Smoothing Distances of KnotsAna Zekovic

University of BelgradeFaculty of Mathematics11000 Belgrade, Serbia

Email: azekovic@matf.bg.ac.rs

Slavik JablanICT College of Vocational Studies

11000 Belgrade, SerbiaEmail: sjablan@gmail.com

Abstract—In the existing tables of knot smoothings knots withsmoothing number 1 are computed by Abe and Kanenobu [1]for knots with at most 𝑛 = 9 crossings, and smoothing knotdistances are computed by Kanenobu [2] for knots with at most𝑛 = 7 crossings. In this work are computed some undecided knotdistances 1 from these papers, and extended the computations bycomputing knots with smoothing number one with at most 𝑛 = 11crossings and smoothing knot distances of knots with at most 𝑛 =9 crossings. All computations are done in the program LinKnot,based on Conway notation and non-minimal representations ofknots.

I. INTRODUCTION

A Gordian knot distance between knots 𝐾1 and 𝐾2 isdefined as the minimum number of crossing changes requiredto convert 𝐾1 into 𝐾2 [3]. Since topoisimerazes are enzymesinvolved in crossing changes in DNA, knot distances canbe used to study topoisomerazes action. In this way, a knotdistance is the minimum number of times needed for topoiso-meraze to mediate strand passage (crossing change) on DNAin order to convert 𝐾1 to 𝐾2, where the minimum is takenover all diagrams of 𝐾1. In particular, the unknotting numberof a knot 𝐾 is its distance from the unknot 01. For unknottingnumber, according to Bernhard-Jablan Conjecture [4], thereis a chance that it can be computed with recursive algorithmfrom minimal knot diagrams, but knot distances are usuallyrealized on non-minimal diagrams.

After the first strand passage metric table [5] containingonly rational knots and composites of rational knots up to theknot 88, I. K. Darcy calculated knot distances and compositesof rational knots up to 𝑛 = 13 crossings using computer pro-gramming based on algebra of rational tangles and continuedfraction representations of rational knots. However, even in theknot distance tables of ”small” rational knots remained someundetermined values. These results are extended to knots up to𝑛 = 10 crossings [6] and improved in the Ph.D. dissertationby H. Moon [7] (containing knots up to 𝑛 = 9 crossings),where the both works include non-rational knots. However,for many non-rational knots results (e.g., the knots 931 − 949and 1046 − 10165) are completely missing, and many valuesof knot distances still remain undetermined. Recent results byauthors of these paper of computation of all knot distances forknots with at most 𝑛 = 9 crossings are presented in the report[8].

Much more difficult for computations, and hence much lesselaborated are smoothing unknotting numbers and smoothing

distances of knots. In this paper are presented new resultsof these computations made in Mathematica based programLinKnot [4] by using Conway notation. After the Introduction(Section 1) different knot codings are considered (Section2) that can be used in the computation of smoothings, e.g.,Gauss codes, compared them with Conway and explained itsadvantages. After brief explanation of the Conway notation(Section 3) smoothing is analyzed as unknotting operationand equivalence of band unknotting number and smoothingnumber (Section 4). In Section 5 computations of knots withsmoothing number 1 based on knots with unknotting number1 is presented, and Section 5 is dedicated to the computationof smoothing distances for knots up to 𝑛 = 9 crossings.

A. Gauss codes and smoothing

The Gauss code was introduced by C. F. Gauss in orderto study closed curves in the plane intersecting themselvesonly in transverse double points called crossings or vertices.Every knot shadow can be represented by such a curve. Gaussmade codes of immersed curves by assigning letters to thecrossing points of a self-intersecting curve and determined”words” defining a closed curve. Orient the curve and label thecrossings by some set of symbols (positive integers or lettersof the alphabet) in one-to-one correspondence with the set ofcrossings. Beginning at an arbitrary point of the curve differentfrom the crossing, following the orientation, write down thesequence of labels of the crossings passed as traversing thecurve. If the number of crossing is 𝑛, the length of the Gausscode is 2𝑛. Hence, every Gauss code is a permutation of 𝑛symbols, where every symbol appears twice. However, someof such permutations are non realizable as plane graphs, e.g.the code (1, 2, 1, 2). The Gauss problem of characterizingcodes with planar immersions (”realizability problem”), firstsolved by M. Dehn, has an extensive literature and manysolutions, leading to the virtual knot theory introduced byL. Kauffman, representing the ”non-realizable part” of Gausscodes. The other way round, knot shadow can be recoveredfrom its Gauss code. Oriented edges of a knot shadow aregiven by the ordered pairs of adjacent numbers in the Gausscode.

