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SMALL EXAMPLES OF CUBE DIAGRAMS OF KNOTS

SCOTT BALDRIDGE AND BEN MCCARTY

Abstract. In this short note we highlight some of the diﬀerences between cube diagrams and grid

diagrams. We also list examples of small cube diagrams for all knots up to 7 crossings and give

some examples of links.

1. Introduction

Cube diagrams, introduced in Baldridge and Lowrance [2], are 3-dimensional representations of

knots or links. We deﬁne cube diagrams carefully in Section 3. The easiest way to imagine a cube

diagram is to think of an embedding of a knot or link in a [0, n]×[0, n]×[0, n] cube (using xyz

coordinates) for some positive integer nsuch that the knot projection of the cube to each axis plane

(x= 0, y= 0, and z= 0) is a grid diagram.

Figure 1: A cube diagram for the Trefoil.

The integer nis called the size of the cube diagram and we will refer to a cube diagram as

“small” if its size is close to the crossing number of the knot. Small cube diagrams are very useful

for computing and testing for knot invariants using computers—calculating certain invariants of

cube diagrams with size n > 20 can be computationally intractable for most computers. In this

paper we discuss why small cube diagrams can be elusive to ﬁnd for a given knot or link and we give

examples of small cube diagrams (n < 20) for knots with seven crossings or less. In particular, the

Appendix gives pictures of the cube diagrams we have generated. A Mathematica program [3] can

be downloaded from http://cubeknots.googlecode.com that can be used to calculate invariants

of cube diagrams and also to rotate in 3-dimensions the examples in this paper.

Grid diagrams and cube diagrams are useful representations in knot theory. In 1996 Cromwell

used grid diagrams as ways to represent embeddings of knots in open books (c.f. [5]). He described

Date: July 30, 2009.

S. Baldridge was partially supported by NSF Grant DMS-0748636.

1

arXiv:0907.5401v1 [math.GT] 30 Jul 2009

2 S. BALDRIDGE AND B. MCCARTY

a set of elementary grid moves that preserve topological knot type that can be used to check for knot

invariants. More recently grid diagrams were used in constructing a combinatorial version of knot

Floer homology (c.f. [9] and [10]) and they provide a natural presentation of the front projection of

a Legendrian knot (c.f. [11] and [12]). Cube diagrams share many similarities with grid diagrams.

Like grid diagrams, there is set of elementary cube moves that preserve the knot type and there

is a combinatorial knot Floer homology that can be computed from a cube diagram [2]. However,

cube diagrams have also appeared independent of grid diagrams in the study of unstable Vassiliev

theory [8].

2. Finding small cube diagrams: the lifting problem

In order to ﬁnd small cube diagrams, it is important to understand how they are diﬀerent from

grid diagrams. In this section we discuss the problem of lifting a grid diagram to a cube diagram.

A grid diagram Gis an n×nsquare grid decorated with Xand Omarkings in such a way that

every row (resp. column) contains exactly one Xand one Omarking. To get an oriented knot or

link projection from a grid diagram one draws edges from Xto Oin each column and from Oto

Xin each row, taking the vertical segment as the over crossing at any intersection (cf. [2, 5]).

Figure 2: Grid diagram: X={2,3,4,5,1}and O={5,1,2,3,4}.

Since each of the three projections of a cube diagram is a grid diagram, it is natural to think

of a cube diagram as a lift of a grid diagram of one of the three projections. Such a lift is clearly

not unique. What is not obvious is that many times such lifts do not exist: a given grid diagram

does not necessarily lift to an embedding of a knot or link in a lattice in an n×n×ncube such

that the projection to each plane is a well-deﬁned knot projection (it can lift to an embedding, but

the intersections in the other two projections are not necessarily isolated double points). For an

example of a grid diagram that can not be the projection of such a lattice knot see [2].

The lifting problem to a true cube diagram is harder than just ﬁnding a lift of a grid diagram in

3-space such that the other two projections are valid knot projections. Even when a grid diagram

does lift to such an embedding, that embedding is rarely a cube diagram and the chances that a

grid diagram lifts to a cube diagram appears to decrease as the grid size increases. To support

this assertion we wrote a brute-force program to get statistics on grid diagrams that lift to cube

diagrams. The program looked at all size n= 5,6,7,8 grid diagrams and tested whether each grid

diagram was a nontrivial knot (we did not consider links) and whether each grid diagram lifted to

a cube diagram. The data for grid diagrams for nontrivial knots are presented below.

SMALL CUBE DIAGRAMS 3

Grid Total grids Number that Percent

Size of nontrivial lift to cube

knots diagrams

5 10 3 30%

6 972 261 27%

7 85,022 19,722 23%

8 8,077,072 1,589,447 19.7%

Note that creating grid diagrams with sizes above 8 is time intensive and the computation to rule

out unknot grid diagrams grows quickly beyond the capabilities of most computers. For example,

the total number of grid diagrams of knots of size 8 including unknots is 101,606,400. It is interesting

to note that unknot grid diagrams do lift to cube diagrams more often: of those 101 million grid

diagrams, 72,109,568 of them or 71 percent lift to cube diagrams. But that fact is because many

unknot grid diagrams do not have any crossings and such grid diagrams often lift to cube diagrams.

