scholarly journals Writhe-like invariants of alternating links

2021 ◽  
Vol 30 (01) ◽  
pp. 2150004
Author(s):  
Yuanan Diao ◽  
Van Pham
Keyword(s):  
Jones Polynomial ◽  
Link Diagram ◽  
Link Invariant ◽  
Link Invariants ◽  
Link Diagrams ◽  
Alternating Links ◽  
Alternating Link

It is known that the writhe calculated from any reduced alternating link diagram of the same (alternating) link has the same value. That is, it is a link invariant if we restrict ourselves to reduced alternating link diagrams. This is due to the fact that reduced alternating link diagrams of the same link are obtainable from each other via flypes and flypes do not change writhe. In this paper, we introduce several quantities that are derived from Seifert graphs of reduced alternating link diagrams. We prove that they are “writhe-like” invariants, namely they are not general link invariants, but are invariants when restricted to reduced alternating link diagrams. The determination of these invariants are elementary and non-recursive so they are easy to calculate. We demonstrate that many different alternating links can be easily distinguished by these new invariants, even for large, complicated knots for which other invariants such as the Jones polynomial are hard to compute. As an application, we also derive an if and only if condition for a strongly invertible rational link.

ON THE NULLIFICATION WRITHE, THE SIGNATURE AND THE CHIRALITY OF ALTERNATING LINKS

2001 ◽  
Vol 10 (04) ◽  
pp. 537-545 ◽  
Author(s):  
CAM VAN QUACH HONGLER
Keyword(s):  
Link Diagram ◽  
Link Invariants ◽  
Link Diagrams ◽  
The Difference ◽  
Alternating Links ◽  
Alternating Link

In this paper, we relate the nullification writhe and the remaining writhe defined by C. Cerf to other link invariants. We prove that the nullification writhe of an oriented reduced alternating link diagram is equal, up to sign, to the signature of the link. Moreover, we relate the difference between the nullication writhe and the remaining writhe to invariants obtained from chessboard-coloured link diagrams such as their numbers of shaded and unshaded regions.

scholarly journals The braid index of reduced alternating links

2019 ◽  
Vol 168 (3) ◽  
pp. 415-434
Author(s):  
YUANAN DIAO ◽  
GÁBOR HETYEI ◽  
PENGYU LIU
Keyword(s):  
Jones Polynomial ◽  
Link Diagram ◽  
Homfly Polynomial ◽  
Link Diagrams ◽  
Braid Index ◽  
Weight One ◽  
Minimum Number ◽  
Remarkable Result ◽  
Alternating Links ◽  
Alternating Link

AbstractIt is well known that the minimum crossing number of an alternating link equals the number of crossings in any reduced alternating link diagram of the link. This remarkable result is an application of the Jones polynomial. In the case of the braid index of an alternating link, Yamada showed that the minimum number of Seifert circles over all regular projections of a link equals the braid index. Thus one may conjecture that the number of Seifert circles in a reduced alternating diagram of the link equals the braid index of the link, but this turns out to be false. In this paper we prove the next best thing that one could hope for: we characterise exactly those alternating links for which their braid indices equal the numbers of Seifert circles in their corresponding reduced alternating link diagrams. More specifically, we prove that if D is a reduced alternating link diagram of an alternating link L, then b(L), the braid index of L, equals the number of Seifert circles in D if and only if GS(D) contains no edges of weight one. Here GS(D), called the Seifert graph of D, is an edge weighted simple graph obtained from D by identifying each Seifert circle of D as a vertex of GS(D) such that two vertices in GS(D) are connected by an edge if and only if the two corresponding Seifert circles share crossings between them in D and that the weight of the edge is the number of crossings between the two Seifert circles. This result is partly based on the well-known MFW inequality, which states that the a-span of the HOMFLY polynomial of L is a lower bound of 2b(L)−2, as well as the result of Yamada relating the minimum number of Seifert circles over all link diagrams of L to b(L).

A geometric proof that alternating knots are non-trivial

1991 ◽  
Vol 109 (3) ◽  
pp. 425-431 ◽  
Author(s):  
William Menasco ◽  
Morwen Thistlethwaite
Keyword(s):  
Jones Polynomial ◽  
Related Result ◽  
Slight Variation ◽  
Geometric Proof ◽  
Link Diagram ◽  
Combinatorial Argument ◽  
Algebraic Invariant ◽  
Almost All ◽  
Alternating Links ◽  
Alternating Link

