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 ο¬ve clas-sical crossings were classiο¬ed by Akimova and Matveev in 2014. In 2018 Kaur,Prabhakar and Vesnin introduced families of πΏ -polynomials and πΉ -polynomialsfor virtual knots which are generalizations of afο¬ne index polynomial. Here weintroduce a notion of totally ο¬at-trivial knots and demonstrate that for such knots πΉ -polynomials and πΏ -polynomials coincide with afο¬ne index polynomial. Weprove that all Akimova β Matveev knots are totally ο¬at-trivial and calculate theirafο¬ne 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, itβsinteresting 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 βMatveev knots. Recall that Kaufman in [7] deο¬ned an aο¬ine index polynomialwhich is an invariant of a virtual knot and possess some important proprties [8].In [9] a generalization of afο¬ne index polynomials was introduced, namely a fam-ily of πΏ -polynomials { πΏ ππΎ ( π‘, β ) } β π =1 and family of πΉ -polynomials { πΉ ππΎ ( π‘, β ) } β π =1 .In [5] authors, using their software, calculated πΉ -polynomials of knots tabulated in[3] and [4]. Here we consider polynomial invariants for knots in a thickened torus. Mathematics Subject Classiο¬cation.
Key words and phrases.
Virtual knot, knot in a thickened torus, afο¬ne 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 deο¬-nitions and facts to use further, in Section 2 we introduce totally ο¬at-trivial knotsand show that for these knots πΏ -polynomials and πΉ -polynomials coincide withafο¬ne index polynomial, in Section 3 we calculate these invariants for Akimova βMatveev knots. In Theorem 3.1 we show that Akimova β Matveev knots are totallyο¬at-trivial. In Corollary 3.2 and Table 2 their afο¬ne index polynomials are given.The investigation of properties of Akimova β Matveev knots leads to the followingQuestion 3.3: Is it true, that every virtual knot of genus one is totally ο¬at-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 ο¬nite 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 ο¬at knot diagram . Equivalence of ο¬at knots is deο¬ned by ο¬at Reidemeister moves ,which are different from virtual Reidemeister moves in having ο¬at crossings insteadof classical ones.Let π· be a diagram of an oriented virtual knot. We denote the set of all classicalcrossings of diagram π· as πΆ ( π· ) . Sign of a classical crossing, denoted by sgn( π ) is deο¬ned as shown in the Fig. 3. (cid:73) (cid:18) sgn( π ) = +1 (cid:73) (cid:18) sgn( π ) = β 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 πΌ of a diagram π· one can assign value π ( πΌ ) = βοΈ π β π ( πΌ ) sgn( π ) ,where π ( πΌ ) is the set of classical crossings, which are ο¬st met as overcrossings,when moving around the knot from πΌ with respect to the orientation. (cid:73) (cid:18) π + 1 π ππ β (cid:73) (cid:18) π + 1 π ππ β (cid:73) (cid:18) (cid:102) ππ ππ F IGURE
4. Cheng coloringIn [2] Cheng and Gao put every classical crossing in correspondence with aninteger value
Ind( π ) , deο¬ned as(1) Ind( π ) = sgn( π )( π β π β where π and π given by Cheng coloring. One can notice, that Cheng coloring doesnot depend on types of classical crossings and hence it is deο¬ned for an orientedο¬at knot diagram. Let us remember that afο¬ne index polynomial from [7] can bewritten in the following form:(2) π π· ( π‘ ) = βοΈ π β πΆ ( π· ) sgn( π )( π‘ Ind( π ) β , MAXIM IVANOV AND ANDREI VESNIN where πΆ ( π· ) is a set of all classical crossings of π· .In [11] Satoh and Taniguchi introduced a notion of π -writhe π½ π ( π· ) . For every π β Z β { } deο¬ne π -writhe of oriented virtual knot diagram as a difference be-tween number of positive crossings and negative crossings of index π . Notice that π½ π ( π· ) is a coefο¬cient of π‘ π in afο¬ne index polynomial and it is an invariant of ori-ented virtual knot. For more information about π -writhe see [11]. Using π -writhein [9] was deο¬ned another invariant β π -dwrithe β π½ π ( π· ) : β π½ π ( π· ) = π½ π ( π· ) β π½ β π ( π· ) . Remark 1.1. β π½ π ( π· ) is an invariant of oriented virtual knot, since π½ π ( π· ) is aninvariant of oriented virtual knot. Moreover, β π½ π ( π· ) = 0 for every classical knot.As it shown in [9], β π½ π ( π· ) represents a ο¬at knot structure. Namely, the follow-ing lemma holds Lemma 1.2. [9, Lemma 2.4]
For every π β N , π -dwrithe β π½ π ( π· ) is an orientedο¬at knot invariant. Let Β― π· be a diagram, obtained from π· by reversing an orientation and π· * isobtained by switching all classical crossings. Lemma 1.3. [9, Lemma 2.5]
Let π· be a diagram of oriented virtual knot, then β π½ π ( π· * ) = β π½ π ( π· ) and β π½ π ( Β― π· ) = ββ π½ π ( π· ) . Consider a smoothing according to the rule, shown in picture 5. We will call thiskind of smoothing by smoothing against orientation . Orientation of π· π is inducedby smoothing. Since π· is a diagram of virtual knot, so π· π is a diagram of virtualknot too. (cid:73) (cid:18) ββ (cid:9) (cid:18) and (cid:73) (cid:18) ββ (cid:73) (cid:82) F IGURE
