Ascending number of knots and links

We shall study knots and links in R^3.
A link diagram of a link on R^2 is

ordered if an order is given to its components.
based if a basepoint (different from the crossing points) is specified on each component.
oriented if an orientation is specified on each component.

Let L be a link and L~ be a based, ordered and oriented link diagram of L.
The descending diagram of L~ is obtained as follows, and it will be denoted by d(L~).

Beginning at the basepoint of the first component of L~ and proceeding in the direction determined by the orientation, change the crossings as necessary so that each crossing is first encountered as an over-crossing.
Continue this procedure with the remaining components in the sequence determined by the ordering, proceeding from the basepoint in the direction determined by the orientation, changing crossing so that ultimately every crossing is first encountered as an over-crossing.
The result is the descending diagram d(L~) obtained from L~.

Definition of ascending number

Let L be a link and L~ be a based ordered oriented link diagram of L.
The ascending number of L~ is defined as the number of different crossings between L~ and d(L~), and denoted by a(L~).
The ascending number of L is defined as the minimum number of a(L~) over all based ordered oriented link diagram L~ of L, and denoted by a(L).

The definition and the above note imply that a(L) >or= u(L), where u(L) denotes the unknotting number of L.
Accodingly, L is a trivial link if and only if a(L)=0.

Proposition 1

For a nontrivial knot K, we have

a(K)=or<[(c(K)-1)/2].
For a link L, we have
a(L)=or<[c(L)/2],
where c(L) denotes the crossing number of L, and [x] denotes the greatest integer which does not exceed x.

The following theorem is the fundamental inequality between the ascending number and the bridge number.

Theorem 2

For a n-component link L, we have

a(L)>or=b(L)-n,
where b(L) is the bridge number of L.

The following corollary asserts the difference between a(K) and u(K).

Corollary 3

For any non-negative integer n, there is a knot K such that a(K)-u(K)>or=n.

For the connected sum of knots, we have the following proposition.

Proposition 4

For knots, the ascending number is sub-additive with respect to connected sum #, i.e. for any knots K1, K2, a(K1#K2)< or =a(K1)+a(K2).

It is natural to ask whether the ascending number is additive with respect to the connected sum.

Conjecture 5

For any knots K1, K2,

a(K1#K2)=a(K1)+a(K2).

The following corollary solves the above question partially.

Corollary 6

Let Ki(i=1,2) be a knot. If Ki satisfies a(Ki)=b(Ki)-1(i=1,2), then

a(K1#K2)=a(K1)+a(K2).

In consequence of this, we have the following corollary.

Corollary 7

For any non-negative integer n, there is a knot K such that a(K)=n.

The next theorem characterizes the ascending number one link.

Theorem 8

Let L be an n-component link. Then a(L)=1 if and only if L is

(twist knot)*O(n-1), if L is completely splittable
or
(Hopf link)*O(n-2), otherwise,
where * denotes the split union, and O(n) denotes an n-component trivial link.
In paticular, the ascending number of a knot K is one if and only if K is a twist knot.

The following theorem determines the ascending number for torus knots.

Theorem 9

Let p and q be coprime positive integers, and let T(p,q) be a (p,q)-torus knot. Then we have

K	a(K)	u(K)	b(K)
3_1	1	1	2
4_1	1	1	2
5_1	2	2	2
5_2	1	1	2
6_1	1	1	2
6_2	2	1	2
6_3	2	1	2
7_1	3	3	2
7_2	1	1	2
7_3	2	2	2
7_4	2	2	2
7_5	2	2	2
7_6	2	1	2
7_7	2	1	2
8_1	1	1	2
8_2	2 or 3	2	2
8_3	2	2	2
8_4	2	2	2
8_5	2 or 3	2	3
8_6	2	2	2
8_7	2 or 3	1	2
8_8	2	2	2
8_9	2 or 3	1	2
8_10	2 or 3	1 or 2	3
8_11	2	1	2
8_12	2	2	2
8_13	2	1	2
8_14	2	1	2
8_15	2 or 3	2	3
8_16	2 or 3	1 or 2	3
8_17	2 or 3	1	3
8_18	2	2	2
8_19	3	3	3
8_20	2	1	3
8_21	2	1	3

a(T(p,q))=(p-1)(q-1)/2.

Table of the ascending number of knots

K a(K) u(K) b(K)

3_1 1 1 2

4_1 1 1 2

5_1 2 2 2

5_2 1 1 2

6_1 1 1 2

6_2 2 1 2

6_3 2 1 2

7_1 3 3 2

7_2 1 1 2

7_3 2 2 2

7_4 2 2 2

7_5 2 2 2

7_6 2 1 2

7_7 2 1 2

8_1 1 1 2

8_2 2 or 3 2 2

8_3 2 2 2

8_4 2 2 2

8_5 2 or 3 2 3

8_6 2 2 2

8_7 2 or 3 1 2

8_8 2 2 2

8_9 2 or 3 1 2

8_10 2 or 3 1 or 2 3

8_11 2 1 2

8_12 2 2 2

8_13 2 1 2

8_14 2 1 2

8_15 2 or 3 2 3

8_16 2 or 3 1 or 2 3

8_17 2 or 3 1 3

8_18 2 2 2

8_19 3 3 3

8_20 2 1 3

8_21 2 1 3

Acknowledgement.

a(8_4)=2 was determined by Miki Okuda, a(8_13)=2 was determined by Sachie Fujimura and a(8_20)=2 was also pointed out by Sachie Fujimura, where they belong to Laboratory of Prof.Takao Matumoto, Department of Mathematics, School of Science, Hiroshima University.

I would like to thank them, their colleague, Prof.Takao Matumoto and Prof. Masakazu Teragaito very much.