There are many proofs in the literature of the non-triviality of alternating, classical links in the 3-sphere, but almost all use a combinatorial argument involving some algebraic invariant, namely the determinant [1], the Alexander polynomial [3], the Jones polynomial [5], and, in [6], the Q-polynomial of Brandt–Lickorish–Millett. Indeed, alternating links behave remarkably well with respect to these and other invariants, but this fact has not led to any significant geometric understanding of alternating link types. Therefore it is natural to seek purely geometric proofs of geometric properties of these links. Gabai has given in [4] a striking geometric proof of a related result, also proved earlier by algebraic means in [3], namely that the Seifert surface obtained from a reduced alternating link diagram by Seifert's algorithm has minimal genus for that link. Here, we give an elementary geometric proof of non-triviality of alternating knots, using a slight variation of the techniques set forth in [7, 8]. Note that if L is a link of more than one component and some component of L is spanned by a disk whose interior lies in the complement of L, then L is a split link, i.e. it is separated by a 2-sphere in S3\L; thus we do not consider alternating links of more than one component here, as it is proved in [7] that a connected alternating diagram cannot represent a split link.

scholarly journals ALL LINK INVARIANTS FOR TWO DIMENSIONAL SOLUTIONS OF YANG–BAXTER EQUATION AND DRESSINGS

2006 ◽  
Vol 15 (10) ◽  
pp. 1279-1301
Author(s):  
N. AIZAWA ◽  
M. HARADA ◽  
M. KAWAGUCHI ◽  
E. OTSUKI
Keyword(s):  
Jones Polynomial ◽  
Alexander Polynomial ◽  
Baxter Equation ◽  
Two Dimensional ◽  
Link Invariant ◽  
Three Solutions ◽  
Polynomial Invariants ◽  
Link Invariants ◽  
Higher Dimensional

All polynomial invariants of links for two dimensional solutions of Yang–Baxter equation is constructed by employing Turaev's method. As a consequence, it is proved that the best invariant so constructed is the Jones polynomial and there exist three solutions connecting to the Alexander polynomial. Invariants for higher dimensional solutions, obtained by the so-called dressings, are also investigated. It is observed that the dressings do not improve link invariant unless some restrictions are put on dressed solutions.

scholarly journals A GEOMETRIC CHARACTERIZATION OF THE UPPER BOUND FOR THE SPAN OF THE JONES POLYNOMIAL

2011 ◽  
Vol 20 (07) ◽  
pp. 1059-1071
Author(s):  
JUAN GONZÁLEZ-MENESES ◽  
PEDRO M. G. MANCHÓN
Keyword(s):  
Upper Bound ◽  
Knot Theory ◽  
Jones Polynomial ◽  
Geometric Proof ◽  
Link Diagram ◽  
Closed Curves ◽  
Number Of Faces ◽  
Alternating Links ◽  
Connected Diagram

Let D be a link diagram with n crossings, sA and sB be its extreme states and |sAD| (respectively, |sBD|) be the number of simple closed curves that appear when smoothing D according to sA (respectively, sB). We give a general formula for the sum |sAD| + |sBD| for a k-almost alternating diagram D, for any k, characterizing this sum as the number of faces in an appropriate triangulation of an appropriate surface with boundary. When D is dealternator connected, the triangulation is especially simple, yielding |sAD| + |sBD| = n + 2 - 2k. This gives a simple geometric proof of the upper bound of the span of the Jones polynomial for dealternator connected diagrams, a result first obtained by Zhu [On Kauffman brackets, J. Knot Theory Ramifications6(1) (1997) 125–148.]. Another upper bound of the span of the Jones polynomial for dealternator connected and dealternator reduced diagrams, discovered historically first by Adams et al. [Almost alternating links, Topology Appl.46(2) (1992) 151–165.], is obtained as a corollary. As a new application, we prove that the Turaev genus is equal to the number k of dealternator crossings for any dealternator connected diagram.

scholarly journals A twisted link invariant derived from a virtual link invariant

2017 ◽  
Vol 26 (09) ◽  
pp. 1743007
Author(s):  
Naoko Kamada
Keyword(s):  
Knot Theory ◽  
Orientable Surface ◽  
Link Diagram ◽  
Link Invariant ◽  
Virtual Knot ◽  
Chord Diagrams ◽  
Link Diagrams ◽  
Virtual Links ◽  
Virtual Knot Theory ◽  
Double Coverings

Virtual knot theory is a generalization of knot theory which is based on Gauss chord diagrams and link diagrams on closed oriented surfaces. A twisted knot is a generalization of a virtual knot, which corresponds to a link diagram on a possibly non-orientable surface. In this paper, we discuss an invariant of twisted links which is obtained from the JKSS invariant of virtual links by use of double coverings. We also discuss some properties of double covering diagrams.

scholarly journals ALTERNATING SUM FORMULAE FOR THE DETERMINANT AND OTHER LINK INVARIANTS