Furthermore, the percentage of unknot grid diagrams that lift to cube diagrams also decreases as

the grid size increases in the examples we have calculated. To get the data for size 8 grid diagrams,

we ran 5 computers simultaneously night and day for 1 week. See the program [3] for other methods

of measuring the probability of ﬁnding a cube diagram when the size is greater than 8.

Therefore building a same-size cube diagram from a given grid diagram cannot be done in general.

However, a grid diagram can always be used to build a larger-sized cube diagram. In fact, given a

grid diagram of size nof a knot, a cube diagram of the same knot always exists of size at most

(2.1) n+ 2(# of bad crossings) + (# of twisted bends).

The bad crossings mentioned above are those crossings in the (y, z)−and (z, x)−projections that

do not follow the convention given in the deﬁnition of a cube diagram. For many grid diagrams,

this leads to large cube diagrams 3 or 4 times the size of the original grid. For example, a grid

diagram for a 7 crossing knot of size 9 may produce a cube diagram of size 35 or more. In this

paper we ﬁnd cube diagram representations for many 7 crossing knots of size 9.

In order to derive the formula above and show how to improve upon it, we need a precise

deﬁnition of cube diagrams, which is the content of the next section.

3. Cube Diagrams of Knots

Let nbe a positive integer and let Γ be the cube [0, n]×[0, n]×[0, n]⊂R3thought of as

3-dimensional Cartesian grid, i.e., a grid with integer valued vertices. A ﬂat of Γ is any cuboid

(a right rectangular prism) with integer vertices in Γ such that there are two orthogonal edges of

length nwith the remaining orthogonal edge of length 1. A ﬂat with an edge of length 1 that is

parallel to the x-axis, y-axis, or z-axis is called an x-ﬂat,y-ﬂat, or z-ﬂat respectively.

The embedding of a link in the cube Γ can be described as follows. A marking is a labeled point

in R3with half-integer coordinates. Mark unit cubes of Γ with either an X,Y, or Zsuch that the

following marking conditions hold:

•each ﬂat has exactly one X, one Y, and one Zmarking;

•the markings in each ﬂat forms a right angle such that each ray is parallel to a coordinate

axis;

4 S. BALDRIDGE AND B. MCCARTY

•for each x-ﬂat, y-ﬂat, or z-ﬂat, the marking that is the vertex of the right angle is an X, Y,

or Zmarking respectively.

An oriented link can be embedded into Γ by connecting pairs of markings with a line segment

whenever two of their corresponding coordinates are the same. Each line segment is oriented to go

from an Xto a Y, from a Yto a Z, or from a Zto an X(note that the cube itself is canonically

oriented by the standard right hand orientation of R3). The markings in each ﬂat deﬁne two

perpendicular segments of the link Ljoined at a vertex, call the union of these segments a cube

bend. If a cube bend is contained in an x-ﬂat, we call it an x-cube bend. Similarly, deﬁne y-cube

bends and z-cube bends.

Arrange the markings in Γ so that the following crossing conditions hold:

•At every intersection point of the (x, y)-projection, the segment parallel to the x-axis has

smaller z-coordinate than the segment parallel to the y-axis.

•At every intersection point of the (y, z)-projection, the segment parallel to the y-axis has

smaller x-coordinate than the segment parallel to the z-axis.

•At every intersection point of the (z, x)-projection, the segment parallel to the z-axis has

smaller y-coordinate than the segment parallel to the x-axis.

If Γ satisﬁes these conditions, then it is called a cube diagram. We say that Γ is a cube diagram

representing the (oriented) link L.

The knot projections of a cube diagram to the three coordinate axis planes are grid diagrams.

What is not obvious, and the deﬁnition above speciﬁes, is how the three projected grid diagrams

are oriented with respect to the cube diagram. In all three cases, the orientation is speciﬁed by

the order of the axes using the standard orientation of R3. For example, in the (x, y)-projection,

the x-axis speciﬁes the ‘row’ and the y-axis speciﬁes the ‘column’ of the grid diagram but in the

(y, z)-projection the y-axis speciﬁes the ‘row’ and the z-axis speciﬁes the ‘column’ of that grid

diagram (see Figure 3).

Figure 3: Crossing conditions for the projections

These orientations matter: if in the deﬁnition above, the ﬁrst crossing condition was modiﬁed

so that the segment parallel to the x-axis has greater z-coordinate than the segment parallel to the

y-axis, then the set of knots types that are represented by modiﬁed ‘cube diagrams’ of a given size

is diﬀerent than the set of knot types using the actual deﬁnition.

SMALL CUBE DIAGRAMS 5

4. Building a cube diagram from a given grid diagram

Section 2 described the problems with lifting a grid diagram to a cube diagram. However, a

grid diagram can always be used to create a cube diagram. There are two issues to overcome: (1)

changing the grid diagram so that it can be the knot-projection of a lattice knot such that the

other two projections are well-deﬁned knot projections and (2) ﬁxing the crossings of that lattice

knot in the other projections so that each crossing satisﬁes the crossing conditions.

