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UNKNOTTING AND ASCENDING NUMBERS OF KNOTS
AND THEIR FAMILIES
Slavik Jablan
Radmila Sazdanovic
Ljiljana Radovic
Ana Zekovic
Unlinking number
Definition: The unlinking number u(L) of a link L is the minimal number of
crossing changes required to obtain an unlink from the link L;
the minimum is taken over all diagrams of L.
Unlinking number
TREFOIL
UNKNOT
“Measure of knottednes” (“unlinking number”) is one from the
most difficult problems in knot theory.
Unlinking number
u=1
u=0
Unlinking number
There are two different approaches for obtaining the unlinking number of L:
1) according to the classical definition, one is allowed to make an ambient
isotopy after each crossing change and then continue the unlinking process
with the newly obtained projection;
2) the standard definition requires all crossing changes to be done simultaneously
in a fixed projection.
Those two definitions are equivalent (see, e.g., Adams, 1994).
Fundamenal question: can we obtain unlinking number from minimal diagrams
of a link? If we can obtain it, it is computable invariant.
The Nakanishi-Bleiler example:
(a) the minimal projection of the knot 5 1 4 that requires at least three crossing
changes
to be unknotted;
(b) the minimal projection of the knot 3 1 2 with the unknotting number 1;
(c) non-minimal projection of the knot 5 1 4 from which we obtain the correct
unknotting number u(5 1 4) = 2.
Consequence: correct unknotting number cannot be always obtained from
fixed minimal diagrams.
Unknotting 5 1 4
514
5 -1 4
312
3 -1 2
u(5 1 4) =2
Bernhard-Jablan Conjecture (1994; 1995)
1) u(L) = 0 for any unlink L;
2) u(L) = min u(L-)+1, where the minimum is taken over all minimal
projections of links L-, obtained from a minimal projection of L by
one crossing change.
This means that we take a minimal projection of a link L, make a crossing
change in every crossing, and minimize all the links L- obtained.
The same algorithm is applied to the first, second, ... k-th generation of
the links obtained. The unlinking number is the number of steps k in this
recursive unlinking process.
Definition: The unlinking number obtained from BJ-conjecture will be called
BJ-unlinking number and denoted by uBJ.
In the case of alternating links, according to the Tait Flyping Theorem,
all minimal projections will give the same result, so it is sufficient to use
only one minimal projection.
Non-minimal diagrams and uBJ
For non-alternating links, we need to work with all minimal projections, since two
minimal diagrams of a same knot or link can give different BJ-unlinking numbers.
This follows from the example found by A. Stoimenow (2001)
Two minimal diagrams projections of the non-alternating knot 1436750,
the first with uBJ=1, and the other with uBJ=2.
Generalization of Stoimenow’s example
Two projections 124*-1.-1.-1.-1.-1:(-1,(-2l)) 0:(2k-1):.-1 and 8*-2 0:(2l+1) 0:-2 0.(-2k) 0.-1.2 0
of the same knot, the first with u=uBJ=k, and the other with uBJ=k+1 (l <2k).
Unlinking gap
The question about unlinking gap originates from the Nakanishi-Bleiler example:
unknotting number of the fixed minimal diagram of the knot 5 1 4 is 3, and it is different
from its BJ-unknotting number and unknotting number uBJ=u=2. The same property
holds for the whole knot family (2k+1) 1 (2k) and link family (2k) 1 (2k) (k>1) .
514
Definition: The minimal diagram unlinking number uM(L) is the minimal number of
simultaneous crossing changes required to unlink a fixed minimal projection M of a
link L.
Definition: The unlinking gap of a link L is the positive difference between the minimal
unlinking number of all minimal projections min{uM(L)} and the unlinking number u(L).
In the case of alternating links, according to Tait Flyping Theorem, all minimal projections
of a link L will have the same unlinking gap δM(L), so a particular value δM(L) will be the
unlinking gap δ(L) of a link L.
D. Garity extended the one-parameter Bernhard knot family (2k+1) 1 (2k) to the
two-parameter family (2k+1) (2l+1) (2k) (k >1,l≥0, k>l) with the unknotting gap δ=1.
