UNKNOTTING AND ASCENDING NUMBERS OF KNOTS AND THEIR FAMILIES

Slavik JablanRadmila Sazdanovic Ljiljana Radovic Ana Zekovic

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

Unlinking number

““Measure of knottednes” (“unlinking number”) is one from the Measure of knottednes” (“unlinking number”) is one from the most difficult problems in knot theory.most difficult problems in knot theory.

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). 

Unlinking number

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 fromfixed minimal diagrams.

5 1 4 5 -1 4

3 1 2 3 -1 2

Unknotting 5 1 4

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. 

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.

Definition: The unlinking number obtained from BJ-conjecture will be called BJ-unlinking number and denoted by uBJ.

Two minimal diagrams projections of the non-alternating knot 1436750, the first with uBJ=1, and the other with uBJ=2.

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)

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) .

5 1 4

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.

Family (2k+1) 1 (2k)

5 1 4 7 1 6 9 1 8

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+1u < l δ ≥ k-1

Unlinking gap• n = 9 10 11 12 13 14 15 16

• 1 1 5 5 23 36 106 180

Alternating knot family with arbitrarilylarge unknotting gap

Non-alternating link family 2k,3,-2kwith arbitrarily large unknotting 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

4 1 4 6 1 4 6 1 6

10 5 8

Multi-parameter families of links with arbitrarily large unlinking gap

Minimal diagrams of non-alternating knots with unknotting gap

.(3,-2).2 .2.(3,-2)

4 1 1,3,-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

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.

(2k),(2k),-3

Rational knots with unknotting number 1

5 4 1 2 1 1 1 1 4 7 2 1 1 6 3

Rational links with unlinking number 1

2 1 2 3 2 1 1 1 3 2 1 2 4 1 1 1 3 1 1 2 1 1 1 4

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.

Conjecture: the knot 2,3,-3 and the link 2,2,-3 are the only pretzel KLs withunlinking number 1 (Buck, Jablan).

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+SL

T, 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 unlinkingnumber 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=4W=3B=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.

9

u = 4

σ = -2p1

u = uBJ = p1

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 numbersof non-alternating knot families

Similar results are obtained for all families of alternating knots derived fromgenerating knots with at most n=10 crossings and links with at most n=9 crossingsfor which unknotting number is equal to a half of signature.

SIGNATURE OF NON-ALTERNATING KNOT FAMILIES

n=10

n=7 n=8

Ascending numbers of knot families

Knowing that unknotting number is a lower bound of ascending number, for certainFamilies 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

3 5 7

2 24 2 6 2

u∞ = 1

u∞ = 2

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).

Splitting number

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)1

2 (212, 61

2, 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 (21

2, 412, 61

2 ...) 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 .7.7 0

Tait series .(2k+1):(2k+1) 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*

1312*.3 0

1312*.5 0

1312*.7 0

Trefoil: 3 = 1,1,1=13

Virtual trefoil: i,1,1=i,12

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)

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.

Information about originalreal knot is preserved.

Conway notation for virtual knots and links

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 knot 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)

Virtual knots and links in Conway notation

Simplest family: i,1n (n=1,2,3,4,…)

i,11 i,1,1=i,12 i,1,1=i,13 i,1,1,1,1=i,14

i,15 i,16i,17 i,18

Families of virtual knots and links in Conway notation

Sawolek:

i,11 1-s-t+s ti,12 -(-1+s) (-1+t) (-1+s t)i,13 1-s-t+s ti,14 -(-1+s)(-1+t)(-1+st-s2t2+s3t3)i,15 1-s-t+s ti,16 -(-1+s)(-1+t)(-1+st-s2t2+s3t3-s4t4+s5t5)i,17 1-s-t+s ti,18 -(-1+s)(-1+t)(-1+st-s2t2+s3t3-s4t4+s5t5-s6t6+s7t7)i,19 1-s-t+s ti,110 -(-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 1+A2

i,12 1+A2-A6

i,13 1+A2-A6+A10

i,14 1+A2-A6+A10-A14

i,15 1+A2-A6+A10-A14+A18

i,16 1+A2-A6+A10-A14+A18-A22

i,17 1+A2-A6+A10-A14+A18-A22+A26

i,18 1+A2-A6+A10-A14+A18-A22+A26-A30

i,19 1+A2-A6+A10-A14+A18-A22+A26-A30+A34

i,110 1+A2-A6+A10-A14+A18-A22+A26-A30+A34-A38

Polynomials of the family i,1p

(i,11) 2

(i,11) 3

(i,11) 4

(i,11) 5

(i,11) 6 (i,12) 6

(i,12) 3

(i,12) 4

(i,12) 5

Virtual knot and link family(i,1p) q

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 realcrossing 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 thereal BJ unlinking number is not always finite, because real crossing change is not necessarily an unlinking operation for virtual links.

(1,i,1) (i,1) (1,i,1) (i,i) = unknot

Example: virtual knot with vuBJ=muBJ=1, and infinite ruBJ.

Unknotting numbers for virtual knots (links)

1) Additional virtualization of crossings – virtual BJ-unknotting number vuBJ

2) Crossing changes and additional virtualizations – mixed BJ-unknotting number muBJ

Unknotting operations:

Crossing change is not necessarily an unknotting operation for virtual knots –infinite real BJ-unknotting number ruBJ

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 thereare 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

Definition: A pseudodiagram P is a knot diagram in which some crossings areundetermined. 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.

PSEUDODIAGRAMS

A trivializing number and knotting number of a knot are realized in non-minimalknot 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) = 1u(2 2) = 1u(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, itis not factorizable for any prime KL.

Conjecture is checked for all rational KLs up to 20 crossings, all knots up to16 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.