The ﬁrst issue can be ﬁxed by removing all of the twisted bends from a grid diagram. A bend in

a grid diagram for a knot Kis a pair of segments in Kthat meet at a common Xor Omarking. If

a bend passes over some other segment of Kand passes under some other segment of K, then call

it twisted (cf. [2]). There are two ways to partition a grid diagram of a knot into non-overlapping

bends, depending on whether the two segments in each bend intersect in an Xor Omarking. A

grid diagram of size ncan be used to construct a lattice knot embedded into an n×n×ncube

such that the other projections are knot projections if a partial order can be put on either of the

two partitions of bends following the convention that if two bends cross the bend crossing over the

other is greater (cf. [2]). If a grid diagram has a partition with no twisted bends, then a partial

order always exists for that partition. If a partition has one twisted bend, then stabilizing at the

vertex of the twisted bend produces a new grid diagram with no twisted bends. The new grid is

one size larger, which explains the third term in the formula above when there are multiple twisted

bends.

If a grid diagram has no twisted bends, the bends can be stacked to form a lattice knot that

projects to valid knot projections in all three planes. While these knots do indeed satisfy the

crossing conditions for the (x, y)-projection they may not satisfy the crossing conditions in one

or both of the other projections. As observed in [2] the invalid crossings may be repaired by the

insertion of a rotated crossing as shown in ﬁgure 4 (c.f. [2]).

Figure 4: Insertion of a rotated crossing

The rotated crossing corrects the crossing condition for that projection while leaving the crossing

data of the other two projections unaﬀected. This procedure has the eﬀect of rotating the crossing

so that the overcrossing is correct, but at the cost of increasing the size of the cube diagram by 2,

which explains the second term in the formula above.

5. Small cube diagrams for knots with small crossing knots

The algorithm in Section 4 tends to produces large cube diagram representations for a given knot

that are computationally intractable for computers to calculate invariants like knot Floer homology.

To ﬁnd small cube diagrams for small crossing knots, we wrote a computer program that searches

6 S. BALDRIDGE AND B. MCCARTY

all size ≤9 grid diagrams looking for grid diagrams that lift to cube diagrams. If a grid diagram

does, we check to see what knot type it is and record the cube diagram. The idea is simple enough,

but virtually all grid diagrams that lift to cube diagrams are unknots, and checking whether each

grid diagram is possibly an unknot involves an O(n3) number of calculations per grid diagram of

size n. The total number of size ngrids (for knots and links) is

(n!)2

41 + 2(1 + n)Γ(1 + n, −1)

eΓ(2 + n)−6Γ(3,−1)

eΓ(4) ,

where Γ(s) is the gamma function and Γ(s, x) is the incomplete gamma function. Clearly, checking

whether each is an unknot is too time intensive for size 9 grids. We describe next how we reduced

the number of calculations to a routine that runs in days rather than years.

Generating grid diagrams of size ninvolves the choice of two size npermutations σ, τ ∈Sn. The

coordinates of the X-markings of the grid diagram are given by (i, σ(i)) and the O-markings are

given by (i, τ(i)). The program generates every possible grid diagram by an outer loop/inner loop

structure. The outer loop generates a new grid diagram by always starting with σ= (1,2,3, . . . , n)

(X’s along the diagonal) and cycling through the choices of τthat generate a grid diagram. The

inner loop then runs through all permutations of the columns using a routine that picks the ‘next

largest’ permutation in lexicographic order. For example, if n= 3, then the order of permutations

from ‘smallest’ to ‘largest’ is σ1= (1,2,3), σ2= (1,3,2), σ3= (2,1,3), σ4= (2,3,1), σ5= (3,1,2),

and σ6= (3,2,1). By setting up the inner and outer loop in this way we reduce the number of times

we need to check for the unknot signiﬁcantly because the diﬀerence between the two diagrams from

σiand σi+1 is often either a (1) column commutation move or (2) a column cyclic permutation

move. Therefore if the grid diagram associated to σiis the unknot and σi+1 is a commutation or

cyclic permutation of σi, then we know that the grid diagram associated to σi+1 is also an unknot

(no time intensive calculations are necessary). Speciﬁcally, if the current grid diagram is merely a

commutation of the previous diagram and the knot determinant of the previous diagram is 1, then

the current diagram is also considered a potential unknot. It is reasonable to throw out determinant

1 knots—according to [4] there are just two nontrivial knots, 10124 and 12242, that have arc index

≤9 and determinant 1.

Once the program ﬁnds a grid diagram that potentially represents a nontrivial knot, the program

looks for several X−Oconﬁgurations known not to lift to a cube diagram including eliminating

links. These conﬁgurations are relatively easy to check for and also signiﬁcantly reduce the number

of diagrams the program attempts to lift to cube diagrams. Next, the knot determinant is computed.