5. Smoothing.
Deο¬nition 1.4. [9] For a diagram π· of a virtual oriented knot πΎ and an integer π ,a polynomial πΏ ππΎ ( π‘, β ) is deο¬ned as:(3) πΏ ππΎ ( π‘, β ) = βοΈ π β πΆ ( π· ) sgn( π ) (οΈ π‘ Ind( π ) β |β π½ π ( π· π ) | β β |β π½ π ( π· ) | )οΈ . ENUS ONE KNOT POLYNOMIALS 5
Note that πΏ -polynomials generalize afο¬ne index polynomial, since π πΎ ( π‘ ) = πΏ ππΎ ( π‘, for every π and every π‘ . Deο¬nition 1.5. [9] For a diagram π· of a virtual oriented knot πΎ and an integer π ,a polynomial πΉ ππΎ ( π‘, β ) is deο¬ned as:(4) πΉ ππΎ ( π‘, β ) = βοΈ π β πΆ ( π· ) sgn( π ) π‘ Ind ( π ) β β π½ π ( π· π ) β βοΈ π β π π ( π· ) sgn( π ) β β π½ π ( π· π ) β βοΈ π/ β π π ( π· ) sgn( π ) β β π½ π ( π· ) , where π π ( π· ) = { π β πΆ ( π· ) : |β π½ π ( π· π ) | = |β π½ π ( π· ) |} . Theorem 1.6. [9]
For every integer π β₯ polynomials πΏ ππΎ ( π‘, β ) and πΉ ππΎ ( π‘, β ) areoriented virtual knot invariants.
2. T
OTALLY FLAT - TRIVIAL KNOTS
Let π· be a diagram of oriented virtual knot πΎ and πΆ ( π· ) a set of all classicalcrossings in π· . Deο¬nition 2.1.
We will call π· totally ο¬at-trivial if diagrams obtained from π· and π· π for all π β πΆ ( π· ) by forgetting over/under crossing information are ο¬atequivalent to unknot. Virtual knot πΎ is said to be totally ο¬at-trivial , if it admits atotally ο¬at-trivial diagram. Lemma 2.2.
If virtual knot πΎ is totally ο¬at-trivial, then (1) For all π β₯ we have πΏ ππΎ ( π‘, β ) = π πΎ ( π‘ ) and πΉ ππΎ ( π‘, β ) = π πΎ ( π‘ ) . (2) π πΎ ( π‘ ) is palindromic.Proof. (1) Let π· be a totally ο¬at-trivial diagram of a knot πΎ , πΆ ( π· ) a set of all itsclassical crossings, and π· π a diagram, obtained by smoothing against orientationin crossing π β πΆ ( π· ) . By the deο¬nition, all these diagrams are ο¬at-equivalent toa trivial knot. By Lemma 1.2 we have β π½ π ( π· ) = 0 and β π½ π ( π· π ) = 0 for all π β πΆ ( π· ) . From these equalities and formulas (2), (3), and (4) we obtain that πΉ ππΎ ( π‘, β ) = πΏ ππΎ ( π‘, β ) = π πΎ (2) It was mentioned above that π½ π ( π· ) is a coefο¬cient of π‘ π in afο¬ne indexpolynomial. By the equality β π½ π ( π· ) = π½ π ( π· ) β π½ β π ( π· ) = 0 , coefο¬cients of π‘ π and π‘ β π coincide and π πΎ ( π‘ ) is palindromic. (cid:3) MAXIM IVANOV AND ANDREI VESNIN
Recall the following properties of afο¬ne index polynomial. Let Β― π· be a diagram,obtained from π· by reversing orientation and π· * is obtained by switching all clas-sical crossings. Lemma 2.3. [7]
The following equalities hold π Β― πΎ ( π‘ ) = π πΎ ( π‘ β ) and π πΎ * ( π‘ ) = β π πΎ ( π‘ ) .
