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### On n-skein isomorphisms of graphs

R.L Hemminger, H.A Jung
1982 Journal of combinatorial theory. Series B (Print)
In this paper we show that for n ) 2, n-skein isomorphisms between 3-connected graphs having (n + I)-skeins are induced by isomorphisms.  ...  Sot. 42 (1967), 254-2561 generalized these results by showing that for n) 2, n-skein isomorphisms between (n + I)-connected graphs are induced by isomorphisms.  ...  If r, and rz are both on the same Pi, then application of the above to the two (n + 1)-skeins in S U R yields the claim. In the remaining cases R is a crossover S-jumper, say with ri E Pi (i = 1,2).  ...

### A short proof and a strengthening of the Whitney 2-isomorphism theorem on graphs

A.K Kelmans
1987 Discrete Mathematics
They proved that for n >t 3 an n-skein isomorphism of an (n + 1)-connected graph on a graph is induced by an isomorphism of the graphs.  ...  Is it true that any n-skein isomorphism of an n-connected graph is induced by an isomorphism of the graphs for n i> 3 and for G =/: K4 if n = 3?  ...  Using Theorem 6.4 we prove A short proof and a strengthening of the Whitney 2-isomorphism theorem on graphs 23 Theorem (A. Kelmans, 1981 [13] ).  ...

### Page 661 of Mathematical Reviews Vol. , Issue 89B [page]

1989 Mathematical Reviews
If a one-to-one map f:E(G) — E(H) has the property that f and f~' preserve edge- sets of k-skeins then / is called a k-skein isomorphism of the graph G onto the graph H.  ...  However, for k > 3, they left open the question of k-skein isomorphisms for k- connected graphs.  ...

### Graph skein modules and symmetry of spatial graphs [article]

Nafaa Chbili
2009 arXiv   pre-print
In this paper, we compute the graph skein algebra of the punctured disk with two holes.  ...  Then, we apply the graph skein techniques developed here to establish necessary conditions for a spatial graph to have a symmetry of order p, where p is a prime.  ...  Since the multiplication depends on the product structure on M, then we will denote the skein algebra of F g,n × I by Y(F g,n ).  ...

### Page 441 of Mathematical Reviews Vol. , Issue 85b [page]

1985 Mathematical Reviews
In the present paper it is proved that for n > 2 any n-skein isomorphism of 3-connected graphs is induced by an isomorphism, provided that each of the graphs contains at least one (n+1)-skein.  ...  On 3-skein isomorphisms of graphs. Combinatorica 2 (1982), no. 4, 373-376. An n-skein in a graph G consists of n internally disjoint paths connecting two vertices.  ...

### Page 2290 of Mathematical Reviews Vol. , Issue 83f [page]

1983 Mathematical Reviews
A principal lemma in the proof of this theorem asserts that, for n>2, an n-skein isomorphism of two graphs G and G* is also an (n+ 1)-skein isomorphism of G and G*.  ...  Math. 54 (1932), 150-158; Zbl 3, 328] by proving that if G and G* are (n+ 1)-connected, where n>2, then any n-skein isomorphism of G and G* is induced by an isomorphism of the two graphs.  ...

### Page 4823 of Mathematical Reviews Vol. , Issue 83m [page]

1983 Mathematical Reviews
In the present paper it is proved that for n>2 any n-skein isomorphism of 3-connected graphs is induced by an isomorphism, provided that each of the graphs in question contains at least one (n+ 1)-skein  ...  A. 83m:05090 On n-skein isomorphisms of graphs. J. Combin. Theory Ser. B 32 (1982), no. 2, 103-111.  ...

### Page 1366 of Mathematical Reviews Vol. 34, Issue 6 [page]

1967 Mathematical Reviews
Two graphs G and H are n-skein isomorphic if there exists a one-to-one mapping between the edges of G and H such that every n-skein of G is mapped onto an n-skein of H and con- versely.  ...  Note on isomorphisms of graphs. J. London Math. Soc. 42 (1967), 254-256. A set of n paths joining two points is an n-skein if no two of the paths have any inner points in common.  ...