As before, if the determinant is 1, the diagram is discarded as the unknot, 10124 knot, or 12242

knot. The program then attempts to lift the grid to a valid cube by choosing a third permutation

ζ∈Snthat determines the order in which the z-cube bends will be stacked in order from smallest

z-coordinate to greatest. For each permutation the program checks to ensure that the stack is

compatible with the crossing conditions determined by the grid. If a valid stack permutation is

found, the crossing conditions are checked in the (y, z)−and (z, x)-projections. If the crossing

conditions are satisﬁed, the tuple of permutations (σ, τ, ζ ) is a cube diagram and the program

computes a variant of the Jones polynomial to determine exactly what knot the cube diagram

represents (c.f. [13]). This ﬁnal and most time intensive calculation involves O(c22c) number of

calculations where cis the crossing number of the diagram. Fortunately, this calculation is not

necessary very often.

Using the routine sketched out above and beginning with grid diagrams of size ≤9 we obtained

most of the list in Appendix A. For the 6 crossing knots, however, a slightly diﬀerent strategy

SMALL CUBE DIAGRAMS 7

was used. Beginning with a valid grid diagram for each knot type the cube stacking algorithm

described above was used to produce lattice knots that projected to valid knot projections in all

three planes. While these knots did indeed satisfy the crossing conditions for the (x, y)-projection

they all had invalid crossings in at least one of the other two projections. As observed above, the

invalid crossings may be repaired by rotating the crossing as shown in Figure 4. It is not known if

these examples are the smallest.

Appendix A. Knot Examples

The following table lists examples of cube knots up to 7 crossings as well as several 8, 9, 10

and 12 crossing knots. Each picture displays the cube knot from the point of view of the (x, y)-

projection. The code presented to the right of each diagram is designed to work with a Mathematica

notebook found at [3]. Since projections of cube diagrams are grid diagrams, the arc index α(K)

of a knot gives a lower bound for the size of a cube diagram. It has been shown in [1] that for

alternating knots α(K) = c(K) + 2 where c(K) is the crossing number of the knot. Therefore in

searching for cube diagrams up to size 9 the program could only be expected to ﬁnd alternating

knots up to 7 crossings (one exception below, 815, was found by running a partial search for size 10

diagrams). However, for non-alternating knots the arc index may be much smaller. This fact helps

to explain why some of the 9, 10 and 12 crossing knots show up in the table below. The examples

that were found were of knots with relatively low arc index. For example 12591 listed below is a

non-alternating knot that has arc index equal to 9 (see [4]).

It is interesting to note that for the 6-crossing knots, 76, and 77the cube diagrams were obtained

using the second method described above. This method was necessary because the program found

no valid cube diagrams of size ≤9.