2010 ◽  
Vol 19 (06) ◽  
pp. 765-782 ◽  
Author(s):  
OLIVER T. DASBACH ◽  
DAVID FUTER ◽  
EFSTRATIA KALFAGIANNI ◽  
XIAO-SONG LIN ◽  
NEAL W. STOLTZFUS
Keyword(s):  
Classical Result ◽  
Spanning Trees ◽  
Jones Polynomial ◽  
Link Invariants ◽  
Alternating Link ◽  
Number Of Spanning Trees

A classical result states that the determinant of an alternating link is equal to the number of spanning trees in a checkerboard graph of an alternating connected projection of the link. We generalize this result to show that the determinant is the alternating sum of the number of quasi-trees of genus j of the dessin of a non-alternating link. Furthermore, we obtain formulas for coefficients of the Jones polynomial by counting quantities on dessins. In particular, we will show that the jth coefficient of the Jones polynomial is given by sub-dessins of genus less or equal to j.

scholarly journals Extending quasi-alternating links

2020 ◽  
Vol 29 (11) ◽  
pp. 2050072
Author(s):  
Nafaa Chbili ◽  
Kirandeep Kaur
Keyword(s):  
Jones Polynomial ◽  
Alternating Links ◽  
Alternating Link ◽  
Torus Links

Champanerkar and Kofman [Twisting quasi-alternating links, Proc. Amer. Math. Soc. 137(7) (2009) 2451–2458] introduced an interesting way to construct new examples of quasi-alternating links from existing ones. Actually, they proved that replacing a quasi-alternating crossing [Formula: see text] in a quasi-alternating link by a rational tangle of same type yields a new quasi-alternating link. This construction has been extended to alternating algebraic tangles and applied to characterize all quasi-alternating Montesinos links. In this paper, we extend this technique to any alternating tangle of same type as [Formula: see text]. As an application, we give new examples of quasi-alternating knots of 13 and 14 crossings. Moreover, we prove that the Jones polynomial of a quasi-alternating link that is obtained in this way has no gap if the original link has no gap in its Jones polynomial. This supports a conjecture introduced in [N. Chbili and K. Qazaqzeh, On the Jones polynomial of quasi-alternating links, Topology Appl. 264 (2019) 1–11], which states that the Jones polynomial of any prime quasi-alternating link except [Formula: see text]-torus links has no gap.

ORIENTED STATE MODEL OF THE JONES POLYNOMIAL AND ITS CONNECTION TO THE DICHROMATIC POLYNOMIAL

2010 ◽  
Vol 19 (01) ◽  
pp. 81-92
Author(s):  
XIAN'AN JIN ◽  
FUJI ZHANG
Keyword(s):  
Jones Polynomial ◽  
Weighted Graph ◽  
The State ◽  
State Model ◽  
Oriented Link ◽  
Model Based ◽  
Link Diagrams ◽  
Alternating Links ◽  
Dichromatic Polynomial ◽  
Appl Math

It is well known that Kauffman constructed a state model of the Jones polynomial based on unoriented link diagrams. In his approach, in order to obtain Jones polynomial one needs to calculate both the writhe and the Kauffman bracket. Stimulated by a paper of Altintas (An oriented state model for the Jones polynomial and its applications to alternating links, Appl. Math. Comput.194 (2007) 168–178), in this paper we present a state sum model based on oriented link diagrams. In our approach, we succeed in adding the writhe to the state sum model and need not to compute the writher any more. We further show that, via our state sum model, Jones polynomial of any link (alternating or not) is a special parametrization of the dichromatic polynomial of a weighted graph with two different edge weights.

scholarly journals THE JONES POLYNOMIAL AND THE PLANAR ALGEBRA OF ALTERNATING LINKS

2010 ◽  
Vol 19 (11) ◽  
pp. 1487-1505 ◽  
Author(s):  
HERNANDO BURGOS-SOTO
Keyword(s):  
Jones Polynomial ◽  
Skein Module ◽  
Single Crossing ◽  
Planar Algebra ◽  
The Right ◽  
Alternating Links ◽  
Alternating Link

It is a well known result that the Jones polynomial of a non-split alternating link is alternating. We find the right generalization of this result to the case of non-split alternating tangles. More specifically: the Jones polynomial of tangles is valued in a certain skein module; we describe an alternating condition on elements of this skein module, show that it is satisfied by the Jones invariant of the single crossing tangles (⤲) and (⤲), and prove that it is preserved by appropriately "alternating" planar algebra compositions. Hence, this condition is satisfied by the Jones polynomial of all alternating tangles. Finally, in the case of 0-tangles, that is links, our condition is equivalent to simple alternation of the coefficients of the Jones polynomial.

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