query: ${\pi_1(\mathbb{R}^3 \setminus A)}$ where $A$ is a circle in ${\mathbb{R}^3}$
positive_passages.docid: 151735
positive_passages.text: $\mathbb{R}^3 \setminus A$ deformation retracts onto $S^1 \vee S^2$
negative_passages.docid: 113915
negative_passages.text: A star shaped open set in $\mathbb{R}^n$ is diffeomorphic to $\mathbb{R}^n$
negative_passages.docid: 111898
negative_passages.text: Is $\{ (e^{2\pi i a n},e^{2\pi i b n }) : n \in \mathbb{Z} \}$ dense in the torus, where $a,b$ irrationals such that $a/b$ irrational?
negative_passages.docid: 14403
negative_passages.text: If $n = p_1^{a_1} ... p_r^{a_r}$ the set of nilpotents of $\mathbb{Z}_n$ is $(p_1 p_2 ... p_{r-1} p_r)\Bbb Z_n$
negative_passages.docid: 11000
negative_passages.text: Fundamental group of $\mathbb{R}^2 \setminus (\mathbb{Z} \times \{0\})$
negative_passages.docid: 31710
negative_passages.text: Show that $\mathcal{J}=\{[a, a+1) \mid a \in \mathbb{R}\}$ generates the Borel set of $\mathbb{R}$
negative_passages.docid: 23721
negative_passages.text: Prove that the Gaussian Integer's ring is a Euclidean domain
negative_passages.docid: 1026
negative_passages.text: Ring homomorphism with $\phi(1_R) \neq1_S$
negative_passages.docid: 14447
negative_passages.text: Direct proof that $\pi$ is not constructible
negative_passages.docid: 41191
negative_passages.text: How to prove this $\pi$ formula?
negative_passages.docid: 27629
negative_passages.text: How to prove that $S^{2n+1}/S^1$ is homeomorphic to $\mathbb CP^n$ under a given identification
negative_passages.docid: 47975
negative_passages.text: Closed cone in the euclidean space $\mathbb{R}^n$
negative_passages.docid: 131185
negative_passages.text: For fixed $z_i$s inside the unit disc, can we always choose $a_i$s such that $\left|\sum_{i=1}^n a_iz_i\right|<\sqrt3$?
negative_passages.docid: 148233
negative_passages.text: Proof that every circle has the ratio of $\pi$
negative_passages.docid: 55070
negative_passages.text: Fundamental group of complement of $n$ lines through the origin in $\mathbb{R}^3$
negative_passages.docid: 158364
negative_passages.text: Are there uncountably many $A\in M_3 (\mathbb {R})$ such that $A^8=I $?
negative_passages.docid: 39256
negative_passages.text: Finding $\det(I+A^{100})$ where $A\in M_3(R)$ and eigenvalues of $A$ are $\{-1,0,1\}$