Region Alternating Knots
James Kreinbihl, Ra’Jene Martin, Colin Murphy, McKenna Renn, Jennifer Townsend
December 15, 2015
Abstract
We examine the number of region crossing changes necessary to produce
alternation of a diagram. This property leads to an invariant similar to
the dealternating number, but which offers new insight. In particular, we
apply the reasoning built for this invariant to prove that (warping) span
of any knot is at most 2.
1
Introduction
Alternating knots have many useful structures and properties, however, most
knots are inherently non-alternating. Adams et al. [2] first introduced the notion
of almost-alternating knots where (for some diagram) a single crossing change
creates an alternating diagram and no alternating projection exists. This notion
was developed by Abe and Kishimoto [1] into the dealternating number of a knot
dalt(K).
Definition 1.1. The dealternating number of a knot diagram D, dalt(D),
equals the minimum number of crossing changes required to obtain an alternating diagram from D.
The dealternating number of the knot K is then
dalt(K) =
min
D, a diagram of K
dalt(D).
As explored by Lowrance[5], dealternating number offers a measure of how far
a knot is from alternation.
By using region-crossing-changes (proven in [6] to be an unknotting operation)
we analogously develop the idea of region almost alternating knots and the region
dealternating number of knots.
1
Definition 1.2. The region dealternating number of a knot diagram D
or daltR (D) equals the minimum number of region crossing changes required to
obtain an alternating diagram from D.
The region dealternating number of the knot K is then
daltR (K) =
min
D, a diagram of K
daltR (D).
We connect region dealternating number to (crossing) dealternating number in
Section 2, where we prove:
Theorem 2.1. For any knot K, daltR (K) ≤ dalt(K).
and
Theorem 2.6. For every non-negative integer n, there exists a knot K such
that dalt(K) − daltR (K) = n.
Together these show that a knot never requires more region crossing changes
than individual crossing changes to produce an alternating diagram. Indeed,
the number of region crossing changes required can be arbitrarily smaller than
the number of crossing changes required.
In Section 3 we investigate (warping) span (an invariant proposed by Shimizu
[7]) in relation to region dealternation.
This investigation leads to a simple, but surprising conclusion to Lowrance’s
open question: Do knots/links with arbitrarily large span exist?.


0 if K is the unknot
Theorem 3.2. For any knot K, spn(K) = 1 if K is alternating


2 otherwise
In Section 4, we examine the impact of composition on dealternating number
and prove that dealternating number is subadditive (with equality possible only
when at most one factor knot is non-alternating):
Theorem 4.1.
daltR (K1 #K2 # . . . #Kn ) ≤ daltR (K1 ) + daltR (K2 ) + . . . + daltR (Kn ) − (s − 1)
where at least one Ki , 1 ≤ i ≤ n, is a non-alternating knot and s is the number
of non-alternating knots in the connect sum.
Finally, in Section 5, we discuss implications of our research and pose open
questions.
2
2
Comparison of dalt(K) and daltR (K)
Before continuing, we note that Shimizu’s proof [6] that region-crossing change
is an unknotting operation demonstrated that an individual crossing change can
be affected by a set of region crossing changes in the same diagram (each region
used at most once). It follows that every diagram can be made alternating using
some collection of region crossing changes.
By observation, it is clear that daltR (D) and dalt(D) are incomparable.
Nonetheless, using an alteration of a diagram via solely Reidemeister II moves
is enough to ensure that:
Theorem 2.1. For any knot K, daltR (K) ≤ dalt(K).
Proof. Given a knot K, consider a diagram D of K such that D realizes dalt(K)
via crossing changes on a minimal set C of crossings.
Centered at each crossing c in C, imagine a disc such that c is the only crossing
inside this disc (this is possible by making the radii of these discs less than 12
the minimal distance between crossings). The disjointness of these discs ensures
no interaction in the following alterations.
Figure 1: The top sequence displays a crossing in C before and after all C
crossing changes produce an alternating knot. The bottom sequence indicates
the corresponding RII move and RCC. Decorations indicate the crossings on
strands exiting the disc (after crossing or region crossing changes on all the
discs).