514
Family (2k+1) 1 (2k)
716
918
Family (2k+1) (2l+1) (2k)
δ=1
Garity also discovered the first two-parameter family of rational links (2k) 1 (2l)
(k≥ 2, l ≥2) with an arbitrarily large unlinking gap.
uM= k+l+1
u<l
δ ≥ k-1
Unlinking gap
•
•
n= 9
1
10
1
11
5
12 13
5 23
14
36
15
106
16
180
Alternating knot family with arbitrarily
large unknotting gap
Non-alternating link family 2k,3,-2k
with arbitrarily large unknotting gap
Multi-parameter families of links with
arbitrarily large unlinking gap
•
•
•
•
(2k)(2l-1)(2m)
3-parameter family with arbitrarily large unlinking gap
The first member is the link 4 1 4 with n=9 crossings
Conditions: k≥2, m≥2, 2k ≥ 2m ≥ 2l-1
10 5 8
414
614
616
Minimal diagrams of non-alternating knots with unknotting gap
.(3,-2).2
4 1 1,3,-2
.2.(3,-2)
3 2,3,-3
Non-alternating links with unlinking gap
Non-alternating minimal projection 3 1,3 1,-3 with the unlinking gap 1 and the family
(2p+1) 1,(2q+1) 1,-3 derived from it.
Multi-parameter families of rational links with arbitrarily large unlinking gap
Knot family of the form
(2k1) ... (2l-1) ...(2k2i+1).
Link family of the form
(2k1) ... (2k2i-1)(2l-1)(2m1) ...(2m2j-1).
Family of non-alternating pretzel link diagrams with
arbitrarily large unlinking gap
(2k),(2k),-3
The family of rational links (2k) 1 (2k) (k>1) with arbitrarily large unlinking gap δ = k-1
can be expressed in pretzel link form as 2k,2k,1. The family of minimal non-alternating
link projections 2k,2k,-3 is derived from it. Every diagram belonging to the family 2k,2k,-3
has uBJ = k, and the fixed minimal projection requires 2k-1simultaneous crossing changes
to be unlinked. Hence, the unlinking gap of the minimal diagram 2k,2k,-3 is k-1.
Rational knots with unknotting number 1
541211114
721163
Rational links with unlinking number 1
212321113212
411131121114
Conjecture: the knot 2,3,-3 and the link 2,2,-3 are the only pretzel KLs with
unlinking number 1 (Buck, Jablan).
A rational tangle which can be transformed into a tangle reducible to -1 by one
crossing change is called t-1-tangle.
For example, by a single crossing change the tangle 2 1 1, gives 2 -1 1= -1,
3 1 1 1 gives 3 -1 1 1= -1, 2 1 1 2 gives 2 1 -1 2 = -1, etc.
Definition: A distance of links L1 and L2 is a minimal number of crossing
changes in L1 required to obtain L2, the minimum taken over all projections of L1.
Definition: A distance of link projection L'1 from a link L2 is a minimal number
of crossing changes in the particular fixed link projection L'1 required to obtain
a projection of L2.
Ordered knots with at most 9 crossings from Rolfsen tables, where unknot is denoted by 1.
Lines in the graph correspond to knots with distance 1 computed from minimal diagrams.
Definition (Murasugi, 1965): signature σ(L) of a link L is the signature of the
matrix SL+SLT, where SLT is the transposed matrix of SL, and SL is the
Seifert matrix of the link L.
Theorem (Murasugi, 1965) The lower bound for unknotting number of a knot K
is |σ(K)|/2, and the lower bound for unlinking number of a link L is (|σ(K)|+1)/2
Theorem: All rational links with unlinking number one have an unlinking
number one minimal diagram.
Present state: BJ-conjecture holds for all two-component links whose unlinking
numbers were computed by P.Kohn (1993). The complete list of BJ-unknotting
numbers for knots with n=11 crossings computed by the authors,
using the program LinKnot, is included in the Table of Knot Invariants by
C.Livingston and J.C.Cha (2005). The recent results by B.Owens (2005)
and Y.Nakanishi (2005) confirmed the unknotting number u=3 for the knot 935,
and the unknotting number u=2 for the knots 1083, 1097, 10105, 10108, 10109, and
10121 computed according to the BJ-conjecture. For many knots with at most 10
crossings results of P.Ozsvath and Z.Szabo confirmed that unknotting number is
equal to 2.