Gauss code of an immersed curve is not unique, and dependson the choice of the beginning point (”basis point”) and theorientation. Out of all permutations representing the Gausscodes of a knot shadow, the minimal one (in the lexicographic

33

SISY 2013 • IEEE 11th International Symposium on Intelligent Systems and Informatics • September 26-28, 2013, Subotica, Serbia

978-1-4799-0305-4/13/$31.00 ©2013 IEEE

order) can be chosen and called minimal Gauss code. In orderto work with knot diagrams that provide sufficient data onpositive and negative crossings, it is possible to use signedGauss codes.

A Gauss diagram or circle chord diagram of degree 𝑛 canbe defined as an oriented circle with a set of a 2𝑛 pointsbelonging to it, labeled by 𝑛 labels, such that each label isused exactly twice. Every two points labeled with the samelabel are joined by an oriented chord, where the orientationof chords goes from the first occurrence of the same label tothe second. Every Gauss diagram of a knot can be representedby a signed chord diagram, where the positive signs of chordscorrespond to positive, and negative signs of chords to thenegative crossings.

To every chord graph its corresponding intersection graphcan be associated: the graph with vertices corresponding tothe labeled chords, where every chord has the label of itsendpoints, and with edges indicating the intersecting chords,where every two intersecting chords define an edge– theunordered pair of the intersecting chord labels.

Every crossing can be resolved by an (oriented) smoothingthat preserves the number of components, by introducing a”two-sided mirror” as in Fig. 1. For making the appropriatechoice of smoothing (”vertical” or ”horizontal”, or Kauffmanstate 𝐴 or 𝐵), in order to preserve the number of components,it is necessary to take care about the orientation of the curve.

An unoriented smoothing can be considered also, wheredepending from the position of a ”two-sided mirror” (marker–”vertical” or ”horizontal”) the number of components can bepreserved or changed. In this case multi-component immersedcurves (or links) are considered. A smoothing of a crossingwhich is intersection of two such curves– components willjoin them into one component, and a mirror placed in the self-intersection of a component according to the oriented case canpreserve the number of components, or break the componentin two closed curves.

Fig. 1. Smoothing in a crossing of (a) two different curves; (b) self-crossingof one curve [4].

Gauss codes of knots and their corresponding chord dia-grams are very suitable tool for computing and representingsmoothing.

Theorem 1: Given a Gauss code 𝐺 of a knot diagram with𝑛 crossings, the Gauss code 𝐺′ of the diagram smoothed incrossing 𝑝 (𝑝 ∈ {1, . . . , 𝑛}) is obtained if the sequence of

labels between two occurrences of 𝑝 in 𝐺 is reversed, and 𝑝is deleted.

In the case of immersed closed curve, when wanted topreserve the information about smoothed crossings, insteaddeleting 𝑝, it should be overlined.

A repetition of smoothing applied to the same Gauss codewill be called smoothing sequence. A part of a Gauss code of aknot equivalent to the word (1, 2, 1, 2) will be called forbiddenword, and to it is forbidden to apply the algorithm describedin Theorem 1. Smoothing sequence preserves the realizabilityof a realizable Gauss code of a knot. Every non-realizableGauss code can be reduced by a smoothing sequence, to oneor several forbidden words.

As a result of the complete smoothing of immersed curvesself-avoiding curves [4] and their corresponding split se-quences (or split codes) [9] are obtained. The Smoothing of theplane immersed curves can be considered as an (unknotting)operation for flat knots (or knot shadows). In this paperconsiderations will be restricted to classical and pseudoknots.