Within every disc, perform a Reidemeister II move from the tail of the overstrand
to the head of the understrand1 such that the introduced bigon containing the
original crossing c is an alternating bigon as illustrated in Figure 2. We denote
the resulting equivalent diagram D0 . It will follow that region crossing changes
1 The specific RII move designation is not necessary here; however, it will be useful in our
proof of Theorem 3.2.
3
on the set of bigons described above (and in Figure 2) induce an alternating
diagram from D0 .
The effect of this RII and single RCC on a strand through the disc is to take
a crossing sequence [e.g., − − u − −] and replace it with a reversed alternating
triplet [e.g., −−uou−−]. Note that performing this process on each disc induces
an alternating diagram if and only if the crossing changes on C [e.g., − − u − −
to − − o − −] induce an alternating diagram. By construction, crossing changes
on C induce an alternating diagram from D. Hence the region crossing changes
(|C| of them) induce an alternating diagram from D0 .
It follows that daltR (D0 ) ≤ |C| = dalt(K). Hence for the knot itself, daltR (K) ≤
dalt(K).
The following corollary regards a special case involving almost-alternating knots.
The notation of almost-alternating knots is from Adams [2], and we extend it
to region-almost alternating knots.
Definition 2.2. A knot K is almost-alternating (resp. region-almost-alternating)
if dalt(K) = 1 (resp. daltR (K) = 1)
Corollary 2.3. If K is a knot with dalt(K) = 1, then daltR (K) = 1.
Using the language of Adams, this means that all almost-alternating knots
are region-almost-alternating.
Proof. As dalt(K) 6= 0, the knot K is non-alternating, so at least one region
crossing change is required to induce an alternating diagram. In conjunction
with the previous theorem, 1 ≤ daltR (K) ≤ dalt(K) = 1.
At this point it is natural to wonder whether the set of almost-alternating knots
is equivalent to the set of region-almost-alternating knots. The following family
of knots will demonstrate that this is not the case: the set of almost alternating
knots (where dalt(K) = 1) is a proper subset of region almost alternating knots.
Theorem 2.4. Let n be a non-negative integer, and let i = 1 or 2. Then
dalt(T3,3n+i ) = n
(1)
daltR (T3,3n+i ) = 1
(2)
and
(1) is proven by Abe-Kishimoto and Lowrance [1] [5]. It implies that (3, q) torus
knots have a dealternating number greater than 1 when q > 5. We prove (2)
below:
4
Figure 2: A T3,q torus with origin and orientation denoted. Note that from
p, we encounter two crossings bordering R1 (shaded) and then two crossings
bordering R2 (dashed).
Proof. Suppose we have a (3, q) torus knot diagram as in Figure 2.
As evident in Figure 2, on a traversal of the diagram originating at p, crossings
bordering R1 are encountered in u − o pairs separated by o − u pairs that border
R2 . Here o denotes a pass over a crossing, and u denotes a pass under a crossing.
The resulting sequence of crossing encounters is
u − o − o − u − u − o − o − u − ··· − u − o − o − u
where crossings bordering R1 are shaded (the pair that terminates the sequence
is necessarily bordering R2 ).
Thus, when we perform a region crossing change on R1 , we yield the following
crossing sequence:
o − u − o − u − o − u − o − u − · · · − o − u − o − u.
The resulting diagram is an alternating diagram. Therefore, every (3, q) torus
knot is a region almost alternating knot.2
This property of the family of T3,q torus knots leads us to the trivial corollaries:
Corollary 2.5. The set of all almost alternating knots is a proper subset of
the set of region almost alternating knots.
Corollary 2.6. For every non-negative integer n, there exists a knot K such
that dalt(K) − daltR (K) = n.
This follows trivially by taking K to be T3,3(n+1)+1 .
2 Observe that the choice of R vs. R was arbitrary: we could have instead performed the
1
2
region crossing change on R2 and still attained an alternating diagram.
5
3
Span is at most 2
In [7], Shimizu introduced the Warping Polynomial of a knot diagram, and the
related invariant of span. The polynomial and span invariant has been used
and developed for various purposes: Cheng [3] used the warping polynomial
and degree to inspire a polynomial invariant of virtual links, Lowrance [5] used
the warping degree as a bound on crossing dealternating number, Higa et al.
