Polynomials of genus one prime knots of complexity at most five
PPOLYNOMIALS OF GENUS ONE PRIME KNOTSOF COMPLEXITY AT MOST FIVE
MAXIM IVANOV AND ANDREI VESNINA
BSTRACT . Prime knots of genus one admitting diagram with at most ๏ฌve clas-sical crossings were classi๏ฌed by Akimova and Matveev in 2014. In 2018 Kaur,Prabhakar and Vesnin introduced families of ๐ฟ -polynomials and ๐น -polynomialsfor virtual knots which are generalizations of af๏ฌne index polynomial. Here weintroduce a notion of totally ๏ฌat-trivial knots and demonstrate that for such knots ๐น -polynomials and ๐ฟ -polynomials coincide with af๏ฌne index polynomial. Weprove that all Akimova โ Matveev knots are totally ๏ฌat-trivial and calculate theiraf๏ฌne index polynomials. I NTRODUCTION
Tabulating of virtual knots and constructing their invariants is one of the keyproblems in mordern low-dimensional topology. Table of virtual knots with dia-grams, having at most four classical crossings may be found in monography [3] andonline [4]. Due to equivalence of virtual knots and knots in thickened surfaces, itโsinteresting to consider tabulation of knots in 3-manifolds, which are thickeningsof surfaces of certain genus. Up to now, there are just few results in this direction.Here we consider prime knots of genus one, admitting diagrams with small numberof classical crossings, tabulated by Akimova and Matveev in [1].We are intrested in behaviour of several polynomial invariants on Akimova โMatveev knots. Recall that Kaufman in [7] de๏ฌned an a๏ฌine index polynomialwhich is an invariant of a virtual knot and possess some important proprties [8].In [9] a generalization of af๏ฌne index polynomials was introduced, namely a fam-ily of ๐ฟ -polynomials { ๐ฟ ๐๐พ ( ๐ก, โ ) } โ ๐ =1 and family of ๐น -polynomials { ๐น ๐๐พ ( ๐ก, โ ) } โ ๐ =1 .In [5] authors, using their software, calculated ๐น -polynomials of knots tabulated in[3] and [4]. Here we consider polynomial invariants for knots in a thickened torus. Mathematics Subject Classi๏ฌcation.
Key words and phrases.
Virtual knot, knot in a thickened torus, af๏ฌne index polynomial.This work was supported by the Laboratory of Topology and Dynamics, Novosibirsk State Uni-versity (contract no. 14.Y26.31.0025 with the Ministry of Science and Education of the RussianFederation). a r X i v : . [ m a t h . G T ] A ug MAXIM IVANOV AND ANDREI VESNIN
The paper has the following structure: in Section 1 we recall some basic de๏ฌ-nitions and facts to use further, in Section 2 we introduce totally ๏ฌat-trivial knotsand show that for these knots ๐ฟ -polynomials and ๐น -polynomials coincide withaf๏ฌne index polynomial, in Section 3 we calculate these invariants for Akimova โMatveev knots. In Theorem 3.1 we show that Akimova โ Matveev knots are totally๏ฌat-trivial. In Corollary 3.2 and Table 2 their af๏ฌne index polynomials are given.The investigation of properties of Akimova โ Matveev knots leads to the followingQuestion 3.3: Is it true, that every virtual knot of genus one is totally ๏ฌat-trivial?1. B ASIC DEFINITIONS
Virtual knots and links were introduced by Louis Kaufman in [6] as an essentialgeneralization of classical knots. Diagrams of virtual knots may have classical andvirtual crossings both. Two virtual knots are equivalent if and only if their diagramscould be transformed in each other by ๏ฌnite sequences of classical (RI, RII, RIII inFig. 1) and virtual (VRI, VRII, VRIII and SV in Fig. 2) Reidemeister moves.F