From the computational point of view, the use of Gausscodes for the computation of smoothings has its advantagesand disadvantages: the main advantage is the very simplesmoothing algorithm preserving the number of components,but the disadvantage is a very large number of Gauss codes(even if minimal Gauss codes are used) and crossings thatneed to be checked for smoothing.

In the case of crossing changes, more concise and suitable isDowker-Thistlethwaite (DT-) code of knots, well known fromthe program Knotscape [10], which represents the standardfor the coding of knots up to 𝑛 = 16 crossings. However, DT-notation, and all similar codings of knots (PD-notation usedin KnotAtlas, pd-notation used in Knot 2000 (K2K), etc.) arebased on the numerical coding, so for all crossing changesin a knot with 𝑛 crossings 2𝑛 codes with single crossingchanges can be obtained. Also, none of the numerical codingsprovide possibility to generalize results obtained for particularpairs consisting from an original knot, and knot obtained bya crossing change. Hence, in this work, Conway notation ofknots is used, their crossing changes and smoothing.

B. Basics of Conway notation

Knots and links can be given in Conway notation [11],[12], [4]. For readers non familiar with it, Conway notationis explained, introduced in Conway’s seminal paper [11]published in 1967, and effectively used since (e.g., [11], [12],[4]). Conway symbols of knots with up to 10 crossings andlinks with at most 9 crossings are given in the Appendix ofthe book [12].

The main building blocks in the Conway notation areelementary tangles. There are three elementary tangles, shownin Fig. 2 and denoted by 0, 1 and −1. All other tangles canbe obtained by combining elementary tangles, while 0 and 1are sufficient for generating alternating knots and links (abbr.𝐾𝐿s). Elementary tangles can be combined by followingoperations: sum, product, and ramification (Figs. 3-4). Giventangles 𝑎 and 𝑏, image of 𝑎 under reflection with mirror line

A. Zeković and S. Jablan • Smoothing Distances of Knots

34

Fig. 2. The elementary tangles.

NW-SE is denoted by −𝑎, and sum is denoted by 𝑎 + 𝑏.Product 𝑎 𝑏 is defined as 𝑎 𝑏 = −𝑎 + 𝑏, and ramification by(𝑎, 𝑏) = −𝑎− 𝑏.

Fig. 3. A sum and product of tangles.

Fig. 4. Ramification of tangles.

Tangle can be closed in two ways (without introducingadditional crossings): by joining in pairs NE and NW, andSE and SW ends of a tangle to obtain a numerator closure;or by joining in pairs NE and SE, and NW and SW ends toobtain a denominator closure (Fig. 5a,b).

Fig. 5. (a) Numerator closure; (b) denominator closure; (c) basic polyhedron1∗.

Definition 1: A rational tangle is any finite product ofelementary tangles. A rational 𝐾𝐿 is a numerator closure ofa rational tangle.

Definition 2: A tangle is algebraic if it can be obtained fromelementary tangles using the operations of sum and product.𝐾𝐿 is algebraic if it is a numerator closure of an algebraictangle.

A Montesinos tangle and the corresponding Montesinoslink, consisting of 𝑛 alternating rational tangles 𝑡𝑖, with at

least three non-elementary tangles 𝑡𝑘 for 𝑘 ∈ {1, 2, . . . , 𝑛},is denoted by 𝑡1, 𝑡2, . . . , 𝑡𝑛, 𝑛 ≥ 3, 𝑖 = 1, ..., 𝑛 (Fig. 6). Thenumber of tangles 𝑛 is called the length of the Montesinostangle. In particular, if all tangles 𝑡𝑖, 𝑛 ≥ 3, 𝑖 = 1, ..., 𝑛 areinteger tangles, pretzel knots and links are obtained.

Fig. 6. Montesinos link 𝑡1, 𝑡2, . . . , 𝑡𝑛.

Definition 3: Basic polyhedron is a 4-regular, 4-edge-connected, at least 2-vertex connected plane graph.

Basic polyhedron [11], [12], [4] of a given 𝐾𝐿 can be iden-tified by recursively collapsing all bigons in a 𝐾𝐿 diagram,until none of them remains.

