scholarly journals Relative Tutte Polynomials of Tensor Products of Coloured Graphs

2013 ◽  
Vol 22 (6) ◽  
pp. 801-828
Author(s):  
Y. DIAO ◽  
G. HETYEI
Keyword(s):  
Tensor Product ◽  
Jones Polynomial ◽  
Tensor Products ◽  
Tutte Polynomial ◽  
Product Formula ◽  
Virtual Knot ◽  
Tutte Polynomials ◽  
Product Of Graphs

The tensor product (G1,G2) of a graph G1 and a pointed graph G2 (containing one distinguished edge) is obtained by identifying each edge of G1 with the distinguished edge of a separate copy of G2, and then removing the identified edges. A formula to compute the Tutte polynomial of a tensor product of graphs was originally given by Brylawski. This formula was recently generalized to coloured graphs and the generalized Tutte polynomial introduced by Bollobás and Riordan. In this paper we generalize the coloured tensor product formula to relative Tutte polynomials of relative graphs, containing zero edges to which the usual deletion/contraction rules do not apply. As we have shown in a recent paper, relative Tutte polynomials may be used to compute the Jones polynomial of a virtual knot.

scholarly journals TUTTE POLYNOMIALS OF TENSOR PRODUCTS OF SIGNED GRAPHS AND THEIR APPLICATIONS IN KNOT THEORY

2009 ◽  
Vol 18 (05) ◽  
pp. 561-589 ◽  
Author(s):  
Y. DIAO ◽  
G. HETYEI ◽  
K. HINSON
Keyword(s):  
Tensor Product ◽  
Knot Theory ◽  
Jones Polynomial ◽  
Tensor Products ◽  
Tutte Polynomial ◽  
Signed Graphs ◽  
Substitution Rule ◽  
Simple Substitution ◽  
Jones Polynomials ◽  
Tutte Polynomials

It is well-known that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman's Tutte polynomial of signed graphs for which describing the effect of taking a signed tensor product of signed graphs is very simple. We show that this Tutte polynomial of a signed tensor product of signed graphs may be expressed in terms of the Tutte polynomials of the original signed graphs by using a simple substitution rule. Our result enables us to compute the Jones polynomials of some large non-alternating knots. The combinatorics used to prove our main result is similar to Tutte's original way of counting "activities" and specializes to a new, perhaps simpler proof of the known formulas for the ordinary Tutte polynomial of the tensor product of unsigned graphs or matroids.

Relative Tutte Polynomials for Coloured Graphs and Virtual Knot Theory

2009 ◽  
Vol 19 (3) ◽  
pp. 343-369 ◽  
Author(s):  
Y. DIAO ◽  
G. HETYEI
Keyword(s):  
Knot Theory ◽  
Jones Polynomial ◽  
Tutte Polynomial ◽  
Ribbon Graph ◽  
Virtual Knot ◽  
Tree Expansion ◽  
The Face ◽  
Tutte Polynomials ◽  
Bracket Polynomial ◽  
Virtual Knot Theory

We introduce the concept of a relative Tutte polynomial of coloured graphs. We show that this relative Tutte polynomial can be computed in a way similar to the classical spanning tree expansion used by Tutte in his original paper on this subject. We then apply the relative Tutte polynomial to virtual knot theory. More specifically, we show that the Kauffman bracket polynomial (and hence the Jones polynomial) of a virtual knot can be computed from the relative Tutte polynomial of its face (Tait) graph with some suitable variable substitutions. Our method offers an alternative to the ribbon graph approach, using the face graph obtained from the virtual link diagram directly.

scholarly journals Colored Tutte polynomials and composite knots

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Gábor Hetyei ◽  
Yuanan Diao ◽  
Kenneth Hinson
Keyword(s):  
Tensor Product ◽  
Jones Polynomial ◽  
Product Formula ◽  
Ongoing Research ◽  
Colored Graphs ◽  
Virtual Knots ◽  
Nous Montrons ◽  
Produit Tensoriel ◽  
Tutte Polynomials ◽  
International Audience

International audience Surveying the results of three recent papers and some currently ongoing research, we show how a generalization of Brylawski's tensor product formula to colored graphs may be used to compute the Jones polynomial of some fairly complicated knots and, in the future, even virtual knots. En faisant une revue de trois articles récents et de la recherche en cours, nous montrons comment une généralisation aux graphes colorés de la formule de Brylawski sur le produit tensoriel peut être utilisée à calculer le polynôme de Jones de quelques nœuds et, dans la future, même de quelques nœuds virtuels, bien compliqués.

scholarly journals Graphs, Links, and Duality on Surfaces

2010 ◽  
Vol 20 (2) ◽  
pp. 267-287 ◽  
Author(s):  
VYACHESLAV KRUSHKAL
Keyword(s):  
Planar Graphs ◽  
Jones Polynomial ◽  
Tutte Polynomial ◽  
Natural Duality ◽  
Polynomial Invariant ◽  
Topological Duality ◽  
Ribbon Graphs ◽  
Virtual Links ◽  
Graphs On Surfaces ◽  
Tutte Polynomials

We introduce a polynomial invariant of graphs on surfaces,PG, generalizing the classical Tutte polynomial. Topological duality on surfaces gives rise to a natural duality result forPG, analogous to the duality for the Tutte polynomial of planar graphs. This property is important from the perspective of statistical mechanics, where the Tutte polynomial is known as the partition function of the Potts model. For ribbon graphs,PGspecializes to the well-known Bollobás–Riordan polynomial, and in fact the two polynomials carry equivalent information in this context. Duality is also established for a multivariate version of the polynomialPG. We then consider a 2-variable version of the Jones polynomial for links in thickened surfaces, taking into account homological information on the surface. An analogue of Thistlethwaite's theorem is established for these generalized Jones and Tutte polynomials for virtual links.