IGURE
1. Classical Reidemeister moves.F
IGURE
2. Virtual Reidemeister moves.
ENUS ONE KNOT POLYNOMIALS 3
Diagram, obtained by forgetting over/under crossing information is said to be ๏ฌat knot diagram . Equivalence of ๏ฌat knots is de๏ฌned by ๏ฌat Reidemeister moves ,which are different from virtual Reidemeister moves in having ๏ฌat crossings insteadof classical ones.Let ๐ท be a diagram of an oriented virtual knot. We denote the set of all classicalcrossings of diagram ๐ท as ๐ถ ( ๐ท ) . Sign of a classical crossing, denoted by sgn( ๐ ) is de๏ฌned as shown in the Fig. 3. (cid:73) (cid:18) sgn( ๐ ) = +1 (cid:73) (cid:18) sgn( ๐ ) = โ F IGURE
3. Signs of classical crossings.For every arc in a diagram of virtual knot we assign an integer value in such waythat relations presented in a picture4 hold. In [7] Kaufman proved, that such color-ing of an oriented virtual knot diagram, called
Cheng coloring , always exists. In-deed, for every arc ๐ผ of a diagram ๐ท one can assign value ๐ ( ๐ผ ) = โ๏ธ ๐ โ ๐ ( ๐ผ ) sgn( ๐ ) ,where ๐ ( ๐ผ ) is the set of classical crossings, which are ๏ฌst met as overcrossings,when moving around the knot from ๐ผ with respect to the orientation. (cid:73) (cid:18) ๐ + 1 ๐ ๐๐ โ (cid:73) (cid:18) ๐ + 1 ๐ ๐๐ โ (cid:73) (cid:18) (cid:102) ๐๐ ๐๐ F IGURE
4. Cheng coloringIn [2] Cheng and Gao put every classical crossing in correspondence with aninteger value
Ind( ๐ ) , de๏ฌned as(1) Ind( ๐ ) = sgn( ๐ )( ๐ โ ๐ โ where ๐ and ๐ given by Cheng coloring. One can notice, that Cheng coloring doesnot depend on types of classical crossings and hence it is de๏ฌned for an oriented๏ฌat knot diagram. Let us remember that af๏ฌne index polynomial from [7] can bewritten in the following form:(2) ๐ ๐ท ( ๐ก ) = โ๏ธ ๐ โ ๐ถ ( ๐ท ) sgn( ๐ )( ๐ก Ind( ๐ ) โ , MAXIM IVANOV AND ANDREI VESNIN where ๐ถ ( ๐ท ) is a set of all classical crossings of ๐ท .In [11] Satoh and Taniguchi introduced a notion of ๐ -writhe ๐ฝ ๐ ( ๐ท ) . For every ๐ โ Z โ { } de๏ฌne ๐ -writhe of oriented virtual knot diagram as a difference be-tween number of positive crossings and negative crossings of index ๐ . Notice that ๐ฝ ๐ ( ๐ท ) is a coef๏ฌcient of ๐ก ๐ in af๏ฌne index polynomial and it is an invariant of ori-ented virtual knot. For more information about ๐ -writhe see [11]. Using ๐ -writhein [9] was de๏ฌned another invariant โ ๐ -dwrithe โ ๐ฝ ๐ ( ๐ท ) : โ ๐ฝ ๐ ( ๐ท ) = ๐ฝ ๐ ( ๐ท ) โ ๐ฝ โ ๐ ( ๐ท ) . Remark 1.1. โ ๐ฝ ๐ ( ๐ท ) is an invariant of oriented virtual knot, since ๐ฝ ๐ ( ๐ท ) is aninvariant of oriented virtual knot. Moreover, โ ๐ฝ ๐ ( ๐ท ) = 0 for every classical knot.As it shown in [9], โ ๐ฝ ๐ ( ๐ท ) represents a ๏ฌat knot structure. Namely, the follow-ing lemma holds Lemma 1.2. [9, Lemma 2.4]
For every ๐ โ N , ๐ -dwrithe โ ๐ฝ ๐ ( ๐ท ) is an oriented๏ฌat knot invariant. Let ยฏ ๐ท be a diagram, obtained from ๐ท by reversing an orientation and ๐ท * isobtained by switching all classical crossings. Lemma 1.3. [9, Lemma 2.5]
Let ๐ท be a diagram of oriented virtual knot, then โ ๐ฝ ๐ ( ๐ท * ) = โ ๐ฝ ๐ ( ๐ท ) and โ ๐ฝ ๐ ( ยฏ ๐ท ) = โโ ๐ฝ ๐ ( ๐ท ) . Consider a smoothing according to the rule, shown in picture 5. We will call thiskind of smoothing by smoothing against orientation . Orientation of ๐ท ๐ is inducedby smoothing. Since ๐ท is a diagram of virtual knot, so ๐ท ๐ is a diagram of virtualknot too. (cid:73) (cid:18) โโ (cid:9) (cid:18) and (cid:73) (cid:18) โโ (cid:73) (cid:82) F IGURE