Conway notation for polyhedral (non-algebraic) 𝐾𝐿s con-tains additionally a symbol of a basic polyhedron. The sym-bol 𝑛∗𝑚 = 𝑛∗𝑚1.1. . . . .1, where ∗𝑚 is a sequence of 𝑚stars, denotes the 𝑚-th basic polyhedron in the list of basicpolyhedra with 𝑛 vertices. A 𝐾𝐿 obtained from a basicpolyhedron 𝑛∗𝑚 by substituting tangles 𝑡1, . . ., 𝑡𝑘, 𝑘 ≤ 𝑛instead of vertices, is denoted by 𝑛∗𝑚𝑡1 . . . 𝑡𝑘, where thenumber of dots between two successive tangles shows thenumber of omitted substituents of value 1. For example,6∗2 : 2 : 2 0 means 6∗2.1.2.1.2 0.1, and 6∗2 1.2.3 2 : −2 2 0means 6∗2 1.2.3 2.1.− 2 2 0.1.

Definition 4: For a link or knot 𝐿 given in an unreduced1

Conway notation 𝐶(𝐿) denote by 𝑆 a set of numbers in theConway symbol excluding numbers denoting basic polyhedronand zeros (determining the position of tangles in the verticesof polyhedron) and let 𝑆 = {𝑎1, 𝑎2, . . . , 𝑎𝑘} be a non-emptysubset of 𝑆. Family 𝐹𝑆(𝐿) of knots or links derived from𝐿 consists of all knots or links 𝐿′ whose Conway symbol isobtained by substituting all 𝑎𝑖 ∕= ±1, by 𝑠𝑔𝑛(𝑎𝑖)∣𝑎𝑖 + 𝑘𝑎𝑖

∣,∣𝑎𝑖 + 𝑘𝑎𝑖

∣ > 1, 𝑘𝑎𝑖∈ 𝑍.

Fig. 7. Hopf link 221 = 2 and its family 𝑝 (𝑝 = 2, 3, . . .).

For making computations of knot distances and smoothingdistances the main advantage of the Conway notation is theuse of twists (or chains of bigons): since crossings in a twistcommute, every crossing change in a positive twist 𝑘 (𝑘 ≥ 0)

1The Conway notation is called unreduced if in symbols of polyhedral linkselementary tangles 1 in single vertices are not omitted.

35

SISY 2013 • IEEE 11th International Symposium on Intelligent Systems and Informatics • September 26-28, 2013, Subotica, Serbia

gives the twist 𝑘− 2, and every crossing change in a negativetwist −𝑘 (𝑘 ≥ 0) gives the twist −𝑘 + 2. Hence, crossingchanges (or smoothings) in particular crossings of a twist arenot necessary: it is sufficient to make a single crossing change(or smoothing) in the twist. Moreover, results obtained forparticular pairs of knots can be extended to knot families.As the only symbolic notation of knots, Conway notation isextremely powerful in all computations with rational knots (2-bridge knots or 4-plats), based on the connection of rationaltangles and rational knots and links with continued fractions[11], [13], [14].

To a rational tangle 𝑛1 𝑛2 . . . 𝑛𝑘 corresponds the continuedfraction

𝑟 =𝑝

𝑞= 𝑛𝑘 +

1

𝑛𝑘−1 +1

⋅⋅⋅+ 1

𝑛2+ 1𝑛1

where the rational number 𝑟 = 𝑝𝑞 (𝐺𝐶𝐷(𝑝, 𝑞) = 1) is the

slope of the rational tangle. A rational knot or link 𝐿(𝑝𝑞 ) is aknot if 𝑝 is odd, and link if 𝑝 is even.

Theorem 2: Unoriented rational links 𝐿(𝑝𝑞 ) and 𝐿(𝑝′

𝑞′ ) areambient isotopic iff:

1) 𝑝 = 𝑝′ and2) either 𝑞 ≡ 𝑞′ (mod 𝑝) or 𝑞𝑞′ ≡ 1 (mod 𝑝) [15].

For rational tangles and rational knots and links the follow-ing theorem holds:

Theorem 3: Two rational tangles are equivalent iff theircontinued fractions yield the same rational number [11], [13],[14].