Open problem (relatively realistic): prove BJ-conjecture for rational knots and links.
Unknotting numbers of alternating knot families
Signature σ(K) can confirm many unknotting numbers, since the lower bound
for unknotting number a knot K is |σ(K)|/2.
Theorem (Traczyk, 2004) If D is a reduced alternating diagram of an oriented
knot, then σD= (-w+W-B)/2, where w is the writhe of D, W is the number of
white regions, and B is the number of black regions in the checkerboard
coloring of D.
w=4
W=3
B=7
σ = (-4+3-7)/2 = -4
3,3,2
Introducing orientation of a knot, every n-twist (chain of bigons) becomes
parallel or antiparallel. For signs of crossings and checkerboard coloring we
use the following convention:
(a) Positive crossing (b) negative crossing
(c) parallel positive twist;
(d) parallel negative twist;
(e) antiparallel positive twist;
(f) antiparallel negative twist.
Lemma: By replacing n-twist (n>1) by (n+2)-twist in the Conway symbol of an
alternating knot K, the signature changes by -2 if the replacement is made in
a parallel twist with positive crossings, the signature changes by +2 if the
replacement is made in a parallel twist with negative crossings, and remains
unchanged if the replacement is made in an antiparallel twist.
Proof: According to the preceding theorem:
1) by adding a full twist in a parallel positive n-twist the writhe changes by +2, the number
of the white regions W remains unchanged, the number of black regions B increases by +2,
and the signature changes by -2;
2) by adding a full twist in a parallel negative n-twist the writhe changes by -2, the number
of white regions W increases by 2, the number of black regions B remains unchanged, and
the signature increases by 2;
3) by adding a full twist in an antiparallel positive n-twist the writhe changes by +2, the
number of white regions W increases by 2, the number of black regions B remains
unchanged, and the signature remains unchanged;
4) by adding a full twist in an antiparallel negative n-twist the writhe changes by -2, the
number of white regions W remains unchanged, the number of black regions B increases
by 2, and the signature remains unchanged.
Theorem: The signature σK of an alternating knot K given by its Conway
symbol is
where the sum is taken over all parallel twists, ci is the sign of crossings
belonging to a parallel twist ni, and 2c0 is a constant which can be computed
from the signature of the generating knot.
Example: The general formula for the signature of the family of pretzel knots
(2p1+1),(2p2+1),(2p3) beginning with the knot 3,3,2 is -2p1-2p2.
σ = -2p1-2p2 +2c0
σ (3,3,2) = -4, p1=1, p2=1
C0 = 0
σ = -2p1-2p2
General formulae for unknotting number of some
alternating knot families
Example (trivial): unknotting number of knots
belonging to the family 2p1+1 (p1>0) is p1.
σ = -2p1
u = uBJ = p1
9
u=4
Example (non-trivial): unknotting number of knots belonging to the family
(2p1+1) 1,(2p2) 1,(2p3) 1 beginning with knot 1066 3 1,2 1,2 1 is p1+p2+p3.
σ = -2p1-2p2-2p3
u = uBJ = p1+p2+p3
Open problem:
signature and unknotting numbers
of non-alternating knot families
Similar results are obtained for all families of alternating knots derived from
generating knots with at most n=10 crossings and links with at most n=9 crossings
for which unknotting number is equal to a half of signature.
SIGNATURE OF NON-ALTERNATING KNOT FAMILIES
n=7
n=10
n=8
Ascending numbers of knot families
Knowing that unknotting number is a lower bound of ascending number, for certain
Families of knots it is possible to find their ascending number by constructing a
(non-minimal) diagram with the ascending number equal to unknotting number.
2p+1
2p 3
(2p+1) 2 (2q)
Every self-crossing point of a link can be turned into an uncrossing by smoothing
(∞-operation )according the following rule:
Analogously as we defined unknotting (unlinking) number, it is possible to define an
∞-unknotting (unlinking) number u∞ (K) by the "classical" and "standard" definition,
to prove that two definitions are equivalent, and make a Conjecture on u∞ number:
1) u∞(K) = 0, where K is the unknot (link without self-crossings);
2) u∞(K) = min u(K' )+1, where the minimum is taken over all minimal projections of
knots (links) K', obtained from a minimal projection of K by an ∞-change.