On the computational complexity of the Jones and Tutte polynomials

1990 ◽  
Vol 108 (1) ◽  
pp. 35-53 ◽  
Author(s):  
F. Jaeger ◽  
D. L. Vertigan ◽  
D. J. A. Welsh
Keyword(s):  
Computational Complexity ◽  
Jones Polynomial ◽  
Tutte Polynomial ◽  
Wide Range ◽  
Tutte Polynomials ◽  
Special Points ◽  
Alternating Link ◽  
Special Case

AbstractWe show that determining the Jones polynomial of an alternating link is #P-hard. This is a special case of a wide range of results on the general intractability of the evaluation of the Tutte polynomial T(M; x, y) of a matroid M except for a few listed special points and curves of the (x, y)-plane. In particular the problem of evaluating the Tutte polynomial of a graph at a point in the (x, y)-plane is #P-hard except when (x − 1)(y − 1) = 1 or when (x, y) equals (1, 1), (−1, −1), (0, −1), (−1, 0), (i, −i), (−i, i), (j, j2), (j2, j) where j = e2πi/3

scholarly journals A note on tensor products of reflexive algebras

2002 ◽  
Vol 66 (1) ◽  
pp. 25-31
Author(s):  
Zhe Dong
Keyword(s):  
Tensor Product ◽  
Simple Proof ◽  
Short Note ◽  
Tensor Products ◽  
Product Formula ◽  
Rank One ◽  
Reflexive Algebras ◽  
Subspace Lattices

In this short note, we obtain a concrete description of rank-one operators in Alg(ℒ1 ⊗…⊗ ℒn). Based on this characterisation, we give a simple proof of the tensor product formula: if Alg(ℒ1 ⊗…⊗ ℒn) is weakly generated by rank-one operators in itself and ℒi(i = 1,…,n) are subspace lattices.

scholarly journals Evaluations of Topological Tutte Polynomials

2014 ◽  
Vol 24 (3) ◽  
pp. 556-583 ◽  
Author(s):  
J. ELLIS-MONAGHAN ◽  
I. MOFFATT
Keyword(s):  
Tensor Product ◽  
General Framework ◽  
Tutte Polynomial ◽  
Ribbon Graph ◽  
Reduction Formula ◽  
Embedded Graphs ◽  
Chromatic Polynomials ◽  
Topological Transition ◽  
Combinatorial Interpretations ◽  
Tutte Polynomials

We find new properties of the topological transition polynomial of embedded graphs, Q(G). We use these properties to explain the striking similarities between certain evaluations of Bollobás and Riordan's ribbon graph polynomial, R(G), and the topological Penrose polynomial, P(G). The general framework provided by Q(G) also leads to several other combinatorial interpretations these polynomials. In particular, we express P(G), R(G), and the Tutte polynomial, T(G), as sums of chromatic polynomials of graphs derived from G, show that these polynomials count k-valuations of medial graphs, show that R(G) counts edge 3-colourings, and reformulate the Four Colour Theorem in terms of R(G). We conclude with a reduction formula for the transition polynomial of the tensor product of two embedded graphs, showing that it leads to additional relations among these polynomials and to further combinatorial interpretations of P(G) and R(G).

INFINITE TENSOR PRODUCTS OF SPATIAL PRODUCT SYSTEMS

2008 ◽  
Vol 11 (03) ◽  
pp. 447-465
Author(s):  
REMUS FLORICEL
Keyword(s):  
Tensor Product ◽  
Tensor Products ◽  
Product Formula ◽  
Countable Family ◽  
Type I ◽  
Product System ◽  
Product Systems ◽  
Normalized Units

The aim of this work is to define the infinite tensor product [Formula: see text] of a countable family {Ei}i ∈ 𝕀 of spatial product systems with respect to a family [Formula: see text] of normalized units, and to analyze the main properties of this construction. Among other things, we show that [Formula: see text] is an amenable product system, respectively a product system of type I, provided that every product system Ei is amenable, respectively of type I.

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

scholarly journals SUBPROJECTIVITY OF PROJECTIVE TENSOR PRODUCTS OF BANACH SPACES OF CONTINUOUS FUNCTIONS

2021 ◽  
pp. 1-14
Author(s):  
R.M. CAUSEY
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Tensor Products ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Spaces Of Continuous Functions

Abstract Galego and Samuel showed that if K, L are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is c0-saturated if and only if it is subprojective if and only if K and L are both scattered. We remove the hypothesis of metrizability from their result and extend it from the case of the twofold projective tensor product to the general n-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces K1, …, K n , $\widehat{\otimes}_{\pi, i=1}^n C(K_i)$ is c0-saturated if and only if it is subprojective if and only if each K i is scattered.

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