ArticlePDF Available

# Small examples of cube diagrams of knots

Authors:

## Abstract and Figures

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. Comment: 15 pages
Content may be subject to copyright.
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
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
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
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 XOconﬁgurations 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
ζSnthat 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
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}]}
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)
D4(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}]}
D4(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.
[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
Department of Mathematics, Louisiana State University
Baton Rouge, LA 70817, USA
... Recall that J is sliced by connected level sets of p. Furthermore, note that there exist integer numbers m 1 and m 2 such that p −1 (t) ∩ J = K 1 for all t ≤ m 1 and p −1 (t) ∩ J = K 2 for all t ≥ m 2 , where K 1 and K 2 are cubic knots which are isotopic to K 1 and K 2 , respectively. Now, we will use the following results which will be proved in 4. Given a cubic knot K we can choose a small cubulation C 1 m fine enough that N (K) = ∪{Q ∈ C 1 m | Q ∩ K = ∅} is a closed tubular neighborhood of K and Q ∩ K is equal to either a vertex, one edge, or two edges sharing a vertex (neighboring edges). ...
... Theorem 4. 6. Given two cubic knots K 1 and K 2 , we obtain K 1 and K 2 as in Lemma 4. 4. Then there exists a finite sequence of cubulated moves that carries K 1 into K 2 . ...
... In this section, we will prove that there exists a finite sequence of cubulated moves that carries K into K . 4. There exists a finite sequence of cubulated moves that carries K into K . ...
Article
Full-text available
In this paper, we prove than given two cubic knots $K_1$, $K_2$ in $\mathbb{R}^3$, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister moves for classical tame knots. We use the fact that a cubic knot is determined by a cyclic permutation of contiguous vertices of the $\mathbb{Z}^3$-lattice (with some restrictions), to describe some of the classic invariants and properties of the knots in terms of such cyclic permutations, by projecting onto a plane such that it is injective when restricted to the $\mathbb{Z}^3$-lattice and the image of the $\mathbb{Z}^3$-lattice is dense.
... The integer n is the size of the cube diagram and the cube number of a knot, denoted c(K), is the smallest n for which there is a cube diagram for the knot of size n. In [4] small examples of cube diagrams of knots were given up to 7 crossings. Some examples given were observed to be minimal but only those knots K for which the cube number equaled the arc index, or α(K). ...
... However, such lifts do not always exist (c.f. [3] and [4]). Before proceeding, we need to establish some terminology and facts about grid diagrams (for more details see [3]). ...
... [3]). Nevertheless, even a partial order doesn't guarantee liftability to a cube diagram as the (y, z)-and (z, x)-projections may not be grid diagrams in such a lift (c.f [3] and [4]). Below, we will introduce some grid configurations that fail to lift, not because of a lack of partial ordering but due to crossings in the (y, z)-or (z, x)-projections that do not satisfy the crossing conditions for a cube diagram. ...
Article
For a knot $K$ the cube number is a knot invariant defined to be the smallest $n$ for which there is a cube diagram of size $n$ for $K$. There is also a Legendrian version of this invariant called the \emph{Legendrian cube number}. We will show that the Legendrian cube number distinguishes the Legendrian left hand torus knots with maximal Thurston-Bennequin number and maximal rotation number from the Legendrian left hand torus knots with maximal Thurston-Bennequin number and minimal rotation number. Comment: 19 pages
... In [3] small examples of cube diagrams of knots were given up to 7 crossings. In some cases it was observed that the examples given were minimal. ...
... However, such lifts do not always exist (c.f. [2] and [3]). ...
... [2]). Nevertheless, even a partial order doesn't guarantee liftability to a cube diagram as the (y, z)-and (z, x)-projections may not be grid diagrams in such a lift (c.f [2] and [3]). Below, we will introduce some grid configurations that fail to lift, not because of a lack of partial ordering but due to crossings in the (y, z)-or (z, x)-projections that do not satisfy the crossing conditions for a cube diagram. ...
Article
Full-text available
For a knot K the cube number is a knot invariant defined to be the smallest n for which there is a cube diagram of size n for K. We will show that the cube number detects chirality in all cases computed thus far, and distinguishes certain legendrian knots.
