scholarly journals Simplicial volume of links from link diagrams

2017 ◽  
Vol 166 (1) ◽  
pp. 75-81 ◽  
Author(s):  
OLIVER DASBACH ◽  
ANASTASIIA TSVIETKOVA
Keyword(s):  
Upper Bound ◽  
Link Diagram ◽  
Simplicial Volume ◽  
Link Diagrams ◽  
Hyperbolic Volume ◽  
Link Complements ◽  
Link Complement ◽  
Half Twist

AbstractThe hyperbolic volume of a link complement is known to be unchanged when a half-twist is added to a link diagram, and a suitable 3-punctured sphere is present in the complement. We generalise this to the simplicial volume of link complements by analysing the corresponding toroidal decompositions. We then use it to prove a refined upper bound for the volume in terms of twists of various lengths for links.

scholarly journals Cyclic coverings of virtual link diagrams

2019 ◽  
Vol 30 (14) ◽  
pp. 1950072 ◽  
Author(s):  
Naoko Kamada
Keyword(s):  
Virtual Link ◽  
Link Diagram ◽  
Cyclic Covering ◽  
Link Diagrams ◽  
Virtual Links ◽  
Cyclic Coverings

A virtual link diagram is called mod [Formula: see text] almost classical if it admits an Alexander numbering valued in integers modulo [Formula: see text], and a virtual link is called mod [Formula: see text] almost classical if it has a mod [Formula: see text] almost classical diagram as a representative. In this paper, we introduce a method of constructing a mod [Formula: see text] almost classical virtual link diagram from a given virtual link diagram, which we call an [Formula: see text]-fold cyclic covering diagram. The main result is that [Formula: see text]-fold cyclic covering diagrams obtained from two equivalent virtual link diagrams are equivalent. Thus, we have a well-defined map from the set of virtual links to the set of mod [Formula: see text] almost classical virtual links. Some applications are also given.

DETERMINING THE COMPONENT NUMBER OF LINKS CORRESPONDING TO LATTICES

2009 ◽  
Vol 18 (12) ◽  
pp. 1711-1726 ◽  
Author(s):  
XIAN'AN JIN ◽  
FENGMING DONG ◽  
ENG GUAN TAY
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Plane Graph ◽  
Link Diagram ◽  
Plane Graphs ◽  
Link Diagrams ◽  
One To One

It is well known that there is a one-to-one correspondence between signed plane graphs and link diagrams via the medial construction. The component number of the corresponding link diagram is however independent of the signs of the plane graph. Determining this number may be one of the first problems in studying links by using graphs. Some works in this aspect have been done. In this paper, we investigate the component number of links corresponding to lattices. Firstly we provide some general results on component number of links. Then, via these results, we proceed to determine the component number of links corresponding to lattices with free or periodic boundary conditions and periodic lattices with one cap (i.e. spiderweb graphs) or two caps.

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 ALMOST POSITIVE LINKS HAVE NEGATIVE SIGNATURE

2010 ◽  
Vol 19 (02) ◽  
pp. 187-289 ◽  
Author(s):  
JÓZEF H. PRZYTYCKI ◽  
KOUKI TANIYAMA
Keyword(s):  
Partial Answer ◽  
Link Diagram ◽  
Link Diagrams ◽  
Left Handed ◽  
Trefoil Knot ◽  
Trivial Link ◽  
Hopf Link

We analyze properties of links which have diagrams with a small number of negative crossings. We show that if a nontrivial link has a diagram with all crossings positive except possibly one, then the signature of the link is negative. If a link diagram has two negative crossings, we show that the signature of the link is nonpositive with the exception of the left-handed Hopf link (possibly, with extra trivial components). We also characterize those links which have signature zero and diagrams with two negative crossings. In particular, we show that if a nontrivial knot has a diagram with two negative crossings then the signature of the knot is negative, unless the knot is a twist knot with negative clasp. We completely determine all trivial link diagrams with two or fewer negative crossings. For a knot diagram with three negative crossings, the signature of the knot is nonpositive except for the left-handed trefoil knot. These results generalize those of Rudolph, Cochran, Gompf, Traczyk and Przytycki, solve [27, Conjecture 5], and give a partial answer to [3, Problem 2.8] about knots dominating the trefoil knot or the trivial knot. We also describe all unknotting number one positive knots.