Every ∞-change transforms an alternating knot to an alternating knot, so the set of all
alternating knots is closed with regard to ∞-changes. According to Tait's Flyping Theorem,
all minimal projections of an alternating knot give the same result, so for every alternating
knot it is sufficient to use only one minimal projection.
∞-unknotting number
u∞ = 1
3
5
7
u∞ = 2
22
42
62
Splitting number
Definition: the splitting number is a minimum number of crossing changes over
all projections of a link required to obtain a split link, that is, a link with split
components, not necessarily unknotted (Adams, 1996).
Link .2 (762) before and after crossing change.
If you are searching for link families with an unlinking number greater then the splitting
number, one of them is the family .(2k), (k≥1). In general, we could define a splitting gap –
difference between unlinking number and splitting number of an alternating link L. Because
for every member of the family .(2k), (k≥1) the unlinking number is k+1 and the splitting
number is always 1 (that is clearly visible from the corresponding figures), the splitting gap
is k, so it can be arbitrarily large.
.2
.4
Borromean links
“No two elements interlock, but all three do interlock”
Theorem: Borromean circles are impossible: Borromean rings cannot be constructed
from three flat circles, but can be constructed from three triangles (Lindstrom,
Zetterstrom, 1991).
Picture-stone from Gotland
(P. Cromwell, 1995)
Intuition by J. Robinson.
Definition: n-Borromean links (n≥3) are n-component non-trivial links such that any two
components form a trivial link.
Among them, those with at least one non-trivial sublink, for which we will keep the name
Borromean links, will be distinguished from Brunnian links in which every sublink is trivial.
It seems surprising that besides the Borromean rings, represented by the link 623 in
Rolfsen notation, no other link with the properties mentioned above can be found in link
Tables. The reason for this is very simple: all existing knot tables contain only links with at
most 9 crossings. In fact, an infinite number of n-Borromean or n-Brunnian links exist,
and they can be derived as infinite series.
Tait series: (3k)-gonal antiprisms (1876-77)
The next infinite series of 3-Borromean links, beginning again with the Borromean rings,
follows from the family of 2-component links (4n-2)12 (212, 612, 1012 ...), where blue and yellow
component make an unlink, by introducing the third component: a red circle intersecting
opposite bigons.
In a similar way, from the family of 2-component links (2n)12 (212, 412, 612 ...) we derive
another infinite series of 3-component Borromean links without bigons.
From such links with a self-intersecting component, new infinite series are obtained. In a
self-crossing point of the oriented component an even chain of bigons is introduced, and
its orientation is used only for choosing the appropriate position of the chain.
Tessellations of an (2n+1)-gonal prism, where in every ring of a prism graph we draw
"left" or "right" diagonals, yield the next infinite series of Borromean rings.
The same method, using "centered" rectangular tessellations, provides another series
of (2n+1)-Borromean links.
Tait series of Borromean links can be recognized as the family of polyhedral
links .(2k+1):(2k+1) 0, (k≥1).
.3.3 0
.5.5 0
Tait series .(2k+1):(2k+1) 0
.7.7 0
The first link in the series of Borromean links without bigons is the the basic polyhedron
1312*, from which originates the complete family of Borromean links with bigons
1312*.(2k+1) 0, (k ≥0)
1312*.3 0
1312*.5 0
1312*
1312*.7 0
Conway notation for virtual knots and links
Gauss Code: U1+O2+O1+U2+
Kamada code: {{{1,3},{2,0}},{1,1}}
PD-notation:
PD[X[3,1,4,2],X[2,4,3,1]]
(Dror Bar Natan)
Trefoil: 3 = 1,1,1=13
Virtual trefoil: i,1,1=i,12
Information about original
real knot is preserved.
In all other codes virtual crossings
are omitted. The information
about real knot from which a virtual
knot is obtained can be recovered,
but in a complicated way.