[4] characterized all possible warping sequences of the trefoil knot, and Shimizu
(along with others) has developed the warp further: most recently into a matrix
form [8].
In this section, we will use the intuition of region crossing changes to prove that
span can take on only the values 0, 1, 2.
The span of a knot, though defined as the span of a warping polynomial, can
also be found directly through the process below.
Let D be a knot diagram with a set orientation and starting point. An edge is
an arc between two crossings. The first edge that we encounter at our starting
point will have d1 = 0 where di is the warping weight of the ith edge encountered
on the traversal. Following the orientation, assign warping weights to edges
consecutively so that
(
di + 1 if the arc went over a crossing between edge i and i + 1
di+1 =
di − 1 if the arc went under a crossing between edge i and i + 1
An example is given in Figure 3.
As we will investigate in the appendix, this assignment of warping weights does
not naturally extend to links, but our bound still holds for any reasonable
definition on links.
Definition 3.1. The span of a diagram D, denoted spn(D), is max{di − dj }
where di and dj are edge weights.
For a knot K, spn(K) =
min
D,Diagrams of K
spn(D).
6
Figure 3: The warping weights of the first edges have been found in this diagram.
It is known that span is related to the (crossing) dealternating number. Indeed,
Shimizu [7] showed:
spn(K) − 1
≤ dalt(K)
2
This inequality peaked our interest in span as a possible bound on the region
dealternating number, as we hoped to find explicit proofs of the existence of
knots with daltR (K) > 1. However, as Theorem 3.2 shows, span cannot differentiate between knots with daltR (K) ≥ 1.


0 if K is the unknot
Theorem 3.2. For any knot K, spn(K) = 1 if K is alternating


2 otherwise
We note that Theorem 3.2 negatively answers the open question posed by
Lowrance [5]: Do knots (or links) with arbitrarily large (warping) span exist?
The first of these two results are trivial and known:
1. The unknot in a diagram with no crossings is solely the edge d1 with
weight 0, hence has span of 0.
2. An alternating diagram has spn(D) = 1 as an orientation can be assigned
so that warping weights alternate between 0 and 1 (see Figure 4).
Our proof is assisted by the following lemma:
Lemma 3.3. If R is an alternating region of a diagram D with an origin and
orientation, then a RCC on R keeps warping weights on edges non-incident to
R (including entering and exiting edges) are fixed.
7
Figure 4: Any alternating diagram has a span of 1.
Here an alternating region is one for which the incident edges are all alternating,
as in Figure 5.
Figure 5: The region R is alternating, as each incident edge is alternating. In
comparison, R0 is not alternating, as the highlighted edges are not alternating.
Proof. Let di be the weight of an edge entering the alternating region R. Then
di+2 is the weight of an edge exiting R. Since R is alternating, the arc goes
both over and under before exiting R. Hence warping weight di = di+2 .
After a RCC is performed on R, R is still alternating. Hence di = di+2 . In
addition, since the first (with respect to the orientation) entering edge (di )
necessarily retains its original warping weight, it follows inductively that all entering/exiting edges retain their original warping weights. This directly implies
that weights of other non-incident edges are also unchanged.
The proof of Theorem 3.2 uses the fixed weight of non-incident edges under a
set of region crossing changes that lead to an alternating diagram in order to
recover original warping weights on both non-incident and incident edges of a
knot’s diagram.
8
Proof. Given a knot K with arbitrary diagram D, let C be a set of crossings
such that crossing changes on C induce an alternating diagram. As in the proof
of Theorem 2.1, imagine a disc around each crossing of C and perform the RII
moves from that proof, creating the equivalent diagram D0 . Now let R be the set
of all bigons created containing the crossings of C. Then, by proof of Theorem
2.1, R is a set of regions that induces an alternating diagram with region-crossing
changes. In addition, each bigon in R is alternating. In particular, due to the
non-intersection of the discs, the sequence of RCCs on regions in R can not
alter the crossings and hence the alternation of other R bigons.
Figure 6: After RII moves in each disc, the bigons containing the original crossing compose R. Each such bigon is also alternating.