Following theorems above, conclusion is that some smooth-ing distance computations with rational knots and links inMathematica can be reduced to the application of very fastMathematica functions ContinuedFraction[] and FromCon-tinuedFraction[]. Thanks to and Cyclic Surgery TheoremI. Darcy and Tanaka completely classified rational knots withGordian distance 1. On the basis of continued fractions,Darcy and her collaborators wrote computer programs forGordian distances 1 of rational knots [16], [17]. The CyclicSurgery Theorem does not hold for smoothings, so there isno equivalent complete classification of rational knots withsmoothing distance 1. However, many smoothing distances1 for rational knots can be computed by using continuedfractions and non-minimal rational representations of rationalknots.

C. Smoothing as unknotting operation

In knot theory smoothing plays a special role since all skeinrelations for computing polynomial knot invariants (Conway,Alexander, Jones, HOMFLYPT, or Kauffman polynomial) usesmoothing.

J. Hoste, Y. Nakanishi, and K. Taniyama [18] introducedan 𝐻(𝑛) move, a deformation of a link diagram. In particular,they distinguished 𝐻(2) move, a band surgery which requiresto preserve the number of components, and proved that an𝐻(2) move is unknotting operation, i.e. that any knot canbe deformed into the unknot by a sequence of 𝐻(2) moves

([18], Theorem 1). The authors defined 𝐻(2)-unknotting num-ber 𝑢2(𝐾) of a knot 𝐾 as the minimal number of 𝐻(2)moves necessary to unknot 𝐾. T. Abe and T. Kanenobu [1]generalized this concept by considering a band surgery ofunoriented knots and permitting the change of the numberof components in the unknotting process. They defined bandunknotting number, 𝑢𝑏(𝐾), as the minimal number of bandsurgeries necessary to unknot 𝐾. The unknotting sequence thatrealizes 𝑢2(𝐾) consists of only knots, whereas the unknottingsequence that realizes 𝑢𝑏(𝐾) may contain links as well, soby definition 𝑢2(𝐾) ≥ 𝑢𝑏(𝐾). This inequality can be strong:𝑢𝑏(818) = 2 and 𝑢2(818) = 3. In fact, there exist infinitelymany knots 𝐾 with 𝑢𝑏(𝐾) = 2 and 𝑢2(𝐾) = 3 ([18],Theorem 7.1). Moreover, Abe and Kanenobu ([18], Corollary3.2) proved the following (in)equalities:

Corollary 4: For a knot 𝐾,

𝑢𝑏(𝐾) = 𝑢2(𝐾)− 1 𝑜𝑟 𝑢2(𝐾).

Furthermore, if 𝑢𝑏(𝐾) is odd, then 𝑢𝑏(𝐾) = 𝑢2(𝐾); equiva-lently, if 𝑢2(𝐾) is either one or even, then 𝑢𝑏(𝐾) = 𝑢2(𝐾).

Using simple topological arguments can be proved that𝐻(2) operation is equivalent with the oriented smoothingin some crossing 𝑐. Namely, if in the crossing 𝑐 (of anarbitrary sign) an 𝐻(2)-move (Fig. 8) is performed followedby a Reidemeister move I, the result is an oriented (verti-cal) smoothing in 𝑐. In the same way, the unoriented bandsurgery in 𝑐 is equivalent to a horizontal smoothing in 𝑐.Hence, the 𝐻(2)-unknotting number 𝑢2(𝐾) of a knot 𝐾and band unknotting number 𝑢𝑏(𝐾) of 𝐾 can be simplycalled component-preserving smoothing number 𝑢2(𝐿) (or𝑢∞-unknotting number [4]), and smoothing number 𝑢𝑏(𝐿),where 𝐿 is a link. In this case, the result of the both unlinkingprocedures can be not only a knot, but a link as well. In thefirst case, the number of components of links in the every stepof unlinking process remains unchanged, and in the other caseit can be changed in any step. In every step the both proceduresproduce a link with one fewer crossings, so the both unlinkingnumbers are finite.

Fig. 8. Equivalence of 𝐻(2)-move and oriented smoothing.