5. Smoothing.
De๏ฌnition 1.4. [9] For a diagram ๐ท of a virtual oriented knot ๐พ and an integer ๐ ,a polynomial ๐ฟ ๐๐พ ( ๐ก, โ ) is de๏ฌned as:(3) ๐ฟ ๐๐พ ( ๐ก, โ ) = โ๏ธ ๐ โ ๐ถ ( ๐ท ) sgn( ๐ ) (๏ธ ๐ก Ind( ๐ ) โ |โ ๐ฝ ๐ ( ๐ท ๐ ) | โ โ |โ ๐ฝ ๐ ( ๐ท ) | )๏ธ . ENUS ONE KNOT POLYNOMIALS 5
Note that ๐ฟ -polynomials generalize af๏ฌne index polynomial, since ๐ ๐พ ( ๐ก ) = ๐ฟ ๐๐พ ( ๐ก, for every ๐ and every ๐ก . De๏ฌnition 1.5. [9] For a diagram ๐ท of a virtual oriented knot ๐พ and an integer ๐ ,a polynomial ๐น ๐๐พ ( ๐ก, โ ) is de๏ฌned as:(4) ๐น ๐๐พ ( ๐ก, โ ) = โ๏ธ ๐ โ ๐ถ ( ๐ท ) sgn( ๐ ) ๐ก Ind ( ๐ ) โ โ ๐ฝ ๐ ( ๐ท ๐ ) โ โ๏ธ ๐ โ ๐ ๐ ( ๐ท ) sgn( ๐ ) โ โ ๐ฝ ๐ ( ๐ท ๐ ) โ โ๏ธ ๐/ โ ๐ ๐ ( ๐ท ) sgn( ๐ ) โ โ ๐ฝ ๐ ( ๐ท ) , where ๐ ๐ ( ๐ท ) = { ๐ โ ๐ถ ( ๐ท ) : |โ ๐ฝ ๐ ( ๐ท ๐ ) | = |โ ๐ฝ ๐ ( ๐ท ) |} . Theorem 1.6. [9]
For every integer ๐ โฅ polynomials ๐ฟ ๐๐พ ( ๐ก, โ ) and ๐น ๐๐พ ( ๐ก, โ ) areoriented virtual knot invariants.
2. T
OTALLY FLAT - TRIVIAL KNOTS
Let ๐ท be a diagram of oriented virtual knot ๐พ and ๐ถ ( ๐ท ) a set of all classicalcrossings in ๐ท . De๏ฌnition 2.1.
We will call ๐ท totally ๏ฌat-trivial if diagrams obtained from ๐ท and ๐ท ๐ for all ๐ โ ๐ถ ( ๐ท ) by forgetting over/under crossing information are ๏ฌatequivalent to unknot. Virtual knot ๐พ is said to be totally ๏ฌat-trivial , if it admits atotally ๏ฌat-trivial diagram. Lemma 2.2.
