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On n-punctured ball tangles | | Jae-Wook Chung
; Xiao-Song Lin
; | | Date: |
9 Feb 2005 | | Subject: | Geometric Topology | math.GT | | Abstract: | We consider a class of topological objects in the 3-sphere $S^3$ which will be called {it $n$-punctured ball tangles}. Using the Kauffman bracket at $A=e^{pi i/4}$, an invariant for a special type of $n$-punctured ball tangles is defined. The invariant $F$ takes values in $PM_{2 imes2^n}(mathbb Z)$, that is the set of $2 imes 2^n$ matrices over $mathbb Z$ modulo the scalar multiplication of $pm1$. This invariant leads to a generalization of a theorem of D. Krebes which gives a necessary condition for a given collection of tangles to be embedded in a link in $S^3$ disjointly. We also address the question of whether the invariant $F$ is surjective onto $PM_{2 imes2^n}(mathbb Z)$. We will show that the invariant $F$ is surjective when $n=0$. When $n=1$, $n$-punctured ball tangles will also be called spherical tangles. We show that $ ext{det} F(S)=0$ or 1 {
m mod} 4 for every spherical tangle $S$. Thus $F$ is not surjective when $n=1$. | | Source: | arXiv, math.GT/0502176 | | Services: | Forum | Review | PDF | Favorites | |
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