After the sequence of RCC moves on all regions of R, the diagram is alternating
and thus has an orientation which produces warping weights of only 0 and 1.
Choose the same origin and orientation in D0 .
We consider performing the RCC moves on R sequentially (the order does not
matter). Since each region in R is alternating and remains so throughout the
RCC sequence, Lemma 3.3 guarantees that each sequential RCC does not alter
the warping weights on edges not incident to R. Thus, after completion of the
RCC sequence, all edges not incident to R retain their original weights and
thus the original warping weights must be either 0 or 1 — identical to the final
weights in the alternating diagram.
It remains to determine the original warping weights on edges incident to R.
The non-intersection of discs guarantees each edge is incident to at most one
region of R, and hence the warping weight of any such edge can only be changed
by the RCC on that single incident region.
Hence we can work backwards from the final alternating diagram to recover the
original warping weights.
Since the warping weights entering a region of R are necessarily di = 0 or 1 and
the warping weights on edges incident to R must be di+1 = di ± 1, the options
for di+1 are {−1, 0, 1, 2}.
As all edges not incident to R have weight 0 or 1 and those incident to R have
weight between -1 and 2, the span for the diagram D0 (and hence the knot K)
is at most 3.
9
Figure 7: The fact that the bigons induce an alternating diagram under RCCs
allows us to predict the warping weights on non-incident edges, and hence fill
in weights on the diagram prior to RCC.
In the interest of refining this bound we recommended a RII move involving the
tail of the overstrand and the head of the understrand to guarantee warping
weights of −1 on the R-incident edges. These moves (and their weights) are
shown in Figure 7 for a positive crossing and Figure 8 for a negative crossing.
Figure 8: The fact that the bigons induce an alternating diagram under RCCs
allows us to predict the warping weights on non-incident edges, and hence fill
in weights on the diagram prior to RCC.
It follows that, given any knot K, we can attain a diagram and assign an
orientation such that each edge of the diagram D0 of K has weight of 0 or
1 (if non-incident to any region of R) or −1 (if incident). Thus the maximum
possible warping degree on an edge of D0 is 1, and the minimum possible warping
degree is −1.
As spn(K) ≤ spn(D0 ) ≤ 2, this concludes our proof.
We discuss repercussions of this finding in the conclusion.
4
Connected Sums and the region dealternating
number
It is an obvious property of alternating knots K1 and K2 that dalt(K1 #K2 ) =
0 = daltR (K1 ) + daltR (K2 ) since K1 #K2 is also alternating.
10
Less obvious is the behavior of non-alternating knots under composition (indeed
even the preservation of non-alternation is unknown). Thus we investigate the
effect of composition on region dealternating number.
Theorem 4.1.
daltR (K1 #K2 # . . . #Kn ) ≤ daltR (K1 ) + daltR (K2 ) + . . . + daltR (Kn ) − (s − 1)
where at least one Ki , 1 ≤ i ≤ n, is a non-alternating knot and s ≥ 1 ≥ 1 is the
number of non-alternating knots in the connect sum.
The proof of this theorem relies on the following lemma:
Lemma 4.2. Given a knot diagram, D and any interior region Ri of the diagram, there is an equivalent diagram (with incidence of regions and crossings
preserved) such that Ri is the exterior region.
Note: this proof is essentially the proof of a lemma for Kuratowski’s Theorem
of criterion for graph planarity.
Proof. Suppose we have a knot diagram, D, on a plane with specified Ri as in
the theorem. Perform an inverse stereographic projection to map D onto the
surface of the unit sphere as in Figure 4. Then, rotate the sphere so that the
north pole is contained within region Ri . Perform a stereographic projection
to map D back onto the plane, and finally (because this produces the mirror
image), reverse all crossings.
Figure 9: For simplicity, we display a standard shadow of the torus knot where
Ri is the central region. The diagram is first moved onto a sphere and repositioned so that Ri the north pole. This guarantees that a stereographic projection
mapping the diagram back on to the plane(not shown) will map the arcs adjacent to Ri furthest away, ensuring Ri is indeed the exterior. Crossings can be
assigned at the end to produce the same knot as the original.