... While cube diagrams project to grid diagrams, grid diagrams rarely lift to cube diagrams (which may be the reason they were not discovered earlier). For example, in Baldridge-McCarty [4] we show that only about 20% of size 8 grid diagrams of nontrivial knots lift to cube diagrams, and that this percentage decreases as the size of the grid increases. This sparsity of cube diagrams leads to strictly stronger invariants than those defined using grid diagrams. ...
... markings and the segments that begin from them continue to remain parallel to axes defined by the markings in the projection. It is not true, in general, that a randomly chosen set of markings in the hypercube will project to data structures C xyz and C wxz that also satisfy the cube diagram crossing conditions (in fact, it is exceedingly rare—cf. [4] for calculations about the sparsity of cube diagrams). Nor is it advantageous to require that the data structures C xyz and C wxz to be cube diagrams—since they both share a projection to the zx-plane, they would share a common grid diagram up to orientation, which means both data structures would essentially describe the same link. Of ...
Article
In this paper we introduce a representation of a embedded knotted (sometimes Lagrangian) tori in $\BR^4$ called a hypercube diagram, i.e., a 4-dimensional cube diagram. We prove the existence of hypercube homology that is invariant under 4-dimensional cube diagram moves, a homology that is based on knot Floer homology. We provide examples of hypercube diagrams and hypercube homology, including using the new invariant to distinguish (up to cube moves) two "Hopf linked" tori. We also give examples of a "Trefoil" torus and an immersed knotted torus that is an amalgamation of the $5_2$ knot and a trefoil knot. Comment: 39 pages, 32 figures
... While cube diagrams project to grid diagrams, grid diagrams rarely lift to cube diagrams. In a recent note, the first author and McCarty [4] show that, for example, only about 20% of size 8 grid diagrams of nontrivial knots lift to cube diagrams, and that this percentage decreases as the size of the grid increases. The sparsity of cube diagrams compared to grid diagrams is advantageous: cube diagrams can be used to develop strictly stronger invariants than grid diagrams. ...
Article
Full-text available
In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under two types of cube diagram operations. A knot homology is constructed from cube diagrams and shown to be equivalent to knot Floer homology.
Article
In this paper we show how to combinatorically compute the rotation class of a large family of embedded Legendrian tori in $\mathbb{R}^5$ with the standard contact form. In particular, we give a formula to compute the Maslov index for any loop on the torus and compute the Maslov number of the Legendrian torus. These formulas are a necessary component in computing contact homology. Our methods use a new way to represent knotted Legendrian tori called Lagrangian hypercube diagrams.
Article
Full-text available
In 1996, Cromwell and Nutt [7] found an upper bound on the arc index which is related to the minimal crossing number and conjectured that the upper bound achieves the lowest possible index for alternating links.
Article
Full-text available
In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under two types of cube diagram operations. A knot homology is constructed from cube diagrams and shown to be equivalent to knot Floer homology.
Article
Full-text available
We use grid diagrams to present a unified picture of braids, Legendrian knots, and transverse knots. Comment: 14 pages, 12 figures, to appear in proceedings of the 2008 Gokova Geometry/Topology Conference; v2: minor changes following referee comments
Article
Full-text available
Link Floer homology is an invariant for links defined using a suitable version of Lagrangian Floer homology. In an earlier paper, this invariant was given a combinatorial description with mod 2 coefficients. In the present paper, we give a self-contained presentation of the basic properties of link Floer homology, including an elementary proof of its invariance. We also fix signs for the differentials, so that the theory is defined with integer coefficients. Comment: Updated to final published version.
Article
In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under two types of cube diagram operations. A knot homology is constructed from cube diagrams and shown to be equivalent to knot Floer homology.
Article
Birman and Menasco recently introduced a new way of presenting knots and links together with a corresponding link invariant. This paper examines the fundamental properties of this arc-presentation. In particular, a set of moves is described which relate two different arc-presentations of the same knot, and the behaviour under the knot operations of distant union and connected sum is established.
Article
Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots in the three-sphere, which takes values in link Floer homology. This invariant can be used to also construct an invariant of transverse knots. Comment: 27 pages, 13 figures; v2: Expand and correct discussion of links