Abe and Kanenobu [18] defined one more important con-cept: band-Gordian distance and 𝐻(2)-Gordian distance ofknots. For two knots 𝐾1 and 𝐾2 the both smoothing distancesare defined as the minimal number of smoothing necessaryto obtain knot 𝐾2 from 𝐾1. Gordian distances play an

A. Zeković and S. Jablan • Smoothing Distances of Knots

36

important role in DNA-knotting, where Topoisomerase I andTopoisomerase II can make crossing changes and smoothingin DNA. In the language of Gordian numbers, smoothingnumbers 𝑢2(𝐿) and 𝑢𝑏(𝐿) are smoothing distances of a link𝐿 from an unlink.

Smoothing distances have the properties of metric:

1) 𝑑2(𝐾1,𝐾2) = 0 iff 𝐾1 = 𝐾2;2) 𝑑2(𝐾1,𝐾2) = 𝑑2(𝐾2,𝐾1);3) 𝑑2(𝐾1,𝐾2) ≤ 𝑑2(𝐾1,𝐾) + 𝑑2(𝐾,𝐾2), where 𝐾 is an

arbitrary knot.

In particular, for smoothing numbers 𝑢2(𝐾1) and 𝑢2(𝐾2)the triangle inequality (3) reduces to 𝑑2(𝐾1,𝐾2) ≤ 𝑢2(𝐾1)+𝑢2(𝐾2).

Smoothing numbers and smoothing distances are mostlyrealized on non-minimal knot diagrams. For example, knots31 = 3 and 74 = 31 3 have smoothing distance 1, but it isrealized on their non-minimal diagrams 2 − 1 − 1 − 2 1 1 1,2 − 1 − 1 0 1 1 1 and not realized on their minimal diagrams.

D. Knots with smoothing number one

Many efforts are made in order to make tables of smoothingnumbers 𝑢𝑏 and 𝑢2 for knots with 𝑛 ≤ 9 crossings, and inthese tables certain smoothing numbers still remained unde-cided. Y. Bao [19] established the criterion for rational knotswith smoothing number one. Let us remind that for a knotwith 𝑢2(𝐾) = 1, the smoothing number 𝑢𝑏(𝐾) is one as well.In the case of unknotting number of a knot 𝑢(𝐾), a simpletool facilitating the confirmation of certain unknotting numbersis the existence of the lower bound of unknotting numbergiven by a half of signature. For smoothing numbers thereis no simply computable lower bound, so every smoothingnumber different from one needs independent confirmation.The simplest case will be considered, knots with smoothingnumber one.

In this paper, all knots with smoothing number one areobtained by the following algorithm:

Algorithm 1:

1) Take a diagram of a knot 𝐾 ′ with unknotting numberone, given by its Conway symbol;

2) find the crossing in 𝐾 ′ which need to be changed from+1 to −1 in order to obtain the unknot;

3) substitute this crossing by (un)tangle (1,−1) and(1,−1) (1,−1). By these substitutions a knot diagram𝐾0 and link diagram 𝐿0 are obtained. Choose thesubstitution which gives a knot;

4) minimize diagram 𝐾0 and recognize the obtained knot𝐾.

Knot 𝐾 has the smoothing number one realized on the non-minimal diagram 𝐾0.

Example 1: Rational knot 𝐾 ′ = 41 1 1 2 has unknottingnumber one. By changing third crossing +1 to −1 unknot4 1 1 (−1) 2 is obtained. The replacement (−1) → (1,−1)gives knot diagram 𝐾0 = 41 1 (1,−1) 2 which reduces to𝐾 = 41 3. Hence, the knot 𝐾 = 41 3 has the smoothingnumber one realized on the non-minimal diagram 𝐾0 =

41 1 (1,−1) 2. Unknot will be obtained by smoothing crossing1 in (1,−1).

Because all minimal diagrams of an alternating knot areflype-equivalent, in the case of alternating knots it is sufficientto take arbitrary minimal diagram of a knot 𝐾 ′ with unknottingnumber one. In the case of non-alternating knots, all minimaldiagrams of 𝐾 ′ needs to be taken. In all cases, implicitly issupposed that unknotting number one is always realized onminimal diagrams, but this is not important, because it canbe started from an arbitrary diagram of 𝐾 ′ which can beunknotted by a single crossing change.

