A NOTE ON THE SHADOW COCYCLE INVARIANT OF A KNOT WITH A BASE POINT

2007 ◽  
Vol 16 (07) ◽  
pp. 959-967 ◽  
Author(s):  
S. SATOH
Keyword(s):  
Base Point ◽  
Laurent Polynomials ◽  
Coloring Number

Fox's shadow p-colorings of a knot K define two kinds of Laurent polynomials, Φp(K) and [Formula: see text], as invariants of K. We prove that the equality [Formula: see text] holds for any knot K. Also we prove that, if the p-coloring number of K is equal to p2, then [Formula: see text] has the form [Formula: see text] for some N ∈ ℤp.

Free subgroups in k(x 1,... ,x n )(X;σ) and k(x,y)(k;σ)

2019 ◽  
Vol 31 (3) ◽  
pp. 769-777
Author(s):  
Jairo Z. Gonçalves
Keyword(s):  
Normal Subgroup ◽  
Polynomial Ring ◽  
Multiplicative Group ◽  
Free Subgroup ◽  
Laurent Polynomials ◽  
Skew Polynomial Ring ◽  
Ring Of Fractions ◽  
Skew Polynomial ◽  
Free Subgroups ◽  
Mild Restriction

Abstract Let k be a field, let {\mathfrak{A}_{1}} be the k-algebra {k[x_{1}^{\pm 1},\dots,x_{s}^{\pm 1}]} of Laurent polynomials in {x_{1},\dots,x_{s}} , and let {\mathfrak{A}_{2}} be the k-algebra {k[x,y]} of polynomials in the commutative indeterminates x and y. Let {\sigma_{1}} be the monomial k-automorphism of {\mathfrak{A}_{1}} given by {A=(a_{i,j})\in GL_{s}(\mathbb{Z})} and {\sigma_{1}(x_{i})=\prod_{j=1}^{s}x_{j}^{a_{i,j}}} , {1\leq i\leq s} , and let {\sigma_{2}\in{\mathrm{Aut}}_{k}(k[x,y])} . Let {D_{i}} , {1\leq i\leq 2} , be the ring of fractions of the skew polynomial ring {\mathfrak{A}_{i}[X;\sigma_{i}]} , and let {D_{i}^{\bullet}} be its multiplicative group. Under a mild restriction for {D_{1}} , and in general for {D_{2}} , we show that {D_{i}^{\bullet}} , {1\leq i\leq 2} , contains a free subgroup. If {i=1} and {s=2} , we show that a noncentral normal subgroup N of {D_{1}^{\bullet}} contains a free subgroup.

scholarly journals Decomposition of sparse graphs, with application to game coloring number

2010 ◽  
Vol 310 (10-11) ◽  
pp. 1520-1523 ◽  
Author(s):  
Mickael Montassier ◽  
Arnaud Pêcher ◽  
André Raspaud ◽  
Douglas B. West ◽  
Xuding Zhu
Keyword(s):  
Sparse Graphs ◽  
Coloring Number ◽  
Game Coloring Number

scholarly journals LAURENT POLYNOMIAL LANDAU–GINZBURG MODELS FOR COMINUSCULE HOMOGENEOUS SPACES

2021 ◽  
Author(s):  
PETER SPACEK
Keyword(s):  
Homogeneous Space ◽  
Homogeneous Spaces ◽  
Laurent Polynomial ◽  
Complete Intersections ◽  
Laurent Polynomials ◽  
Young Diagrams ◽  
Similar Shape ◽  
Algebraic Torus ◽  
The One ◽  
Polynomial Potentials

AbstractIn this article we construct Laurent polynomial Landau–Ginzburg models for cominuscule homogeneous spaces. These Laurent polynomial potentials are defined on a particular algebraic torus inside the Lie-theoretic mirror model constructed for arbitrary homogeneous spaces in [Rie08]. The Laurent polynomial takes a similar shape to the one given in [Giv96] for projective complete intersections, i.e., it is the sum of the toric coordinates plus a quantum term. We also give a general enumeration method for the summands in the quantum term of the potential in terms of the quiver introduced in [CMP08], associated to the Langlands dual homogeneous space. This enumeration method generalizes the use of Young diagrams for Grassmannians and Lagrangian Grassmannians and can be defined type-independently. The obtained Laurent polynomials coincide with the results obtained so far in [PRW16] and [PR13] for quadrics and Lagrangian Grassmannians. We also obtain new Laurent polynomial Landau–Ginzburg models for orthogonal Grassmannians, the Cayley plane and the Freudenthal variety.

scholarly journals Colored HOMFLY-PT for hybrid weaving knot $$ {\hat{\mathrm{W}}}_3 $$(m, n)

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Vivek Kumar Singh ◽  
Rama Mishra ◽  
P. Ramadevi
Keyword(s):  
Closed Form ◽  
Topological Strings ◽  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Infinite Family ◽  
Closed Form Expression ◽  
Two Dimensional ◽  
Laurent Polynomials ◽  
Form Expression ◽  
Torus Knots

Abstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as $$ {\hat{W}}_3 $$ W ̂ 3 (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $$ \mathrm{\mathcal{R}} $$ ℛ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional $$ \hat{\mathrm{\mathcal{R}}} $$ ℛ ̂ -matrices can be written in terms of infinite family of Laurent polynomials $$ {\mathcal{V}}_{n,t}\left[q\right] $$ V n , t q whose absolute coefficients has interesting relation to the Fibonacci numbers $$ {\mathrm{\mathcal{F}}}_n $$ ℱ n . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.

scholarly journals ON THE STABILITY OF SYZYGY BUNDLES

2011 ◽  
Vol 22 (04) ◽  
pp. 515-534 ◽  
Author(s):  
IUSTIN COANDĂ
Keyword(s):  
Projective Space ◽  
Complete Intersection ◽  
Vector Bundles ◽  
Elementary Proof ◽  
Base Point ◽  
Vector Spaces ◽  
Similar Argument ◽  
Vanishing Theorem ◽  
The Stability ◽  
Koszul Cohomology

We are concerned with the problem of the stability of the syzygy bundles associated to base-point-free vector spaces of forms of the same degree d on the projective space of dimension n. We deduce directly, from M. Green's vanishing theorem for Koszul cohomology, that any such bundle is stable if its rank is sufficiently high. With a similar argument, we prove the semistability of a certain syzygy bundle on a general complete intersection of hypersurfaces of degree d in the projective space. This answers a question of H. Flenner [Comment. Math. Helv.59 (1984) 635–650]. We then give an elementary proof of H. Brenner's criterion of stability for monomial syzygy bundles, avoiding the use of Klyachko's results on toric vector bundles. We finally prove the existence of stable syzygy bundles defined by monomials of the same degree d, of any possible rank, for n at least 3. This extends the similar result proved, for n = 2, by L. Costa, P. Macias Marques and R. M. Miro-Roig [J. Pure Appl. Algebra214 (2010) 1241–1262]. The extension to the case n at least 3 has been also, independently, obtained by P. Macias Marques in his thesis [arXiv:0909.4646/math.AG (2009)].

Orderings on Graphs and Game Coloring Number

Order ◽  
2003 ◽  
Vol 20 (3) ◽  
pp. 255-264 ◽  
Author(s):  
H. A. Kierstead ◽  
Daqing Yang
Keyword(s):  
Coloring Number ◽  
Game Coloring Number

CMV Biorthogonal Laurent Polynomials: Perturbations and Christoffel Formulas

2018 ◽  
Vol 140 (3) ◽  
pp. 333-400 ◽  
Author(s):  
Gerardo Ariznabarreta ◽  
Manuel Mañas ◽  
Alfredo Toledano
Keyword(s):  
Laurent Polynomials

scholarly journals OPTIMIZATION OF COLLIMATOR APERTURE GEOMETRY FOR BNCT KARTINI RESEARCH REACTOR USING MCNPX

2019 ◽  
Vol 21 (1) ◽  
pp. 9
Author(s):  
Ramadhan Valiant Gill S.B. ◽  
Yohannes Sardjono
Keyword(s):  
Neutron Source ◽  
Research Reactor ◽  
Irradiation Time ◽  
Base Point ◽  
Cell Diameter ◽  
Cylindrical Shape ◽  
Trace Distance ◽  
Optimal Output ◽  
Boron Neutron Capture ◽  
High Ionization

Boron Neutron Capture Therapy (BNCT) is one of the promising cancer therapy modalities due to its selectivity which only kills the cancer cells and does not damage healthy cells around cancer. In principle, BNCT utilizes the high ionization properties of alpha (4He) and lithium (7Li) particles derived from the reaction between epithermal and boron-10 neutrons (10B + n → 7Li + 4He) in cells, where trace distance of alpha and lithium particles is equivalent with cell diameter. The neutron source used in BNCT can come from a reactor, as a condition for conducting BNCT therapy tests, there are five standard parameters that must be met for a neutron source to be used as a source, and the standards come from IAEA. This research is based on simulation using the MCNPX program which aims to optimize IAEA parameters that have been obtained in previous studies by changing the shape of the collimator geometry from cone shape to cylinder with variations diameter from 3, 5 and 10 cm and also the simulation divided into two schemes namely first moderator Al is placed in a position 9.5 cm behind the collimator and the second is the moderator Al is pressed into the base point of the aperture in the collimator. In this work, neutrons originated from Yogyakarta Kartini research reactor have the energy range in the continuous form. The results of the optimization on each scheme of the collimator are compared with the outputs that have been obtained in previous studies where the aperture of the collimator is in the cone shape. The most optimal output obtained from the results is a collimator with a diameter of 5 cm in the second scheme where the results of IAEA parameters that are produced (n/cm2 s) = 2.18E+8, / (Gy-cm2/n) = 6.69E-13, / (Gy-cm2/n) = 2.44E-13,  = 4.03E-01, and J/ = 6.31E-01. These results can still be used for BNCT experiments but need a long irradiation time and when compared to previous studies, the output of the collimator with the diameter of 5 cm is more optimal.Keywords: BNCT, Collimator, IAEA Parameters, MCNPX, Cylindrical shape 

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