scholarly journals Computation of Gordian distances and H2-Gordian distances of knots

2015 ◽  
Vol 25 (1) ◽  
pp. 133-152 ◽  
Author(s):  
Ana Zekovic
Keyword(s):  
Knot Theory ◽  
Np Hard ◽  
Minimal Representations ◽  
Rational Knots ◽  
Knot Diagrams

One of the most complicated problems in Knot theory is to compute unknotting number. Hass, Lagarias and Pippenger proved that the unknotting problem is NP hard. In this paper we discuss the question of computing unknotting number from minimal knot diagrams, Bernhard-Jablan Conjecture, unknown knot distances between non-rational knots, and searching for minimal distances by using a graph with weighted edges, which represents knot distances. Since topoizomerazes are enzymes involved in changing crossing of DNA, knot distances can be used to study topoizomerazes actions. In the existing tables of knot smoothing, knots with smoothing number 1 are computed by Abe and Kanenobu [27] for knots with at most n = 9 crossings, and smoothing knot distances are computed by Kanenobu [26] for knots with at most n = 7 crossings. We compute some undecided knot distances 1 from these papers, and extend the computations by computing knots with smoothing number one with at most n = 11 crossings and smoothing knot distances of knots with at most n = 9 crossings. All computations are done in LinKnot, based on Conway notation and non-minimal representations of knots.

A PARITY AND A MULTI-VARIABLE POLYNOMIAL INVARIANT FOR VIRTUAL LINKS

2013 ◽  
Vol 22 (13) ◽  
pp. 1350073 ◽  
Author(s):  
YOUNG HO IM ◽  
KYOUNG IL PARK
Keyword(s):  
Knot Theory ◽  
Virtual Link ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Link Diagrams ◽  
Virtual Links ◽  
Knot Diagrams

We introduce a parity of classical crossings of virtual link diagrams which extends the Gaussian parity of virtual knot diagrams and the odd writhe of virtual links that extends that of virtual knots introduced by Kauffman [A self-linking invariants of virtual knots, Fund. Math.184 (2004) 135–158]. Also, we introduce a multi-variable polynomial invariant for virtual links by using the parity of classical crossings, which refines the index polynomial introduced in [Index polynomial invariants of virtual links, J. Knot Theory Ramifications19(5) (2010) 709–725]. As consequences, we give some properties of our invariant, and raise some examples.

COMBINATORIC MASSEY–MILNOR LINKING THEORY

2011 ◽  
Vol 20 (06) ◽  
pp. 927-938 ◽  
Author(s):  
CHUN-CHUNG HSIEH
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Configuration Space ◽  
Invariant Theory ◽  
Knot Theory ◽  
Quantum Field ◽  
Perturbative Quantum ◽  
Knot Diagrams ◽  
Chern Simons ◽  
Milnor Invariant

In this paper following the scheme of Massey–Milnor invariant theory [C. C. Hsieh, Combinatoric and diagrammatic studies in knot theory J. Knot Theory Ramifications16 (2007) 1235–1253; C. C. Hsieh, Massey-Milnor linking = Chern-Simons-Witten graphs, J. Knot Theory Ramifications17 (2008) 877–903; C. C. Hsieh and S. W. Yang, Chern-Simons-Witten configuration space integrals in knot theory, J. Knot Theory Ramifications14 (2005) 689–711], we studied the first non-vanishing linkings of knot theory in ℝ3 and also derived the combinatorial formulae from which we could read out the invariants directly from the knot diagrams. Though the theme is calculus, the idea comes from perturbative quantum field theory.

scholarly journals INVARIANTS OF KNOT DIAGRAMS AND RELATIONS AMONG REIDEMEISTER MOVES

2001 ◽  
Vol 10 (08) ◽  
pp. 1215-1227 ◽  
Author(s):  
OLOF-PETTER ÖSTLUND
Keyword(s):  
Knot Theory ◽  
Knot Invariants ◽  
Reidemeister Moves ◽  
Knot Invariant ◽  
Knot Diagrams