Open question: Can be proved that the proposed algorithmis exhaustive, i.e. that all knots with smoothing number onecan be obtained in this way. For knots with 𝑛 ≤ 9 crossingsthis seems to be true.

The results obtained in this paper for smoothing numberone knots with 𝑛 ≤ 9 crossings almost coincide with theresults from tables of smoothing numbers given in the Table 2of the paper [18]. The only disagreements are the following:in this table are omitted smoothing number one knots 52

2,923 (Fig. 9), and 925 (shown in Fig. 15 [18] as one of thesmoothing number one knots), and knot 924 is included intwo lists of knots, those with 𝑢𝑏 = 𝑢2 = 1, and those with𝑢𝑏 = 𝑢2 = 2. Its smoothing number should not be one, and itsexpected smoothing number is 2. Also, to the list of compositeknots with smoothing number one the knots !31#41, 31#52,41#!52, 41#52, and 31#!62 (Fig. 10) are added, where ! isused to denote the mirror-image of a knot.

Fig. 9. (a) Knot 52 and its non-minimal diagram 2 (1,−1) 1 2 with smoothingnumber one; (b) knot 925 and its non-minimal diagram 6∗2 1 0.(1,−1) 0.2. :2 0 with smoothing number one.

Fig. 10. Composite knots (a) !31#41 (b) 31#52 (c) 41#!52 (d) 41#52 (e)31#!62 and their non-minimal diagrams with smoothing number one.

In the paper [18], Figure 15, are represented rational 𝐻(2)-unknotting number one knots with 𝑛 ≤ 9 crossings, and tothem, the non-minimal diagrams of the rational knots 921, 923,926, and 931 (Fig. 11) with smoothing number one followingfrom the results of Bao [1] should be added.

2The correct result 𝑢2(52) = 1 is given in [2], Table 2.

37

SISY 2013 • IEEE 11th International Symposium on Intelligent Systems and Informatics • September 26-28, 2013, Subotica, Serbia

Fig. 11. Knots (a) 921 (b) 923 (c) 926 (d) 931 and their non-minimal diagramswith smoothing number one.

By using Algorithm 1, non-minimal diagrams of smoothingnumber one knots with 𝑛 ≤ 11 crossings are derived. Theirlist can be downloaded from the address:http://www.mi.sanu.ac.rs/vismath/smoothing.pdf.

In this list, every knot is given in Alexander-Briggs notation(used for knots with 𝑛 ≤ 10 crossings) [12], or Knotscapenotation (used for knots with 𝑛 = 11 crossings) [10], itsConway symbol [12], [11], and the Conway symbol of itsnon-minimal diagram with smoothing number one.

E. Smoothing distances of knots

In this section 𝐻2-smoothing will be considered, i.e.smoothing of a knot that preserves the number of componentsand induced metrics of knot 𝑑2-distances (or 𝐻2-Gordiandistances [2]) where 𝑑2(𝐾,𝐾1) is the minimal number of𝐻2-smoothings necessary to transform knot 𝐾 to 𝐾1. In thepaper [2] are computed 𝐻2-Gordian distances for knots withat most 𝑛 = 7 crossings. The final results of computations aregiven in Tables 3,4,5,6 (pages 826-827, [2]). As the partialresult of our computations undecided (1 or 2) distances 1 areconfirmed for the pairs of knots (51, !51), (51, 72), (52, !71),(62, 72), (63, 71), (71, !75), and (71, 77) (Fig. 12). Moreover,knot 𝑑2-distances are computed for knots with at most 𝑛 = 9crossings and obtained 732 knot pairs with 𝑑2-distance equal1. These pairs served as the basis for the complete tablesof 𝑑2-distances for all knots with at most 𝑛 = 9 crossingswhich contain 9427 knot pairs. Among them, together with732 knot pairs with 𝑑2-distance equal one, smoothing distance2 is confirmed for 2913 pairs, and distance 3 for 243 pairs.In order to confirm these distances the criteria I-VII from [2](page 828) is used, which is implemented in the Mathematicaprogram with different obstructions proving that some knots 𝐾and 𝐾1 with undecided 𝑑2-distance 1 or 2 cannot have distance1, and that certain knots have distance 3. The complete resultswill be presented in the Ph.D. dissertation of the first author.

