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On n-skein isomorphisms of graphs

1982
*
Journal of combinatorial theory. Series B (Print)
*

In this paper we show that for

doi:10.1016/0095-8956(82)90027-2
fatcat:nakqkwfb2ffytgka45allr76zu
*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). ...##
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A short proof and a strengthening of the Whitney 2-isomorphism theorem on graphs

1987
*
Discrete Mathematics
*

They proved that for

doi:10.1016/0012-365x(87)90236-6
fatcat:krukdv2o5jcuhc4is5xzun45gq
*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] ). ...##
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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*. ...##
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Graph skein modules and symmetry of spatial graphs
[article]

2009
*
arXiv
*
pre-print

In this paper, we compute the

arXiv:0911.3766v1
fatcat:bo2m4weozrbulalbkavj2ykicy
*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*). ...##
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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

2014
*
Discrete Mathematics
*

Several matroids can be defined

doi:10.1016/j.disc.2014.03.022
fatcat:vycxgg375bdtvosvi6e5ootthi
*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 ...##
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On 4-isomorphisms of graphs

1984
*
Discrete Mathematics
*

We show that an edge-bijection between 4-connected

doi:10.1016/0012-365x(84)90153-5
fatcat:koapa4wo4rennez6rs6uztznj4
*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*. ...##
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Skein Algebras of the solid torus and symmetric spatial graphs
[article]

2006
*
arXiv
*
pre-print

The proof

arXiv:math/0601403v1
fatcat:3imobtb7zvggbglhyyxy5yeb3a
*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. ...##
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Skein algebras of the solid torus and symmetric spatial graphs

2006
*
Fundamenta Mathematicae
*

The proof

doi:10.4064/fm190-0-1
fatcat:nngjysgnyraynclautwng5i6fe
*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. ...##
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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. ...##
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Relating stated skein algebras and internal skein algebras
[article]

2021
*
arXiv
*
pre-print

We prove excision properties

arXiv:2104.13848v2
fatcat:mrt6dgzyavcojf7qnx44asw5ou
*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*]. ...##
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Skein theories for virtual tangles
[article]

2020
*
arXiv
*
pre-print

In this paper, we use

arXiv:2008.04294v1
fatcat:em5yidd3tbhe3htb5sh75jmare
*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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