If virtual knot ๐พ is totally ๏ฌat-trivial, then (1) For all ๐ โฅ we have ๐ฟ ๐๐พ ( ๐ก, โ ) = ๐ ๐พ ( ๐ก ) and ๐น ๐๐พ ( ๐ก, โ ) = ๐ ๐พ ( ๐ก ) . (2) ๐ ๐พ ( ๐ก ) is palindromic.Proof. (1) Let ๐ท be a totally ๏ฌat-trivial diagram of a knot ๐พ , ๐ถ ( ๐ท ) a set of all itsclassical crossings, and ๐ท ๐ a diagram, obtained by smoothing against orientationin crossing ๐ โ ๐ถ ( ๐ท ) . By the de๏ฌnition, all these diagrams are ๏ฌat-equivalent toa trivial knot. By Lemma 1.2 we have โ ๐ฝ ๐ ( ๐ท ) = 0 and โ ๐ฝ ๐ ( ๐ท ๐ ) = 0 for all ๐ โ ๐ถ ( ๐ท ) . From these equalities and formulas (2), (3), and (4) we obtain that ๐น ๐๐พ ( ๐ก, โ ) = ๐ฟ ๐๐พ ( ๐ก, โ ) = ๐ ๐พ (2) It was mentioned above that ๐ฝ ๐ ( ๐ท ) is a coef๏ฌcient of ๐ก ๐ in af๏ฌne indexpolynomial. By the equality โ ๐ฝ ๐ ( ๐ท ) = ๐ฝ ๐ ( ๐ท ) โ ๐ฝ โ ๐ ( ๐ท ) = 0 , coef๏ฌcients of ๐ก ๐ and ๐ก โ ๐ coincide and ๐ ๐พ ( ๐ก ) is palindromic. (cid:3) MAXIM IVANOV AND ANDREI VESNIN
Recall the following properties of af๏ฌne index polynomial. Let ยฏ ๐ท be a diagram,obtained from ๐ท by reversing orientation and ๐ท * is obtained by switching all clas-sical crossings. Lemma 2.3. [7]
The following equalities hold ๐ ยฏ ๐พ ( ๐ก ) = ๐ ๐พ ( ๐ก โ ) and ๐ ๐พ * ( ๐ก ) = โ ๐ ๐พ ( ๐ก ) .
3. P
OLYNOMIAL INVARIANTS OF A KIMOVA -M ATVEEV KNOTS
Prime knots in thickened torus ๐ ร ๐ผ , that is a product of 2-dimensional torus ๐ and the unit interval ๐ผ = [0 , , admitting diagrams with at most ๏ฌve classicalcrossings were tabulated by Akimova and Matveev in [1]. The total number ofthese diagrams is equal to . Due to Kuperbergโs result [10], it is equivalent totabulating prime virtual knots of genus one. To distinguish the knots, Akimova andMatveev introduced for diagrams on a torus an analogue of bracket polynomial.These diagrams, pictured on a plane using virtual crossing are given in [1, Fig.17]. For readerโs convenience we present them in Tables 3 and 4. Theorem 3.1.
Every Akimova โMatveev knot is totally ๏ฌat-trivial.Proof.
Itโs easy to see, that forgetting the information of over/under crossings indiagrams from Tables 3 and 4 leads us to 40 different diagrams of ๏ฌat knots as inTable 1. Further we consider each of these classes separately and prove them to beT
ABLE
1. Classes of diagrams.knot knot knot knot knot1 2.1 9 4.10, 4.11 17 5.10 25 5.23, 5.24 33 5.40-5.422 3.1 10 4.12-4.14 18 5.11 26 5.25, 5.26 34 5.43-5.463 3.2, 3.3 11 4.15-4.17 19 5.12 27 5.27-5.29 35 5.47-5.504 4.1 12 5.1, 5.2 20 5.13 28 5.30 36 5.51-5.535 4.2 13 5.3, 5.4 21 5.14 29 5.31 37 5.54-5.596 4.3 14 5.5 22 5.15, 5.16 30 5.32, 5.33 38 5.60-5.657 4.4, 4.5 15 5.6, 5.7 23 5.17, 5.18 31 5.34-5.37 39 5.66-5.688 4.6-4.9 16 5.8, 5.9 24 5.19-5.22 32 5.38, 5.39 40 5.69totally ๏ฌat-trivial. Changing type of a crossing leads to changing in orientation ofa knot, obtained by smoothing at the crossing. Thereby it is suf๏ฌcient to considerjust one member from each of 40 classes to prove the theorem for the all 90 knots.