II. CONCLUSION

Having in mind the importance of smoothing operation inthe analysis of the structure of knotted DNA, the existingresults for Gordian distances [8] and smoothing distances ofknots up to 𝑛 = 9 crossings are extended. The proposedapproach based on the Conway notation and non-minimal knotrepresentations can be efficiently extended to knots with morethan 𝑛 = 9 crossings, as it was done, for example, with the

Fig. 12. Knots (a) (51, !51), (b) (51, 72), (c) (52, !71), (d) (62, 72), (e)(63, 71), (f) (71, !75), and (g) (71, 77) with smoothing distances one.

computation of smoothing unknotting number 1 knots with𝑛 ≤ 11 crossings.

ACKNOWLEDGMENT

This paper was supported by project No 174012 financed bythe Serbian Ministry of Education, Science and TechnologicalDevelopment.

REFERENCES

[1] T. Abe and T. Kanenobu, Unoriented Band Surgery on Knots and Links,arXiv:1112.2449v1 [math.GT]

[2] T. Kanenobu, H(2)-Gordian Distance of Knots, J. Knot Theory Ramifi-cations, 20, 6 (2011) 813–835.

[3] H. Murakami, Some metrics on classical knots, in Knot Theory, Math.Ann., 20(1), (1985) 35–45.

[4] S. V. Jablan and R. Sazdanovic, LinKnot – Knot Theory byComputer, World Scientific, New Jersey, London, Singapore, 2007;http://math.ict.edu.rs/

[5] I. K. Darcy, D. W. Sumners, Applicatios of topology to DNA, in KnotTheory, (Eds.: V.F.R. Jones, J. Kania-Bartoszynska, J.H. Przytycki, P.Traczyk, and V.G. Turaev), Banach Center Publications, 42, 1998, 65–75.

[6] I. Darcy, Strand passage metric table,http://www.math.uiowa.edu/∼idarcy/TAB/tabnov.pdf

[7] H. Moon, Calculating knot distances and solving tangle equations in-volving Montesinos tangles. dissertation, University of Iowa,http://ir.uiowa.edu/etd/859

[8] A. Zekovic and S. Jablan, Algorithms for computation of unknottingnumbers and knot distances, ETRAN 2013.

[9] T. Fleming, B. Mellor, Chord Diagrams and Gauss Codes for Graphs,arXiv:0508269v2 [math.CO]

[10] M. Thistlethwaite, (1999) Knotscape 1.01,http://www.math.utk.edu/∼morwen/knotscape.html

[11] J. Conway, An enumeration of knots and links and some of their relatedproperties, in Computational Problems in Abstract Algebra, ed. J. Leech,Proc. Conf. Oxford 1967, (Pergamon Press, New York, 1970), 329–358.

[12] D. Rolfsen, Knots and Links, (Publish & Perish Inc., Berkeley, 1976);American Mathematical Society, AMS Chelsea Publishing, 2003.

[13] L.H. Kauffman, and S. Lambropoulou, On the classification of rationaltangles, arXiv:math.GT/0311499v2

[14] L.H. Kauffman, and S. Lambropoulou, On the classification of rationalknots, arXiv:math.GT/0212011v2

[15] H. Schubert, Knoten mit zwei Brucken, Math. Zeit., 65, (1956) 133–170.[16] R. Scharein, KnotPlot, http://www.knotplot.com/[17] T. Kawasaki, Java distance program,

http://www.ma.utexas.edu/users/kawasaki/darcy/mainFr.html

[18] J. Hoste, Y. Nakanishi, and K. Taniyama, Unknotting Operations Involv-ing Trivial Tangles, Osaka J. Math. 27 (1990), 555–566.

[19] Y. Bao, H(2)-unknotting operation related to 2-bridge links,arXiv:1104.4435v1 [math.GT]

A. Zeković and S. Jablan • Smoothing Distances of Knots

38