Let T be a pearl necklace consisting of the union of n consecutive tangent closed 3-balls \(B_i\) (\(i=1,2,\ldots , n\)) and consider the Kleininan group \(\Gamma _T\) generated by the reflections on the boundaries \(\partial B_i\). Let \(\Lambda (\Gamma _T)\) be a wild knot obtained as the limit set of \(\Gamma _T\) acting on the 3-sphere \(\mathbb {S}^3\). We say that a n-pearl necklace V consisting of the union of consecutive tangent closed 3-balls \(C_i\) (\(i=1,2,\ldots , n\)) is equivalent to T if there exists a homeomorphism \(\varphi :\mathbb {S}^3\rightarrow \mathbb {S}^3\) such that \(\varphi (V)=T\), \(\varphi (C_i)=B_i\). and \(\varphi (C_i\cap C_{i+1})=B_i\cap B_{i+1}\). In this paper, we prove that the Hausdorff dimension of \(\Lambda (\Gamma _T)\) varies continuously for equivalent pearl necklaces.

设T是由n个连续切线闭合 3 球\(B_i\) ( \(i=1,2,\ldots , n\) ) 的并组成的珍珠项链，并考虑 Kleininan 群\(\Gamma _T\ )由边界\(\partial B_i\)上的反射生成。设\(\Lambda (\Gamma _T)\)是一个野生结，作为作用在 3 球体\(\mathbb {S}^3\)上的\(\Gamma _T\)的极限集而获得。我们说由连续切线闭合 3 球\(C_i\) ( \(i=1,2,\ldots , n\) )的并集组成的n珍珠项链V等价于T如果存在同胚\(\varphi :\mathbb {S}^3\rightarrow \mathbb {S}^3\)使得\(\varphi (V)=T\) , \(\varphi (C_i) =B_i\)。和\(\varphi (C_i\cap C_{i+1})=B_i\cap B_{i+1}\)。在本文中，我们证明了\(\Lambda (\Gamma _T)\)的 Hausdorff 维数对于等效的珍珠项链是连续变化的。