Linking invariants of even virtual links

2017 ◽  
Vol 26 (12) ◽  
pp. 1750072 ◽  
Author(s):  
Haruko A. Miyazawa ◽  
Kodai Wada ◽  
Akira Yasuhara
Keyword(s):  
Lower Bound ◽  
The Other ◽  
Virtual Link ◽  
Link Diagram ◽  
Linking Number ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Classical Link ◽  
Virtual Links ◽  
The Difference

A virtual link diagram is even if the virtual crossings divide each component into an even number of arcs. The set of even virtual link diagrams is closed under classical and virtual Reidemeister moves, and it contains the set of classical link diagrams. For an even virtual link diagram, we define a certain linking invariant which is similar to the linking number. In contrast to the usual linking number, our linking invariant is not preserved under the forbidden moves. In particular, for two fused isotopic even virtual link diagrams, the difference between the linking invariants of them gives a lower bound of the minimal number of forbidden moves needed to deform one into the other. Moreover, we give an example which shows that the lower bound is best possible.

Supporting Visual Queries on Medium-Sized Node–Link Diagrams

2005 ◽  
Vol 4 (1) ◽  
pp. 49-58 ◽  
Author(s):  
Colin Ware ◽  
Robert Bobrow
Keyword(s):  
Information Visualization ◽  
Error Rates ◽  
Link Diagram ◽  
Breadth First Search ◽  
Underlying Graph ◽  
Network Diagram ◽  
Complex Information ◽  
Link Diagrams ◽  
Interactive Diagrams ◽  
Static Technique

For reasons of clarity, a typical node–link diagram statically displayed on paper or a computer screen contains fewer than 30 nodes. However, many problems would benefit if far more complex information could be diagrammed. Following Munzner et al., we suggest that with interactive diagrams this may be possible. We describe an interactive technique whereby a subset of a larger network diagram is highlighted by being set into oscillatory motion when a node is selected with a mouse. The subset is determined by a breadth first search of the underlying graph starting from the selected node. This technique is designed to support visual queries on moderately large node-link diagrams containing up to a few thousand nodes. An experimental evaluation was carried out with networks having 32, 100, 320, 1000, and 3200 nodes respectively, and with four highlighting techniques: static highlighting, motion highlighting, static+ motion highlighting, and none. The results show that the interactive highlighting methods support rapid visual queries of nodes in close topological proximity to one another, even for the largest diagrams tested. Without highlighting, error rates were high even for the smallest network that was evaluated. Motion highlighting and static highlighting were equally effective. A second experiment was carried out to evaluate methods for showing two subsets of a larger network simultaneously in such a way that both are clearly distinct. The specific task was to determine if the two subsets had nodes in common. The results showed that this task could be performed rapidly and with few errors if one subset was highlighted using motion and the other was highlighted using a static technique. We discuss the implications for information visualization.

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.

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.

Splitting number of augmented links

2018 ◽  
Vol 27 (06) ◽  
pp. 1850038
Author(s):  
Darlan Girao
Keyword(s):  
Link Diagram ◽  
Link Diagrams ◽  
Linking Numbers ◽  
Splitting Number ◽  
Knot Diagrams

We completely determine the splitting number of augmented links arising from knot and link diagrams in which each twist region has an even number of crossings. In the case of augmented links obtained from knot diagrams, we show that the splitting number is given by the size of a maximal collection of Boromean sublinks, any two of which have one component in common. The general case is stablished by considering the linking numbers between components of the augmented links. We also discuss the case when the augmented link arises from a link diagram in which twist regions may have an odd number of crossings.

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