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.)

THE JACOBI DIAGRAM FOR A Cn-MOVE AND THE HOMFLY POLYNOMIAL

2007 ◽  
Vol 16 (02) ◽  
pp. 227-242 ◽  
Author(s):  
SUMIKO HORIUCHI
Keyword(s):  
Homfly Polynomial ◽  
Vassiliev Invariant ◽  
Knot Diagrams

The HOMFLY polynomial, which is an invariant of an oriented knot or link L, is the two-variable polynomial P(L; t, z) ∈ ℤ[t±1, z±1]. If we arrange the HOMFLY polynomial by the variable z, then the coefficient of zk is the polynomial of t, which is called the Pk polynomial. Let [Formula: see text] be the nth derivative of the Pk polynomial of a link L evaluated at 1. It is a Vassiliev invariant of order less than or equal to max {n + k, 0}. Let K and K′ be oriented knots such that K′ is obtained from K by a Cn-move (n ≥ 3). We show that the value [Formula: see text] is 0 or ±n! · 2n by using Jacobi diagrams and moreover it is determined by the planarity of the Jacobi diagram corresponding to the Cn-move. By the result, we can decide the above value only by the knot diagrams of K and K′.

SATANIC AND THELEMIC CIRCLES ON KNOTS

2013 ◽  
Vol 22 (05) ◽  
pp. 1350017 ◽  
Author(s):  
G. FLOWERS
Keyword(s):  
Geometric Interpretation ◽  
Second Order ◽  
Geometric Properties ◽  
Vassiliev Invariants ◽  
Vassiliev Invariant ◽  
Knot Diagrams

While Vassiliev invariants have proved to be a useful tool in the classification of knots, they are frequently defined through knot diagrams, and fail to illuminate any significant geometric properties the knots themselves may possess. Here, we provide a geometric interpretation of the second-order Vassiliev invariant by examining five-point cocircularities of knots, extending some of the results obtained in [R. Budney, J. Conant, K. P. Scannell and D. Sinha, New perspectives on self-linking, Adv. Math. 191(1) (2005) 78–113]. Additionally, an analysis on the behavior of other cocircularities on knots is given.

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.

scholarly journals INVARIANTS OF GENUS 2 MUTANTS

2009 ◽  
Vol 18 (10) ◽  
pp. 1423-1438 ◽  
Author(s):  
H. R. MORTON ◽  
N. RYDER
Keyword(s):  
Jones Polynomial ◽  
Homfly Polynomial ◽  
Vassiliev Invariant ◽  
Genus 2 ◽  
Homfly Polynomials

Pairs of genus 2 mutant knots can have different Homfly polynomials, for example some 3-string satellites of Conway mutant pairs. We give examples which have different Kauffman 2-variable polynomials, answering a question raised by Dunfield et al. in their study of genus 2 mutants. While pairs of genus 2 mutant knots have the same Jones polynomial, given from the Homfly polynomial by setting v = s2, we give examples whose Homfly polynomials differ when v = s3. We also give examples which differ in a Vassiliev invariant of degree 7, in contrast to satellites of Conway mutant knots.

DISTINGUISHING MUTANTS BY KNOT POLYNOMIALS

1996 ◽  
Vol 05 (02) ◽  
pp. 225-238 ◽  
Author(s):  
HUGH R. MORTON ◽  
PETER R. CROMWELL
Keyword(s):  
Explicit Calculation ◽  
Homfly Polynomial ◽  
Quantum Invariants ◽  
Quantum Invariant ◽  
Vassiliev Invariant ◽  
Simple Quantum ◽  
The Difference ◽  
Homfly Polynomials

We consider the problem of distinguishing mutant knots using invariants of their satellites. We show, by explicit calculation, that the Homfly polynomial of the 3-parallel (and hence the related quantum invariants) will distinguish some mutant pairs. Having established a condition on the colouring module which forces a quantum invariant to agree on mutants, we explain several features of the difference between the Homfly polynomials of satellites constructed from mutants using more general patterns. We illustrate this by our calculations; from these we isolate some simple quantum invariants, and a framed Vassiliev invariant of type 11, which distinguish certain mutants, including the Conway and Kinoshita-Teresaka pair.

scholarly journals Minimal diagrams of free knots

2014 ◽  
Vol 23 (06) ◽  
pp. 1450032
Author(s):  
Tomas Boothby ◽  
Allison Henrich ◽  
Alexander Leaf
Keyword(s):  
Equivalence Class ◽  
Virtual Knots ◽  
Knot Diagrams

Manturov recently introduced the idea of a free knot, i.e. an equivalence class of virtual knots where equivalence is generated by crossing change and virtualization moves. He showed that if a free knot diagram is associated to a graph that is irreducibly odd, then it is minimal with respect to the number of classical crossings. Not all minimal diagrams of free knots are associated to irreducibly odd graphs, however. We introduce a family of free knot diagrams that arise from certain permutations that are minimal but not irreducibly odd.

scholarly journals THE HOMFLY POLYNOMIAL FOR LINKS IN RATIONAL HOMOLOGY 3-SPHERES

Topology ◽  
1999 ◽  
Vol 38 (1) ◽  
pp. 95-115 ◽  
Author(s):  
Efstratia Kalfagianni ◽  
Xiao-Song Lin
Keyword(s):  
Homfly Polynomial ◽  
Rational Homology
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