Virtual knot and links in Conway notation
PD[X[1,6,2,7],X[2,11,3,12],X[8,3,9,1],X[4,11,5,10],X[9,6,10,5],X[12,8,4,7]]
Conway notation: 6*i.-1.(2,i)
Virtual knots and links in Conway notation
PD[X[12,6,1,7],X[10,2,11,1],X[2,10,3,9],X[11,3,12,4],X[8,4,9,5],X[5,7,6,8]] (6 crossings)
Gauss Code: O1+U2-O3-U4+U5+O6+U1+U6+O5+U3-O2-O4+ (6 crossings)
Conway notation: (1,i) 2,(i,2),(i,1,i) (10 crossings)
Families of virtual knots and links in Conway notation
Simplest family: i,1n (n=1,2,3,4,…)
i,11
i,15
i,1,1=i,12
i,16
i,1,1=i,13
i,17
i,1,1,1,1=i,14
i,18
Polynomials of the family i,1p
Sawolek:
i,11
i,12
i,13
i,14
i,15
i,16
i,17
i,18
i,19
i,110
1-s-t+s t
-(-1+s) (-1+t) (-1+s t)
1-s-t+s t
-(-1+s)(-1+t)(-1+st-s2t2+s3t3)
1-s-t+s t
-(-1+s)(-1+t)(-1+st-s2t2+s3t3-s4t4+s5t5)
1-s-t+s t
-(-1+s)(-1+t)(-1+st-s2t2+s3t3-s4t4+s5t5-s6t6+s7t7)
1-s-t+s t
-(-1+s)(-1+t)(-1+st-s2t2+s3t3-s4t4+s5t5-s6t6+s7t7-s8t8+s9t9)
Polynomials of the family i,1p
Kauffman bracket (Jones):
i,11
i,12
i,13
i,14
i,15
i,16
i,17
i,18
i,19
i,110
1+A2
1+A2-A6
1+A2-A6+A10
1+A2-A6+A10-A14
1+A2-A6+A10-A14+A18
1+A2-A6+A10-A14+A18-A22
1+A2-A6+A10-A14+A18-A22+A26
1+A2-A6+A10-A14+A18-A22+A26-A30
1+A2-A6+A10-A14+A18-A22+A26-A30+A34
1+A2-A6+A10-A14+A18-A22+A26-A30+A34-A38
Virtual knot and link family
(i,1p) q
(i,11) 2
(i,11) 3
(i,12) 3
(i,11) 4
(i,12) 4
(i,11) 5
(i,12) 5
(i,11) 6
(i,12) 6
BJ-unlinking numbers for virtual links
All unlinking numbers are computed from minimal diagrams, where after every
crossing change KL diagram is minimized.
Crossing changes:
1) virtualization of a crossing;
2) real crossing change.
Definition:
Virtual BJ-unlinking number vuBJ is the minimal number of vertex virtualizations
(crossing changes 1) necessary to obtain unlink.
Mixed BJ-unlinking number muBJ is the minimal number of virtualizations and real
crossing changes (crossing changes 1 and 2) necessary to obtain unlink.
Real BJ-unlinking number ruBJ is the minimal number of real crossing changes
(crossing changes 2) necessary to obtain unlink.
For real knots all three BJ unlinking numbers are finite, but for virtual links the
real BJ unlinking number is not always finite, because real crossing change
is not necessarily an unlinking operation for virtual links.
Unknotting numbers for virtual knots (links)
Unknotting operations:
1) Additional virtualization of crossings – virtual BJ-unknotting number vuBJ
2) Crossing changes and additional virtualizations – mixed BJ-unknotting
number muBJ
Crossing change is not necessarily an unknotting operation for virtual knots –
infinite real BJ-unknotting number ruBJ
Example: virtual knot with vuBJ=muBJ=1, and infinite ruBJ.
(1,i,1) (i,1)
(1,i,1) (i,i) = unknot
Infinite real BJ unknotting number
Family of virtual knots with infinite real BJ unknotting number
Equivalent of Nakanishi-Bleiler example for virtual unknotting number: fixed
projection can be unknotted with at least 3 virtual crossing changes, and vuBJ=2
Virtual knot with vuBJ=ruBJ=3, and mixed unknotting number muBJ=2
Four-state Conjecture
A vertex coloring of a graph G is the coloring of its vertices where every two
adjacent vertices have different color. A shadow of a knot or link L is its
diagram without overcrossings and undercrossings (just a 4-valent graph,
obtained as a projection of L).