Diagram Mathematica Code

K31={X[{1, 5, 4},{4, 3, 2},{5, 4, 3},{2, 1, 5},{3, 2, 1}],

Y[{1, 5, 1},{2, 1, 2},{3, 2, 3},{4, 3, 4},{5, 4, 5}],

Z[{1, 2, 1},{2, 3, 2},{3, 4, 3},{4, 5, 4},{5, 1, 5}]}

31

K41={X[{1, 1, 2},{2, 2, 4},{3, 3, 3},{4, 7, 5},{5, 8, 6},

{6, 4, 8},{7, 5, 7},{8, 6, 1}],

Y[{1, 1, 8},{2, 2, 1},{3, 3, 6},{4, 7, 4},{5, 8, 7},

{6, 4, 5},{7, 5, 2},{8, 6, 3}],

Z[{1, 5, 2},{2, 7, 4},{3, 6, 3},{4, 4, 5},{5, 3, 6},

{6, 1, 8},{7, 8, 7},{8, 2, 1}]}

41

8 S. BALDRIDGE AND B. MCCARTY

Diagram Mathematica Code

K51={X[{1, 7, 1},{2, 1, 2},{3, 2, 3},{4, 3, 4},{5, 4, 5},

{6, 5, 6},{7, 6, 7}],

Y[{1, 7, 6},{2, 1, 7},{3, 2, 1},{4, 3, 2},{5, 4, 3},

{6, 5, 4},{7, 6, 5}],

Z[{1, 2, 1},{2, 3, 2},{3, 4, 3},{4, 5, 4},{5, 6, 5},

{6, 7, 6},{7, 1, 7}]}

51

K52={X[{2, 1, 6},{1, 7, 7},{3, 2, 1},{4, 4, 3},{5, 5, 4},

{6, 3, 2},{7, 6, 5}],

Y[{2, 1, 2},{1, 7, 3},{3, 2, 4},{4, 4, 5},{5, 5, 6},

{6, 3, 7},{7, 6, 1}],

Z[{2, 3, 2},{1, 4, 3},{3, 5, 4},{4, 6, 5},{5, 1, 6},

{6, 7, 7},{7, 2, 1}]}

52

K61={X[{1, 16, 7},{2, 1, 8},{3, 13, 11},{4, 5, 14},{5, 15, 13},

{6, 14, 10},{7, 7, 9},{8, 12, 12},{9, 9, 3},{10, 2, 4},

{11, 3, 2},{12, 4, 5},{13, 6, 6},{14, 8, 1},{15, 10, 15},

{16, 11, 16}],

Y[{1, 16, 14},{2, 1, 2},{3, 13, 13},{4, 5, 12},{5, 15, 15},

{6, 14, 8},{7, 7, 11},{8, 12, 10},{9, 9, 6},{10, 2, 7},

{11, 3, 3},{12, 4, 4},{13, 6, 16},{14, 8, 5},{15, 10, 1},

{16, 11, 9}],

Z[{1, 2, 7},{2, 14, 8},{3, 7, 11},{4, 16, 14},{5, 13, 13},

{6, 12, 10},{7, 11, 9},{8, 5, 12},{9, 3, 3},{10, 4, 4},

{11, 1, 2},{12, 8, 5},{13, 9, 6},{14, 10, 1},{15, 15, 15},

{16, 6, 16}]}

61

K62={X[{1, 9, 5},{2, 12, 8},{3, 1, 9},{4, 7, 3},{5, 4, 2},

{6, 3, 6},{7, 2, 1},{8, 5, 10},{9, 8, 11},{10, 10, 12},

{11, 6, 7},{12, 11, 4}],

Y[{1, 9, 7},{2, 12, 12},{3, 1, 5},{4, 7, 4},{5, 4, 1},

{6, 3, 3},{7, 2, 8},{8, 5, 6},{9, 8, 9},{10, 10, 10},

{11, 6, 2},{12, 11, 11}],

Z[{1, 1, 5},{2, 2, 8},{3, 8, 9},{4, 3, 3},{5, 6, 2},

{6, 5, 6},{7, 4, 1},{8, 10, 10},{9, 11, 11},{10, 12, 12},

{11, 9, 7},{12, 7, 4}]}

62

SMALL CUBE DIAGRAMS 9

Diagram Mathematica Code

K63={X[{1, 1, 4},{2, 3, 1},{3, 13, 9},{4, 2, 14},{5, 15, 13},

{6, 14, 5},{7, 8, 6},{8, 9, 7},{9, 12, 8},{10, 7, 3},

{11, 4, 2},{12, 6, 10},{13, 10, 11},{14, 11, 12},{15, 5, 15}],

Y[{1, 1, 14},{2, 3, 9},{3, 13, 13},{4, 2, 12},{5, 15, 15},

{6, 14, 8},{7, 8, 4},{8, 9, 5},{9, 12, 11},{10, 7, 1},

{11, 4, 7},{12, 6, 3},{13, 10, 6},{14, 11, 10},{15, 5, 2}],

Z[{1, 8, 4},{2, 7, 1},{3, 3, 9},{4, 1, 14},{5, 13, 13},

{6, 9, 5},{7, 10, 6},{8, 4, 7},{9, 14, 8},{10, 6, 3},

{11, 5, 2},{12, 11, 10},{13, 12, 11},{14, 2, 12},{15, 15, 15}]}

63

K71={X[{1, 9, 1},{2, 1, 2},{3, 2, 3},{4, 3, 4},{5, 4, 5},

{6, 5, 6},{7, 6, 7},{8, 7, 8},{9, 8, 9}],

Y[{1, 9, 8},{2, 1, 9},{3, 2, 1},{4, 3, 2},{5, 4, 3},

{6, 5, 4},{7, 6, 5},{8, 7, 6},{9, 8, 7}],

Z[{1, 2, 1},{2, 3, 2},{3, 4, 3},{4, 5, 4},{5, 6, 5},

