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UNKNOTTING AND ASCENDING NUMBERS OF KNOTS AND THEIR FAMILIES

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

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UNKNOTTING AND ASCENDING NUMBERS OF KNOTS AND THEIR FAMILIES

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  1. UNKNOTTING AND ASCENDING NUMBERS OF KNOTS AND THEIR FAMILIES Slavik Jablan Radmila Sazdanovic Ljiljana Radovic Ana Zekovic

  2. Unlinking number Definition: The unlinking numberu(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. 

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

  4. Unlinking number u=1 u=0

  5. Unlinking number • There are two different approaches for obtaining the unlinking number of L: • 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.

  6. The Nakanishi-Bleiler example: • 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.

  7. Unknotting 5 1 4 5 1 4 5 -1 4 3 1 2 3 -1 2 u(5 1 4) =2

  8. 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.

  9. 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.

  10. 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).

  11. 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 linkL is the positive difference between the minimal unlinking number of all minimal projections min{uM(L)} and the unlinking number u(L). 

  12. 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 gapδ(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. 5 1 4 7 1 6 9 1 8 Family (2k+1) 1 (2k)

  13. 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

  14. Unlinking gap • n = 9 10 11 12 13 14 15 16 • 1 1 5 5 23 36 106 180 Alternating knot family with arbitrarily large unknotting gap Non-alternating link family 2k,3,-2k with arbitrarily large unknotting gap

  15. (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 Multi-parameter families of links with arbitrarily large unlinking gap 10 5 8 4 1 4 6 1 4 6 1 6

  16. Minimal diagrams of non-alternating knots with unknotting gap .(3,-2).2 .2.(3,-2) 4 1 1,3,-2 3 2,3,-3

  17. 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.

  18. Multi-parameter families of rational links with arbitrarily large unlinking gap Link family of the form (2k1) ... (2k2i-1)(2l-1)(2m1) ...(2m2j-1). Knot family of the form (2k1) ... (2l-1) ...(2k2i+1).

  19. 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.

  20. Rational knots with unknotting number 1 5 4 1 2 1 1 1 1 4 7 2 1 1 6 3

  21. 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

  22. 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.

  23. Definition: A distance of linksL1 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 projectionL'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.

  24. Definition (Murasugi, 1965): signature σ(L) of a link L is the signature of the matrix SL+SLT, where SLTis 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.

  25. 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

  26. 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: • Positive crossing (b) negative crossing • (c) parallel positive twist; • (d) parallel negative twist; • (e) antiparallel positive twist; • (f) antiparallel negative twist.

  27. 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: • 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; • 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; • 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; • 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.

  28. 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

  29. 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

  30. 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.

  31. SIGNATURE OF NON-ALTERNATING KNOT FAMILIES

  32. n=7 n=8 n=10

  33. 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.

  34. 2p+1 2p 3

  35. (2p+1) 2 (2q)

  36. 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. 

  37. ∞-unknotting number u∞ = 1 3 5 7 u∞ = 2 2 2 6 2 4 2

  38. 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.

  39. 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. .4 .2

  40. 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).

  41. 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.

  42. 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)

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