多面体亏格修改(2)

Where for each v L(G) ,
d (v ) 3
简要证明
Seifert 环数增加1或减少1
因此
n 2 p s( L(G)) n 2 p
e1
e2
ek
引理2:
p
G
If G is obtained from C n by adding a new point then
n If e1 , e2 ,, e pis a set of p ( p 2 ) chords
of C n ，let G be a graph obtained by adding
.
p
chords to C n . Then n 2 p s( L(G)) n 2 p
（1）Biblioteka （2）（3）如果一个具有B个边界的定向曲面的亏 格为n，则它同胚于去掉B个圆盘的n亏格环面。
Genus of links
Definition: The genus of an oriented link L is the minimum genus of any connected orientable surface that spans L. The genus of an unoriented link is the minimum taken over all possible choices of orientation. We denote the genus of L by g (L )
i 1 i 1 k
i 1
k
c( L(G )) c( L(Gi ))
k i 1
(G ) (G) (k 1)
i 1 i
k
so s( L(G )) 1 k s( L(Gi ))
For each Gi ,take out a cycle C p from Gi Assume v1 ,, v V ( Gi ) p C p
2、For a polyhedral link diagram L( D)
c( L( D)) g ( L) 1 6
Where c( L( D)) is the number of crossings.
3、The Braid index of polyhedral links equals
c( L( D)) 2 6
DNA cube
DNA-Truncated octahedron
Seeman synthesize them in 1991 and 1994 ,separately.
Bacteriophage HK97
Wen-Yuan Qiu and Xin-Dong Zhai, Molecular design of Goldberg polyhedral links J. Mol. Struct. (Theochem), 756 (2005) 163–166
最小Seifert 环数 4 6 7 8
亏格 1 3 4 5
十二面体链 环 二十面体链 环
32 56
12 20
9 17
The main conclusions 1、For a polyhedral link diagram L( D)
c( L( D)) c( L( D)) 2 s( L( D)) 2 6 2
三叶结覆盖
G
L(G )
(Seifert’s algorithm) Every link bounds an orientable surface.
So, polyhedral links have orientable surface. call it Seifert surface.
Seifert surface of trefoil knot
3
A link is a finite disjoint union of knots: L K1 K n
Borremean Rings Hopf Links with Hopf Links with with three two components ten components components.
Then
2 V (G) 2k E (G) s( L(G)) 2 E (G)
定理1:
For a polyhedral link diagram
L(D)
c( L( D)) c( L( D)) 2 s( L( D)) 2 6 2
Where c( L( D)) is the number of crossings of L(D) .
Then
c( L( D)) g ( L) 1 6
Braid
辫子
闭辫子
Theorem :
[Yamada,1987] The minimal number of Seifert circles equals the braid index of a link.
ˆ ˆ ˆ b1 , b2 , , bk
Construction of Polyhedral Links
The method for ‘three-cross-curve and double-line covering’.
对于任意的连通简单图G ,可以通过上面三种操作 变成链环图 L(G), 其中:△(G)=3
我们称 L(G) 为三叶结覆盖链环图
定理2:
For a polyhedral link diagram
L(D)
c( L( D)) g ( L) 1 6
Where c( L( D)) is the number of crossings of L(D) .
Proof: Because
2 g 2 s ( L( D)) c( L( D)) ( L( D)) c( L( D )) 2 2 c( L( D)) ( L( D )) 2 c( L( D)) ( L( D)) 2 However 6
Seifert surface of hopf link
Seifert surface
Seifert circle
根据 seifert 代数,对于无向链环,各个分 支的走向不同, seifert 环数也不同.
纽结或链环和亏格的联系？
什么是亏格？
亏格0
亏格1 曲面的亏格就是环柄的数目
亏格2
Then
2 s( L(G )) c( L(G )) ( L(G )) 2k s( L(Gi )) c( L(Gi )) ( L(Gi ))
i 1 i 1 i 1 k k k
However
k
( L ) ( L # # L ) (k 1)
p with at least 3 neighbors in G
,
n 4 d ( p) s( L(G)) n 2 d ( p)
e1
e2
ek
Proof:
p
G
e1
ek
e2
Let e1 , e2 ,, ek (k 3) is adjacent to p . s( L(G)) s( L(G))
因此
g ( L(G)) 1
max s( L(G)) 8
2 g ( L(G )) 2 s ( L(G )) c( L(G )) ( L(G )) 2 8 12 4 2
所以
g ( L(G)) 1
最大Seifert 环数 四面体链环 六面体链环 七面体链环 八面体链环 8 14 17 20
Proof: Because polyhedron is 3-regual,so
3n 2e
By lemma 4
2 V ( D) 2k E ( D) s( L( D)) 2 E ( D)
Then
c( L( D)) c( L( D)) 2 s( L( D)) 2 6 2
G
s( L(G)) s( L(G))
Then e1 , e2 ,, ek 1 is d ( p) 1 chords of Cn1 . So,
ek e 2 e1
n 4 d ( p) s( L(G)) n 2 d ( p)
G
p
引理3：
对于三叶结覆盖链环图 L(G )
Borremean links
1
2
3
4
5
6
7
8
令G为连通简单图，L(G)是从G得到 的三叶结覆盖链环图，其中△(G)=3。
The genus of a projection surface F constructed from a connected diagram satisfies
2 g ( F ) [1 s( L(G)) c( L(G))] [1 ( L(G))]
相信他正在创造一个元素表。1887年迈克耳 逊－莫雷实验证明“以太”不存在，“以太 漩涡论”成为过时理论。十九世纪末，产生 拓扑学，纽结理论再此成为热点研究课题。 今日纽结理论的应用包括弦理论、DNA复制 和统计力学等领域。
A knot K R is a subset of points homeomorphic to a circle
ˆ ˆ ˆ b( L) min{n(b1 ), n(b2 ),, n(bk )}
The braid index of a link is the least number of strings in a braid corresponding to a closed-braid representation of the link.
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