{6, 7, 6},{7, 8, 7},{8, 9, 8},{9, 1, 9}]}

71

K72={X[{2, 1, 2},{1, 9, 3},{3, 2, 4},{4, 4, 5},{5, 5, 6},

{6, 6, 7},{7, 7, 8},{8, 3, 9},{9, 8, 1}],

Y[{2, 1, 8},{1, 9, 9},{3, 2, 1},{4, 4, 3},{5, 5, 4},

{6, 6, 5},{7, 7, 6},{8, 3, 2},{9, 8, 7}],

Z[{2, 3, 2},{1, 4, 3},{3, 5, 4},{4, 6, 5},{5, 7, 6},

{6, 8, 7},{7, 1, 8},{8, 9, 9},{9, 2, 1}]}

72

K73={X[{1, 1, 2},{2, 2, 3},{3, 3, 4},{4, 8, 5},{5, 4, 6},

{6, 5, 8},{7, 6, 7},{8, 7, 9},{9, 9, 1}],

Y[{1, 1, 9},{2, 2, 1},{3, 3, 2},{4, 8, 7},{5, 4, 3},

{6, 5, 5},{7, 6, 4},{8, 7, 6},{9, 9, 8}],

Z[{1, 3, 2},{2, 4, 3},{3, 6, 4},{4, 5, 5},{5, 7, 6},

{6, 9, 8},{7, 8, 7},{8, 1, 9},{9, 2, 1}]}

73

10 S. BALDRIDGE AND B. MCCARTY

Diagram Mathematica Code

K74={X[{1, 1, 4},{2, 5, 5},{3, 2, 2},{4, 3, 6},{5, 6, 7},

{6, 4, 8},{7, 7, 9},{8, 8, 3},{9, 9, 1}],

Y[{1, 1, 9},{2, 5, 3},{3, 2, 7},{4, 3, 1},{5, 6, 4},

{6, 4, 2},{7, 7, 6},{8, 8, 8},{9, 9, 5}],

Z[{1, 6, 4},{2, 9, 5},{3, 4, 2},{4, 7, 6},{5, 2, 7},

{6, 8, 8},{7, 1, 9},{8, 5, 3},{9, 3, 1}]}

74

K75={X[{1, 1, 2},{2, 2, 3},{3, 3, 4},{4, 9, 5},{5, 4, 6},

{6, 5, 9},{7, 6, 7},{8, 7, 8},{9, 8, 1}],

Y[{1, 1, 9},{2, 2, 1},{3, 3, 2},{4, 9, 8},{5, 4, 3},

{6, 5, 5},{7, 6, 4},{8, 7, 6},{9, 8, 7}],

Z[{1, 3, 2},{2, 4, 3},{3, 6, 4},{4, 5, 5},{5, 7, 6},

{6, 1, 9},{7, 8, 7},{8, 9, 8},{9, 2, 1}]}

75

K76={X[{1, 1, 2},{2, 14, 5},{3, 2, 6},{4, 3, 11},{5, 6, 7},

{6, 5, 12},{7, 4, 10},{8, 10, 3},{9, 11, 4},{10, 12, 9},

{11, 7, 8},{12, 9, 13},{13, 8, 14},{14, 13, 1}],

Y[{1, 1, 14},{2, 14, 7},{3, 2, 1},{4, 3, 5},{5, 6, 10},

{6, 5, 11},{7, 4, 13},{8, 10, 6},{9, 11, 2},{10, 12, 3},

{11, 7, 12},{12, 9, 9},{13, 8, 8},{14, 13, 4}],

Z[{1, 11, 2},{2, 3, 5},{3, 10, 6},{4, 5, 11},{5, 14, 7},

{6, 7, 12},{7, 6, 10},{8, 12, 3},{9, 13, 4},{10, 9, 9},

{11, 8, 8},{12, 4, 13},{13, 1, 14},{14, 2, 1}]}

76

K77={X[{1, 1, 7},{2, 2, 13},{3, 6, 8},{4, 5, 12},{5, 4, 9},

{6, 14, 6},{7, 8, 5},{8, 13, 11},{9, 7, 10},{10, 16, 14},

{11, 12, 15},{12, 15, 16},{13, 17, 17},{14, 10, 3},{15, 9, 4},

{16, 3, 2},{17, 11, 1}],

Y[{1, 1, 16},{2, 2, 1},{3, 6, 9},{4, 5, 13},{5, 4, 14},

{6, 14, 3},{7, 8, 7},{8, 13, 6},{9, 7, 12},{10, 16, 11},

{11, 12, 10},{12, 15, 8},{13, 17, 15},{14, 10, 2},{15, 9, 5},

{16, 3, 17},{17, 11, 4}],

Z[{1, 8, 7},{2, 5, 13},{3, 15, 8},{4, 7, 12},{5, 6, 9},

{6, 13, 6},{7, 9, 5},{8, 16, 11},{9, 12, 10},{10, 4, 14},

{11, 17, 15},{12, 1, 16},{13, 3, 17},{14, 14, 3},{15, 11, 4},

{16, 10, 2},{17, 2, 1}]}

77

SMALL CUBE DIAGRAMS 11

Diagram Mathematica Code

K815 ={X[{1, 1, 5},{2, 2, 2},{3, 3, 4},{4, 5, 6},{5, 7, 7},

{6, 8, 8},{7, 4, 9},{8, 6, 10},{9, 9, 3},{10, 10, 1}],

Y[{1, 1, 10},{2, 2, 8},{3, 3, 1},{4, 5, 3},{5, 7, 5},

{6, 8, 6},{7, 4, 2},{8, 6, 4},{9, 9, 7},{10, 10, 9}],

Z[{1, 7, 5},{2, 4, 2},{3, 6, 4},{4, 8, 6},{5, 9, 7},

{6, 2, 8},{7, 10, 9},{8, 1, 10},{9, 5, 3},{10, 3, 1}]}

815

K819 ={X[{1, 7, 2},{2, 1, 3},{3, 2, 4},{4, 3, 5},{5, 4, 6},