ENUS ONE KNOT POLYNOMIALS 7 F IGURE
6. Diagram of a virtual knot ๐พ = 5 . .As an example we consider a virtual knot ๐พ = 5 . pictured in Fig. 6.Itโs easy to see, that it is ๏ฌat-trivial. Then we consider all the diagrams obtainedby smoothings in classical crossings. There are ๏ฌve classical crossings denoted as ๐ , ๐ , ๐ , ๐ , ๐ . Diagrams, obtained by smoothings at ๐ , ๐ and ๐ are shown inthe picture 7.F IGURE
7. Diagrams, obtained by smoothings at crossings ๐ , ๐ and ๐ , respectively.As one can see, all diagrams in Fig. 7 are also ๏ฌat-trivial. Similarly, diagramsobtained by smoothing at ๐ and ๐ are also ๏ฌat-trivial. Hence, virtual knot ๐พ =5 . is totally ๏ฌat-trivial. Analogous considerations for knots from other classesshow that they are all totally ๏ฌat-trivial, and thus all Akimova-Matveev knots aretotally ๏ฌat-trivial. (cid:3) Theorem 3.1 and Lemma 2.2 allow us o obtain the following properties of ๐ฟ -polynomials, ๐น -polynomials and af๏ฌne index polynomial of Akimova โ Matveevknots. Corollary 3.2.
Let ๐พ be a genus one knot admitting a diagram with at most ๏ฌvecrossings. Then for every ๐ โฅ its ๐ฟ polynomials and ๐น -polynomials coin-cide with af๏ฌne index polynomial, presented in Table 2, where knots are splittedin groups with respect to the value of polynomials for the knot ๐พ or its mirrorimage ๐พ * . Question 3.3.
Is it true, that every virtual knot of genus one is totally ๏ฌat trivial?
MAXIM IVANOV AND ANDREI VESNIN T ABLE
2. Polynomial invariants.knot ๐พ polynomial ๐ ๐พ ( ๐ก ) ๐ก โ โ ๐ก ๐ก โ โ ๐ก ๐ก โ โ ๐ก ๐ก โ โ ๐ก ๐ก โ โ ๐ก โ โ ๐ก + ๐ก ๐ก โ + ๐ก โ โ ๐ก + ๐ก ๐ก โ โ ๐ก ๐ก โ โ ๐ก โ โ ๐ก + ๐ก ๐ก โ + ๐ก โ โ ๐ก + ๐ก ๐ก โ โ ๐ก ๐ก โ โ ๐ก โ โ ๐ก + ๐ก ๐ก โ + ๐ก โ โ ๐ก + ๐ก R EFERENCES [1] A.A. Akimova, S.V. Matveev,
Classi๏ฌcation of genus 1 virtual knots having at most ๏ฌve classi-cal crossings , Journal of Knot Theory and Its Rami๏ฌcatons (2014).1450031 (19 pages).[2] Z. Cheng, H. Gao,
A polynomial invariant of virtual links , Journal of Knot Theory and ItsRami๏ฌcations (2013), 1341002.[3] H. Dye,
An invitation to knot theory: virtual and classical.
A table of virtual knots ๐น -polynomials of tabulated virtual knots , preprint available atarXiv:1906.01976.[6] L. Kauffman, Virtual knot theory , European Journal of Combinatorics (1999), 663โ691.[7] L. Kauffman,
An af๏ฌne index polynomial invariant of virtual knots , Journal of Knot Theory andIts Rami๏ฌcations (2013), 1340007.[8] L. Kauffman,
Virtual knot cobordism and the af๏ฌne index polynomial , Journal of Knot Theoryand Its Rami๏ฌcations, (2018), 1843017.[9] K. Kaur, M. Prabhakar, A. Vesnin,
Two-variable polynomial invariants of virtual knots arisingfrom ๏ฌat virtual knot invariants , Journal of Knot Theory and Its Rami๏ฌcations, (2018),1842015.[10] G. Kuperberg,
What is a virtual knot?
Algebr. Geom. Topol., (2003), 587โ591.[11] S. Satoh, K. Taniguchi, The writhes of a virtual knot , Fundamenta Mathematicae (2014),327โ341.
ENUS ONE KNOT POLYNOMIALS 9 L ABORATORY OF T OPOLOGY AND D YNAMICS , N
OVOSIBIRSK S TATE U NIVERSITY , N
OVOSI - BIRSK
E-mail address : [email protected] L ABORATORY OF T OPOLOGY AND D YNAMICS , N
OVOSIBIRSK S TATE U NIVERSITY , N
OVOSI - BIRSK ; S
OBOLEV I NSTITUTE OF M ATHEMATICS OF
SB RAS, N
OVOSIBIRSK ; T
OMSK S TATE U NIVERSITY , T
OMSK
E-mail address : [email protected] T ABLE
3. Diagrams of Akimova โ Matveev knots (I).