The resulting diagram has the original Gaussian Code, sufficing to show this is
a diagram of the same knot with the same incidence of regions and crossings.
11
Observe that the borders of region Ri were the closest to the north pole and
therefore are projected the furthest on the plane. Thus, Ri is now the outer
region of the diagram on the plane.
Lemma 4.3. For a non-alternating knot K, let D be a diagram that realizes
daltR (K) via a minimal set of regions R that induces an alternating diagram
via region crossing change on R. Then D contains both
1. an arc which borders a region in R and a region in RC
2. an arc which borders two regions in RC .
Proof. Case I: D is a reduced diagram.
We note an important result from Shimizu [6]: In a reduced knot diagram,
performing region crossing changes on R produces the same crossing changes as
performing RCCs on RC . Hence |R| ≤ b 2r c where r is the number of regions in
the diagram.
1. As D is non-alternating, R is nonempty, as is RC , by the note above. It
follows directly that there exists an arc of the knot that borders a region of each
type.
2. The only way to choose R so that no regions in RC share a border is to do a
checkerboard shading and ensure that R contains (at minimum) all shaded or
unshaded regions, so at minimum b 2r c regions, where r is the number of regions.
With the note above, this implies |R| = b 2r c, and contains exactly the shaded
or unshaded regions. But, as noted by Shimizu, this would yield no changes to
the original diagram since each crossing is changed exactly twice.
We conclude that R cannot contain all of the shaded (resp. unshaded) regions,
ergo the existence of the desired arc exists.
Case II: D contains nugatory crossings.
1. As above, R is nonempty and RC is nonempty as well: performing region
crossings on all regions preserves the non-nugatory crossings (as in Case I).
Though nugatory crossings have changed, the resulting diagram is still a diagram
of K and hence cannot be a diagram of a non-alternating knot. Thus there exists
an arc bordering a region of each type.
2. As with a reduced diagram, R must contain all of the shaded (resp. unshaded)
regions in a checkerboard coloring to ensure no arc borders two RC regions.
Consider the diagram D0 modified from D by performing RCCs on the shaded
regions. As with reduced diagrams, the non-nugatory crossings of D are preserved in D0 : hence D0 is still a diagram of K and so cannot be alternating. Thus
R = {shaded regions} t U for some nonempty subset U of unshaded regions.
12
As D0 is a diagram obtained from D via RCCs on the shaded regions, RCC
moves on U convert D0 to an alternating diagram. Since D0 is also a diagram
of K, this contradicts the minimality of |R|.
Hence, by minimality, R does not contain all shaded (resp. unshaded) regions
in a checkerboard coloring: and there exists an arc bordering two RC regions.
Although there are subtleties, the proof of Theorem 4.1 uses the following idea:
Given two knot diagrams we use Lemma 4.2 and Lemma 4.3 to specify exterior
regions, and then connect along the guaranteed arcs from Lemma 4.3. As in
Figure 10 this merges the R regions.
Figure 10: For these T3,8 torus diagrams, we can consider R to be solely the
exterior regions (striped). This example forgoes the need to perform stereographic projection. In the connected sum, the R regions are merged, as are the
RC regions. The exterior region of the connected sum now suffices to make the
new diagram alternating.
Theorem 4.1.
daltR (K1 #K2 # . . . #Kn ) ≤ daltR (K1 ) + daltR (K2 ) + . . . + daltR (Kn ) − (s − 1)
where at least one Ki , 1 ≤ i ≤ n, is a non-alternating knot and s ≥ 1 ≥ 1 is the
number of non-alternating knots in the connect sum.
Proof. We first examine the composition of two knots K1 and K2 .
13
Without loss of generality, we assume K1 is non-alternating with diagram D1
which realizes daltR (K1 ) via the set R1 .
Base Case: Consider (K1 #K2 ).
Case I: K2 is alternating with an alternating diagram D2 (R2 is empty)
Using Lemma 4.3, there exists an arc e bordering two regions of RC
1 for
diagram D1 . By Lemma 4.2, we can make one of these regions exterior.
Take the connected sum along e and any arc of D2 (or the exterior arc
on a RI move if necessary to make alternation consistent between knots).
The resulting diagram can be made alternating using only R1 .