{6, 5, 7},{7, 6, 1}],

Y[{1, 7, 6},{2, 1, 7},{3, 2, 1},{4, 3, 2},{5, 4, 3},

{6, 5, 4},{7, 6, 5}],

Z[{1, 3, 2},{2, 4, 3},{3, 5, 4},{4, 6, 5},{5, 7, 6},

{6, 1, 7},{7, 2, 1}]}

819

K821 ={X[{1, 1, 3},{2, 2, 2},{3, 3, 4},{4, 5, 5},{5, 7, 6},

{6, 6, 8},{7, 10, 7},{8, 4, 9},{9, 8, 10},{10, 9, 1}],

Y[{1, 1, 8},{2, 2, 6},{3, 3, 1},{4, 5, 3},{5, 7, 7},

{6, 6, 4},{7, 10, 10},{8, 4, 2},{9, 8, 5},{10, 9, 9}],

Z[{1, 5, 3},{2, 4, 2},{3, 6, 4},{4, 8, 5},{5, 2, 6},

{6, 1, 8},{7, 7, 7},{8, 9, 9},{9, 10, 10},{10, 3, 1}]}

821

K949 ={X[{1, 9, 2},{2, 1, 3},{3, 8, 4},{4, 2, 5},{5, 3, 8},

{6, 4, 6},{7, 5, 9},{8, 6, 7},{9, 7, 1}],

Y[{1, 9, 8},{2, 1, 9},{3, 8, 7},{4, 2, 1},{5, 3, 4},

{6, 4, 2},{7, 5, 5},{8, 6, 3},{9, 7, 6}],

Z[{1, 4, 2},{2, 6, 3},{3, 3, 4},{4, 5, 5},{5, 9, 8},

{6, 7, 6},{7, 1, 9},{8, 8, 7},{9, 2, 1}]}

949

12 S. BALDRIDGE AND B. MCCARTY

Diagram Mathematica Code

K10124 ={X[{1, 1, 3},{2, 2, 4},{3, 3, 5},{4, 9, 6},{5, 4, 7},

{6, 5, 8},{7, 6, 9},{8, 7, 2},{9, 8, 1}],

Y[{1, 1, 9},{2, 2, 1},{3, 3, 2},{4, 9, 8},{5, 4, 3},

{6, 5, 4},{7, 6, 5},{8, 7, 6},{9, 8, 7}],

Z[{1, 4, 3},{2, 5, 4},{3, 6, 5},{4, 7, 6},{5, 8, 7},

{6, 9, 8},{7, 1, 9},{8, 3, 2},{9, 2, 1}]}

10124

K10128 ={X[{1, 9, 2},{2, 8, 3},{3, 1, 4},{4, 2, 5},{5, 3, 6},

{6, 4, 8},{7, 5, 9},{8, 6, 7},{9, 7, 1}],

Y[{1, 9, 8},{2, 8, 7},{3, 1, 9},{4, 2, 1},{5, 3, 3},

{6, 4, 4},{7, 5, 5},{8, 6, 2},{9, 7, 6}],

Z[{1, 6, 2},{2, 3, 3},{3, 4, 4},{4, 5, 5},{5, 7, 6},

{6, 9, 8},{7, 1, 9},{8, 8, 7},{9, 2, 1}]}

10128

K10139 ={X[{1, 1, 2},{2, 2, 4},{3, 9, 5},{4, 3, 6},{5, 4, 7},

{6, 5, 8},{7, 6, 9},{8, 7, 3},{9, 8, 1}],

Y[{1, 1, 9},{2, 2, 1},{3, 9, 8},{4, 3, 2},{5, 4, 3},

{6, 5, 4},{7, 6, 5},{8, 7, 6},{9, 8, 7}],

Z[{1, 3, 2},{2, 5, 4},{3, 6, 5},{4, 7, 6},{5, 8, 7},

{6, 9, 8},{7, 1, 9},{8, 4, 3},{9, 2, 1}]}

10139

K10145 ={X[{1, 1, 2},{2, 2, 4},{3, 9, 5},{4, 4, 6},{5, 3, 7},

{6, 5, 9},{7, 6, 8},{8, 7, 3},{9, 8, 1}],

Y[{1, 1, 9},{2, 2, 1},{3, 9, 8},{4, 4, 3},{5, 3, 2},

{6, 5, 5},{7, 6, 4},{8, 7, 7},{9, 8, 6}],

Z[{1, 3, 2},{2, 6, 4},{3, 5, 5},{4, 8, 6},{5, 7, 7},

{6, 1, 9},{7, 9, 8},{8, 4, 3},{9, 2, 1}]}

10145

SMALL CUBE DIAGRAMS 13

Diagram Mathematica Code

K10161 ={X[{1, 1, 2},{2, 2, 4},{3, 9, 5},{4, 4, 6},{5, 3, 7},

{6, 5, 8},{7, 6, 9},{8, 7, 3},{9, 8, 1}],

Y[{1, 1, 9},{2, 2, 1},{3, 9, 8},{4, 4, 3},{5, 3, 2},

{6, 5, 4},{7, 6, 5},{8, 7, 7},{9, 8, 6}],

Z[{1, 3, 2},{2, 5, 4},{3, 6, 5},{4, 8, 6},{5, 7, 7},

{6, 9, 8},{7, 1, 9},{8, 4, 3},{9, 2, 1}]}

10161

K12591 ={X[{1, 9, 2},{2, 1, 4},{3, 2, 3},{4, 3, 5},{5, 4, 6},

{6, 6, 7},{7, 7, 8},{8, 5, 9},{9, 8, 1}],

Y[{1, 9, 7},{2, 1, 9},{3, 2, 8},{4, 3, 1},{5, 4, 2},

{6, 6, 4},{7, 7, 5},{8, 5, 3},{9, 8, 6}],

Z[{1, 4, 2},{2, 6, 4},{3, 5, 3},{4, 7, 5},{5, 8, 6},

{6, 9, 7},{7, 2, 8},{8, 1, 9},{9, 3, 1}]}

12591

14 S. BALDRIDGE AND B. MCCARTY

Appendix B. Link Examples

Diagram Mathematica Code