According to Appel and Haken, every plane graph can be vertex colored
by 4 or less colors, so the same holds for KL shadows. Because there
are four possible Kauffman states of a crossing in a diagram, we can pose
the following question:
Let's color a shadow of a link L in such a way that every pair of adjacent
crossings have different states (in fact, this is a vertex coloring of a shadow
by states). Prove (or disprove) that every obtained knot or link L' is the
unknot or unlink.
If you disprove this conjecture, find a smallest knot or link obtained.
A. HENRICH, R. HANAKI
PSEUDODIAGRAMS
Definition: A pseudodiagram P is a knot diagram in which some crossings are
undetermined. Such crossings are called precrossings. A precrossing is
represented as a flat crossing in a drawing. We resolve a precrossing by
assigning the local writhe of that crossing. In other words, a precrossing of a
diagram is resolved by converting it to a traditional crossing.
A trivializing number and knotting number of a knot are realized in non-minimal
knot diagrams.
Prime and composite links
Given any two links L1 and L2 we can define their composition (connected sum,
or direct product) denoted by L1#L2. Suppose that a sphere in R3 intersects a
link L in exactly two points. This splits a link L into two arcs. The endpoints of
either of those arcs can be joined by an arc lying on the sphere. That construction
results in two links, L1 and L2. The links L1 and L2 that make up the composite
link L are called factor links (or simply, factors). A link is called prime if in every
decomposition into a connected sum, one of the factors is unknotted.
Prime and composite links
Open problems about composite knots and links:
1) if c(L) is the crossing number of a knot or link L, is it true that
c(L1#L2) = c(L1)+c(L2)?
(For alternating links proof follows from the Kauffman-Murasugi Theorem)
2) for unknotting (unlinking) numbers, is it true that u(L1#L2) = u(L1)+u(L2)?
The conjecture is true in the case u(L1#L2) = 1, Scharlemann, 1985)
u(3) = 1
u(2 2) = 1
u(3#2 2) = 2 ?
Standard theorem: If P is a polynomial KL invariant, P(K1#K2)=P(K1)P(K2)
Example: Alexander, Jones, Conway, HOMFLYPT, colored Jones polynomial, etc.
Reverse statement: A KL is prime if its polynomial invariant is not factorizable
Counter-examples: Alexander, Jones, Conway, HOMFLYPT, colored Jones
polynomial, etc.
Conjecture: Kauffman two-variable polynomial distinguishes prime KLs, i.e, it
is not factorizable for any prime KL.
Conjecture is checked for all rational KLs up to 20 crossings, all knots up to
16 crossings, and all links up to 12 crossings.
References:
Nakanishi, Y. (1983) Unknotting numbers and knot diagrams with the minimum
crossings, Math. Sem. Notes Kobe Univ., 11, 257-258.
Bleiler, S.A. (1984) A note on unknotting number, Math. Proc. Camb. Phil. Soc., 96,
469-471.
Bernhard, J.A. (1994) Unknotting numbers and their minimal knot diagrams, J. Knot
Theory Ramifications, 3, 1, 1-5.
Stoimenow, A. (2001) Some examples related to 4-genera, unknotting numbers, and
knot polynomials, Jour. London Math. Soc., 63, 2, 487-500.
Murasugi, K. (1965) On a certain numerical invariant of link types, Trans. Amer. Math.
Soc., 117, 387-422.
Kohn, P. (1993) Unlinking two component links, Osaka J. Math., 30, 741-752.
Livingston, C. and Cha J.C. (2005) Table of knot invariants,
http://www.indiana.edu/~knotinfo/
Owens,B. (2008) Unknotting information from Heegaard Floer homology, Advances in
Mathematics, 217, 5, 2353-2376, arXiv:math/0506485v1 [math.GT].
Nakanishi,Y. (2005) A note on unknotting number, II, J. Knot Theory Ramifications,
14, 1, 3-8.