### Bicircular signed-graphic matroids

Vaidy Sivaraman
2014 Discrete Mathematics
Several matroids can be defined on the edge set of a graph.  ...  A theorem of Matthews from late 1970s gives a characterization of graphs whose bicircular matroids are graphic. We give a characterization of graphs whose bicircular matroids are signed-graphic.  ...  Luis Goddyn gave an excellent talk on the excluded minors for the class of bicircular matroids at the Third Workshop on Graphs and Matroids at Maastricht (organized by Bert Gerards), and that was the starting  ...

### On 4-isomorphisms of graphs

G Lassmann
1984 Discrete Mathematics
We show that an edge-bijection between 4-connected graphs preserving homeomorphs of K a in both directions is induced by an isomorphism.  ...  Halin and Jung [1] generalized Whitney's result to n-skein preserving bijections, where an n-skein is a graph spanned by n openly disjoint paths between two vertices.  ...  As a first step of their proof they show that each 3-skein isomorphism is a 4-isomorphism. Note that our result does not hold for 3-connected graphs.  ...

### Skein Algebras of the solid torus and symmetric spatial graphs [article]

Nafaa Chbili
2006 arXiv   pre-print
The proof of the main result is based on computing the Yamada skein algebra of the solid torus then proving that this algebra injects into the Kauffman bracket skein algebra of the solid torus.  ...  We use the topological invariant of spatial graphs introduced by S. Yamada to find necessary conditions for a spatial graph to be periodic with a prime period.  ...  This ends the proof of part (1) of Theorem 1.1 Similar arguments are used to prove the second part of Theorem 1.1.  ...

### Skein algebras of the solid torus and symmetric spatial graphs

Nafaa Chbili
2006 Fundamenta Mathematicae
The proof of the main result is based on computing the Yamada skein algebra of the solid torus and then proving that it injects into the Kauffman bracket skein algebra of the solid torus. 2000 Mathematics  ...  We use the topological invariant of spatial graphs introduced by S. Yamada to find necessary conditions for a spatial graph to be periodic with a prime period.  ...  Finally, the skein algebra of the solid torus is isomorphic to the polynomial algebra R[z]. This ends the proof of Theorem 3.2.  ...

### Page 2276 of Mathematical Reviews Vol. , Issue 85f [page]

1985 Mathematical Reviews
As a first step in their proof they showed that each 3-skein isomorphism is a 4-isomorphism (it and its inverse preserve homeomorphs of K4).  ...  (D-TUB) 85f:05106 On 4-isomorphisms of graphs. Discrete Math. 49 (1984), no. 1, 79-81. Starting with H. Whitney’s paper [Amer. J.  ...

### Relating stated skein algebras and internal skein algebras [article]

Benjamin Haïoun
2021 arXiv   pre-print
We prove excision properties of multi-edges internal skein algebras using excision properties of skein categories, and agreeing with excision properties of stated skein algebras when 𝒱 = 𝒰_q^2(SL_2)–  ...  Stated skein algebras are defined on surfaces with multiple boundary edges and we generalise internal skein algebras in this context.  ...  One has to choose an isomorphism from each object of Ribbon V (R 2 ) to one of the form [n].  ...

### Skein theories for virtual tangles [article]

Joshua R. Edge
2020 arXiv   pre-print
In this paper, we use skein-theoretic techniques to classify all virtual knot polynomials and trivalent graph invariants with certain smallness conditions.  ...  In the second half of the paper, we classify all non-trivial invariants of (perhaps non-planar) trivalent graphs coming from symmetric trivalent categories.  ...  In this project, we care about a particular class of graphs, which we call * n : Definition 2.3.4. * n is the complete bipartite graph with U − = { } and U + = S, a set of n even vertices.  ...
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