In this paper a classification of Reidemeister moves, which is the most refined, is introduced. In particular, this classification distinguishes some Ω3-moves that only differ in how the three strands that are involved in the move are ordered on the knot. To transform knot diagrams of isotopic knots into each other one must in general use Ω3-moves of at least two different classes. To show this, knot diagram invariants that jump only under Ω3-moves are introduced. Knot diagrams of isotopic knots can be connected by a sequence of Reidemeister moves of only six, out of the total of 24, classes. This result can be applied in knot theory to simplify proofs of invariance of diagrammatical knot invariants. In particular, a criterion for a function on Gauss diagrams to define a knot invariant is presented.

scholarly journals NP–hard problems naturally arising in knot theory

2021 ◽  
Vol 8 (15) ◽  
pp. 420-441
Author(s):  
Dale Koenig ◽  
Anastasiia Tsvietkova
Keyword(s):  
Knot Theory ◽  
Np Hard ◽  
Hard Problems ◽  
Np Hard Problems

scholarly journals On unknotting numbers and knot trivadjacency

2004 ◽  
Vol 94 (2) ◽  
pp. 227 ◽  
Author(s):  
A. Stoimenow
Keyword(s):  
Homfly Polynomial ◽  
Vassiliev Invariant ◽  
Rational Knots ◽  
Knot Diagrams ◽  
Form Condition

We prove for rational knots a conjecture of Adams et al. that an alternating unknotting number one knot has an alternating unknotting number one diagram. We use this then to show a refined signed version of the Kanenobu-Murakami theorem on unknotting number one rational knots. Together with a similar refinement of the linking form condition of Montesinos-Lickorish and the HOMFLY polynomial, we prove a condition for a knot to be $2$-trivadjacent, improving the previously known condition on the degree-2-Vassiliev invariant. We finally show several partial cases of the conjecture that the knots with everywhere $1$-trivial knot diagrams are exactly the trivial, trefoil and figure eight knots. (A knot diagram is called everywhere $n$-trivial, if it turns into an unknot diagram by switching any set of $n$ of its crossings.)

scholarly journals Virtual Mosaic Knot Theory

2020 ◽  
pp. 2050091
Author(s):  
Sandy Ganzell ◽  
Allison Henrich
Keyword(s):  
Knot Theory ◽  
Computational Results ◽  
Tile Type ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Link Type ◽  
Knot Diagrams

Mosaic diagrams for knots were first introduced in 2008 by Lomanoco and Kauffman for the purpose of building a quantum knot system. Since then, many others have explored the structure of these knot mosaic diagrams, as they are interesting objects of study in their own right. Knot mosaics have been generalized by Garduño to virtual knots, by including an additional tile type to represent virtual crossings. There is another interpretation of virtual knots, however, as knot diagrams on surfaces, which inspires this work. By viewing classical mosaic diagrams as [Formula: see text]-gons and gluing edges of these polygons, we obtain knots on surfaces that can be viewed as virtual knots. These virtual mosaics are our present objects of study. In this paper, we provide a set of moves that can be performed on virtual mosaics that preserve knot and link type, we show that any virtual knot or link can be represented as a virtual mosaic, and we provide several computational results related to virtual mosaic numbers for small classical and virtual knots.

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

scholarly journals Optimizing Transmit Sequence and Instrumental Variables Receiver for Dual-Function Complexity System

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Yu Yao ◽  
Junhui Zhao ◽  
Lenan Wu
Keyword(s):  
Approximate Solution ◽  
Instrumental Variables ◽  
Performance Index ◽  
Dual Function ◽  
Np Hard ◽  
Match Filter ◽  
Filter Output ◽  
Phase Code ◽  
Discrete Phase ◽  
Simulation Results

This correspondence deals with the joint cognitive design of transmit coded sequences and instrumental variables (IV) receive filter to enhance the performance of a dual-function radar-communication (DFRC) system in the presence of clutter disturbance. The IV receiver can reject clutter more efficiently than the match filter. The signal-to-clutter-and-noise ratio (SCNR) of the IV filter output is viewed as the performance index of the complexity system. We focus on phase only sequences, sharing both a continuous and a discrete phase code and develop optimization algorithms to achieve reasonable pairs of transmit coded sequences and IV receiver that fine approximate the behavior of the optimum SCNR. All iterations involve the solution of NP-hard quadratic fractional problems. The relaxation plus randomization technique is used to find an approximate solution. The complexity, corresponding to the operation of the proposed algorithms, depends on the number of acceptable iterations along with on and the complexity involved in all iterations. Simulation results are offered to evaluate the performance generated by the proposed scheme.

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