3. P
OLYNOMIAL INVARIANTS OF A KIMOVA -M ATVEEV KNOTS
Prime knots in thickened torus π Γ πΌ , that is a product of 2-dimensional torus π and the unit interval πΌ = [0 , , admitting diagrams with at most ο¬ve classicalcrossings were tabulated by Akimova and Matveev in [1]. The total number ofthese diagrams is equal to . Due to Kuperbergβs 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 readerβs convenience we present them in Tables 3 and 4. Theorem 3.1.
Every Akimova βMatveev knot is totally ο¬at-trivial.Proof.
Itβs easy to see, that forgetting the information of over/under crossings indiagrams from Tables 3 and 4 leads us to 40 different diagrams of ο¬at 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 ο¬at-trivial. Changing type of a crossing leads to changing in orientation ofa knot, obtained by smoothing at the crossing. Thereby it is sufο¬cient 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 πΎ = 5 . .As an example we consider a virtual knot πΎ = 5 . pictured in Fig. 6.Itβs easy to see, that it is ο¬at-trivial. Then we consider all the diagrams obtainedby smoothings in classical crossings. There are ο¬ve classical crossings denoted as π , π , π , π , π . Diagrams, obtained by smoothings at π , π and π are shown inthe picture 7.F IGURE
7. Diagrams, obtained by smoothings at crossings π , π and π , respectively.As one can see, all diagrams in Fig. 7 are also ο¬at-trivial. Similarly, diagramsobtained by smoothing at π and π are also ο¬at-trivial. Hence, virtual knot πΎ =5 . is totally ο¬at-trivial. Analogous considerations for knots from other classesshow that they are all totally ο¬at-trivial, and thus all Akimova-Matveev knots aretotally ο¬at-trivial. (cid:3) Theorem 3.1 and Lemma 2.2 allow us o obtain the following properties of πΏ -polynomials, πΉ -polynomials and afο¬ne index polynomial of Akimova β Matveevknots. Corollary 3.2.
Let πΎ be a genus one knot admitting a diagram with at most ο¬vecrossings. Then for every π β₯ its πΏ polynomials and πΉ -polynomials coin-cide with afο¬ne index polynomial, presented in Table 2, where knots are splittedin groups with respect to the value of polynomials for the knot πΎ or its mirrorimage πΎ * . Question 3.3.
Is it true, that every virtual knot of genus one is totally ο¬at trivial?
MAXIM IVANOV AND ANDREI VESNIN T ABLE
2. Polynomial invariants.knot πΎ polynomial π πΎ ( π‘ ) π‘ β β π‘ π‘ β β π‘ π‘ β β π‘ π‘ β β π‘ π‘ β β π‘ β β π‘ + π‘ π‘ β + π‘ β β π‘ + π‘ π‘ β β π‘ π‘ β β π‘ β β π‘ + π‘ π‘ β + π‘ β β π‘ + π‘ π‘ β β π‘ π‘ β β π‘ β β π‘ + π‘ π‘ β + π‘ β β π‘ + π‘ R EFERENCES [1] A.A. Akimova, S.V. Matveev,
Classiο¬cation of genus 1 virtual knots having at most ο¬ve classi-cal crossings , Journal of Knot Theory and Its Ramiο¬catons (2014).1450031 (19 pages).[2] Z. Cheng, H. Gao,
A polynomial invariant of virtual links , Journal of Knot Theory and ItsRamiο¬cations (2013), 1341002.[3] H. Dye,
An invitation to knot theory: virtual and classical.
A table of virtual knots πΉ -polynomials of tabulated virtual knots , preprint available atarXiv:1906.01976.[6] L. Kauffman, Virtual knot theory , European Journal of Combinatorics (1999), 663β691.[7] L. Kauffman,
An afο¬ne index polynomial invariant of virtual knots , Journal of Knot Theory andIts Ramiο¬cations (2013), 1340007.[8] L. Kauffman,
Virtual knot cobordism and the afο¬ne index polynomial , Journal of Knot Theoryand Its Ramiο¬cations, (2018), 1843017.[9] K. Kaur, M. Prabhakar, A. Vesnin,
Two-variable polynomial invariants of virtual knots arisingfrom ο¬at virtual knot invariants , Journal of Knot Theory and Its Ramiο¬cations, (2018),1842015.[10] G. Kuperberg,
What is a virtual knot?
Algebr. Geom. Topol., (2003), 587β591.[11] S. Satoh, K. Taniguchi, The writhes of a virtual knot , Fundamenta Mathematicae (2014),327β341.
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 β Matveev knots (I).