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 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 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.

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).

Some links with non-adequate minimal-crossing diagrams

1992 ◽  
Vol 111 (2) ◽  
pp. 283-289 ◽  
Author(s):  
Masao Hara ◽  
Makoto Yamamoto
Keyword(s):  
Crossing Number ◽  
Link Diagram ◽  
Polynomial Invariants ◽  
Crossing Numbers ◽  
Almost All ◽  
Alternating Link ◽  
Torus Links

To investigate invariants of links derived from their diagrams, the recent new polynomial invariants of links play important roles. Murasugi6, 7, Kauffman3 and Thistlethwaite 9 independently showed that the number of crossings in a proper connected alternating diagram of a link is the minimal-crossing number of the link and that the writhe of the diagram is invariant. Murasugi 8 also determined the minimal-crossing number of torus links. In 5, Lickorish and Thistlethwaite introduced the concept of an adequate link diagram and showed that the number of crossings in an adequate diagram of a semi-alternating link is the minimal-crossing number of the link. They also determined the minimal-crossing number of almost all Montesinos links. In this paper we show that for some links represented by plats and braids which are not adequate, the numbers of crossings in the diagrams are the minimal-crossing numbers of the links.

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 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.

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.

scholarly journals The colored Kauffman Skein relation and the head and tail of the colored Jones polynomial

2017 ◽  
Vol 26 (03) ◽  
pp. 1741002 ◽  
Author(s):  
Mustafa Hajij
Keyword(s):  
Natural Extension ◽  
Jones Polynomial ◽  
Colored Jones Polynomial ◽  
Alternating Links ◽  
Skein Relation

Using the colored Kauffman skein relation, we study the highest and lowest [Formula: see text] coefficients of the [Formula: see text] unreduced colored Jones polynomial of alternating links. This gives a natural extension of a result by Kauffman in regard with the Jones polynomial of alternating links and its highest and lowest coefficients. We also use our techniques to give a new and natural proof for the existence of the tail of the colored Jones polynomial for alternating links.

POSITIVE ALTERNATING LINKS ARE POSITIVELY ALTERNATING

2000 ◽  
Vol 09 (01) ◽  
pp. 107-112 ◽  
Author(s):  
TAKUJI NAKAMURA
Keyword(s):  
Alternating Links ◽  
Alternating Link

By using a result of P. R. Cromwell on homogeneous links which contain both positive links and alternating links, we prove that any reduced alternating diagram of a positive alternating link is positive.

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