HL = {X[{1, 4, 1},{2, 1, 2},{3, 2, 3},{4, 3, 4}],

Y[{1, 4, 3},{2, 1, 4},{3, 2, 1},{4, 3, 2}],

Z[{1, 2, 1},{2, 3, 2},{3, 4, 3},{4, 1, 4}]}

Hopf Link

D0(41) = {X[{1, 1, 16},{2, 2, 15},{3, 3, 1},{4, 4, 2},{5, 5, 12},

{6, 6, 11},{7, 14, 8},{8, 13, 7},{10, 15, 13},{9, 16, 14},

{11, 7, 10},{12, 8, 9},{14, 10, 4},{13, 9, 3},{15, 11, 5},

{16, 12, 6}],

Y[{1, 1, 4},{2, 2, 3},{3, 3, 7},{4, 4, 8},{5, 5, 6},

{6, 6, 5},{7, 14, 10},{8, 13, 9},{10, 15, 11},{9, 16, 12},

{11, 7, 15},{12, 8, 16},{14, 10, 14},{13, 9, 13},{15, 11, 2},

{16, 12, 1}],

Z[{1, 10, 4},{2, 9, 3},{3, 13, 7},{4, 14, 8},{5, 12, 6},

{6, 11, 5},{7, 7, 10},{8, 8, 9},{10, 6, 11},{9, 5, 12},

{11, 2, 15},{12, 1, 16},{14, 16, 14},{13, 15, 13},{15, 4, 2},

{16, 3, 1}]}

D0(41)

D−4(52) = {X[{1, 1, 4},{2, 2, 3},{3, 3, 7},{4, 4, 8},{5, 5, 5},

{6, 6, 6},{7, 8, 9},{8, 7, 10},{9, 11, 11},{10, 12, 12},

{11, 9, 13},{12, 10, 14},{13, 13, 2},{14, 14, 1}],

Y[{1, 1, 14},{2, 2, 13},{3, 3, 1},{4, 4, 2},{5, 5, 12},

{6, 6, 11},{7, 8, 4},{8, 7, 3},{9, 11, 8},{10, 12, 7},

{11, 9, 6},{12, 10, 5},{13, 13, 10},{14, 14, 9}],

Z[{1, 8, 4},{2, 7, 3},{3, 12, 7},{4, 11, 8},{5, 10, 5},

{6, 9, 6},{7, 14, 9},{8, 13, 10},{9, 6, 11},{10, 5, 12},

{11, 2, 13},{12, 1, 14},{13, 4, 2},{14, 3, 1}]}

D−4(52)

References

[1] Y. Bae, C. Park. An upper bound of arc index of links. Mathematical Proceedings of the Cambridge Philosophical

Society. Volume 129, Issue 3, Nov 2000, 491-500.

[2] S. Baldridge, A. Lowrance. Cube diagrams and a homology theory for knots. arXiv:0811.0225v1.

[3] S. Baldridge, A. Lowrance. Cube Knot Calculator, http://cubeknots.googlecode.com.

[4] J. C. Cha, C. Livingston. KnotInfo: Table of Knot Invariants, http://www.indiana.edu/∼knotinfo, July 10,

2009.

[5] P. Cromwell. Embedding knots and links in an open book. I. Basic properties. Topology Appl., 64 (1995), no. 1,

37–58.

[6] M. Culler. Gridlink. http://www.math.uic.edu/∼culler/gridlink/.

[7] W.B.R. Lickorish. An Introduction to Knot Theory. Springer, 1997.

[8] C. Giusti. Plumbers’ knots. arXiv:0811.2215v1.

SMALL CUBE DIAGRAMS 15

[9] C. Manolescu, P. Ozsvath, S. Sarkar. A combinatorial description of knot Floer homology. arXiv:math/0607691v2.

[10] C. Manolescu, P. Ozsvath, Z. Szabo, D. Thurston. On combinatorial link Floer homology. arXiv:math/0610559v2.

[11] P. Ozvath, Z. Szabo, D. Thurston. Legendrian knots, transverse knots and combinatorial Floer homology.

arXiv:math/0611841v2.

[12] L. Ng, D. Thurston. Grid Diagrams, Braids, and Contact Geometry. arXiv:0812.3665v2.

[13] L. Zulli. A matrix for computing the Jones polynomial of a knot. Topology, 34, 717-729.

Department of Mathematics, Louisiana State University

Baton Rouge, LA 70817, USA

E-mail address: sbaldrid@math.lsu.edu

Department of Mathematics, Louisiana State University

Baton Rouge, LA 70817, USA

E-mail address: benm@math.lsu.edu