This yields daltR (K1 #K2 ) ≤ daltR (K1 ). As daltR (K2 ) = 0 and s = 1 in
this case, we have the inequality:
daltR (K1 #K2 ) ≤ daltR (K1 ) + 0 = daltR (K1 ) + daltR (K2 ) − (s − 1)
Case II: Let K2 be non-alternating with diagram D2 realizing daltR (K1 )
via RCCs on R2 . By Lemma 4.3, there are arcs e1 of D1 and e2 of D2
that border regions in their respective Ri and RC
i .
Using Lemma 4.2 we make the Ri regions bordering each ei exterior. Connecting along e1 and e2 (or the exterior arc on a RI move of e1 if necessary
to make alternation consistent between knots post-RCC moves). The set
R1 ∪ R2 now suffices to make the composed diagram alternating, and by
merging a region we guarantee |R1 ∪ R2 | = daltR (K1 ) + daltR (K2 ) − 1.
As s = 2 in this case, we obtain our desired inequality:
daltR (K1 #K2 ) ≤ daltR (K1 ) + daltR (K2 ) − (s − 1)
Though we cannot prove the resulting composition diagram with R = R1 ∪ R2
meets the hypothesis of Lemma 4.3, the edges from K1 that met each criterion
either (1) were not cut in the composition and still satisfy the criterion or (2)
were cut and merged into a new edge that now meets the same criterion. Thus
the diagram D1 #D2 built via this process has daltR (D1 #D2 ) ≤ daltR (K1 ) +
daltR (K2 ) − (s − 1) and contains one edge bordering R and RC and one edge
bordering two RC regions.
The general inequality follows via induction with (K1 # . . . Kn−1 )#Kn , where
the proof exactly mirrors the base case.
14
5
Conclusion
The region crossing change is a natural variation of crossing changes. It opens
up many possibilities for new research pathways in knot theory, in particular in
examining how the region crossing change operation impacts various invariants.
The region dealternating number has interesting and sometimes surprising connections to more established invariants, as evidenced by the unexpected result
that all knots have a span of at most 2.
Sadly this discovery trivializes some previous results that seemed promising,
such as Shimizu’s
spn(K) − 1
≤ dalt(K)
2
and the results by others (including Lowrance [5]) based upon this result. With
Theorem 3.2, this result becomes equivalent to saying that it requires at least
one crossing change to make a non-alternating knot into an alternating diagram.
It may also have implications for other invariants and structures based upon
the warping polynomial, such as Cheng and Gao’s [3] odd writhe polynomial
for virtual links.
This work has only scratched the surface of the properties and classification of
region almost alternating knots. In particular, it remains an open question if
the region dealternating number of a knot can be arbitrarily large — or indeed,
if a knot with daltR (K) greater than 1 exists.
Acknowledgements
This work was conducted under the NSF grant DMS 1460537 as part of the
Seattle University 2015 SUmMER REU. We would also like to recognize the
help and advice of Allison Henrich and Steven Klee, as well as the useful correspondence of Adam Lowrance.
References
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[2] C. C. Adams, J. F. Brock, J. Bugbee, T. D. Comar, K. A. Faigin,
A. M. Huston, A. M. Joseph, and D. Pesikoff, Almost alternating
links, Topology and its Applications, 46 (1992), pp. 151–165.
[3] Z. Cheng, A polynomial invariant of virtual knots, Proceedings of the American Mathematical Society, 142 (2014), pp. 713–725.
15
[4] R. Higa, Y. Nakanishi, S. Satoh, and T. Yamamoto, Crossing information and warping polynomials about the trefoil knot, Journal of Knot
Theory and Its Ramifications, 21 (2012), p. 1250117.
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of
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Appendix
The minor problem with the extension of warping span on links from [5] is
visible in Figure 11
Figure 11: The warping span is not well-defined (at least for non-reduced link
diagrams). As demonstrated here the span of warping weights for a diagram
can be dependent on the origin definition in a link diagram. Also, if continued,
the warping weight does not return to 0.
The warping span can be extended to links by taking the maximum (over every component) of the minimum span among all possible orientations of that
component. This is certainly a messy definition, and we encourage the reader
to create a more elegant extension.
16