
Theorem 2.2. If K is a positive knot, s(K) = 2g4 (K) = 2g(K), where g(K) is the ordinary genus of K.
Theorem 2.3. If K is a knot, then we have s(K) = −s(K!).
Proof of Theorem 1.1. Since the unknot is positive and negative, we have the right inequalities of (1) and (2),uP ≤ u(K) and uN (K) ≤ u(K). The equation (3) follows from the deﬁnition immediately. The left inequalityof (2), s(K)/2 ≤ uN (K), follows from that of (1), −s(K!)/2 ≤ uP (K!), the equality (3) and Theorem 2.3.
  thentheequalityholds.
So
weassumeHence, it is suﬃce to show that −s(K)/2 ≤ uP (K). If s(K) is positive, nK . (The integer s(K) is always even.) Then we need to perform at least nK
that s(K) ≤ 0 and let s(K)/2=
crossing changes to have a knot with non-negative Rasmussen invariant by Theorem 2.1(4). By Theorem 2.2,
if K is a positive knot, then s(K) ≥ 0. Thus by the deﬁnition, we have nK ≤ uP .

Proof of Corollary 1.2. The torus knot T (p, q) is a positive knot. Hence uP (T (p, q)) = 0. By a result in [4], it isshown that s(K) = 2u(K) = (p−1)(q−1). Thus, by Theorem 1.1(2), we know that uN (T (p, q)) = (p−1)(q−1)/2.
Proof of Proposition 1.3. Let K be a slice knot with unknotting number one. Then we know that s(K) = 0
and K is non-trivial. By Theorem 2.2, if K is non-trivial positive knot, then s(K) is positive. Thus, we needto perform at least one crossing change to make K into a positive knot by Theorem 2.1(4). Therefore, we   have uP (K) ≥ 1. By Theorem 2.2 and 2.3, we know that non-trivial negative knot have negative Rasmussen
invariant. So by using the same method as above, we know that uN (K) ≥ 1. Then by Theorem 1.1, we haveuP (K) = uN (K) = 1.
Proof of Corollary 1.4. Let K be a Stevedore’s knot as shown in Figure 2. Then we know that K is a slice knotand u(K) = 1. Thus, Corollay 1.4 follows from Proposition 1.3.

Stevedore's Knot

Figure 2
3. Examples
In this section, we calculate the positive index and the negative index for some knots. Finally, we give a list ofcomputations for knots with crossing number less than 8.
We need the following proposition.
Proposition 3.1. Let K be a positive knot and let T B(K) be the maximal Thurston-Bennequin invariant of aknot K. If K be a positive knot then T B(K) + 1 = s(K).
Proof. The result follows from results in [6] and [4].
Example 3.2. Let K be the knot 82 in Rolfsen’s table [5] (See Figure 3). Then s(K) = σ(K) = −4 and−s(K)/2 = 2 ≤ uP (K) ≤ u(K) = 2. Thus up (K) = 2. The knot K is transformed into a negative knot byperforming a single crossing changes as shown in Figure 2. (Change the crossing marked by a dotted circle.)
Hence we know that uN (K) ≤ 1. However, we can show that T B(K!) = s(K!) − 1. Thus we have uN (K) = 1by Proposition 3.1.

82Figure 3

Example 3.3. Next, let K be the knot 816 in Rolfsen’s table (See Figure 4). Then s(K) = σ(K) = −2 andu(K) = 2. The knot K can be transformed into a negative knot as shown in Figure 3 by a single crossingchange. So we know that uN (K) ≤ 1. On the other hand, we can show that T B(K!) = s(K!) − 1. ThusuN (K) = 1 by Proposition 3.1.

8 16

Figure 4
The knot K can be also transformed into a positive knot as shown in Figure 5 by two crossing changes. Sowe know that uP (K) ≤ 2. Now let K ∗ be a knot obtained from K by a single crossing change. If s(K ∗ ) = 0,then K ∗ is not a positive knot since K has unknotting number two and non-trivial positive knot has positive
Rasmussen invariant by Theorem 2.2. If s(K ∗ ) = 0, then s(K ∗ ) is still negative, because of Theorem 2.1(4).
Hence K ∗ is not a positive knot by Theorem 1.1(1). Thus uP (K) ≥ 2 and we have uP (K) = 2.

8 16

Figure 5
Finally, we show a list of the positive indices and the negative indices of knots with crossing number less thanor equal 8 in Rolfsen’s table. (The character X means that the number is one or two.)
Acknowledgements. This research is partially supported by the Ministry of Education, Science, Sports andCulture, Grant-in-Aid for Young Scientists (B), 2011-2014（23740046).
References
1. R. H. Fox, A quick trip through knot theory, 1962 Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia
Institute, 1961) 120-167 Prentice-Hall, Englewood Cliﬀs, N.J.
2. M. Khovanov, A categoriﬁcation of the Jones polynomial, Duke Math. J. 101 (2000) 359–426.
3. E. S. Lee, An endomorphism of the Khovanov invariant, Adv. Math., 197 (2005), 554–586.
4. J. Rasmussen, Khovanov homology and the slice genus, arXiv:math.GT/0402131.
5. D. Rolfsen, Knots and Links, Math. Lecture Ser. 7. Publish or Perish, Berkeley, Calif., 1976.
6. T. Tanaka, Maximal Bennequin numbers and Kauﬀman polynomials of positive links, Proc. Amer. Math. Soc. 127 (1999), no.
11, 3427–3432.

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