For the handlebody Hg of genus g, Przytycki studied the (Kauffman bracket) skein module 𝒮q(Hn#Hm) of the connected sum Hn#Hm at q. One of his results is that, in the case when 1 − qk is invertible for any k≠0, a homomorphism φ:𝒮q(Hn ⊔ Hm) →𝒮q(Hn#Hm) is an isomorphism, which is induced by a natural way. In this paper, in the case when n = m = 1, the ground ring is ℂ, and q ∈ ℂ is a 4k-th root of unity (k ≥ 2), we show that φ is not injective.

中文翻译：

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Kauffman 支架绞纱模块的连接总和和非注入性

用于把手HG属G, Przytycki 研究了（考夫曼支架）绞纱模块𝒮q(Hn#H米)连接总和的Hn#H米在q. 他的结果之一是，在这种情况下，1 - qķ是可逆的任何ķ≠0, 同态φ：𝒮q(Hn ⊔ H米) →𝒮q(Hn#H米)是同构，由自然方式诱导而来。在本文中，当n = 米 = 1，接地环为ℂ， 和q ∈ ℂ是一个4ķ- 统一的根 (ķ ≥ 2)，我们证明了φ不是单射的。