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Recently I came across this mathoverflow question, in which the number of homeomorphism classes of 3-dimensional lens spaces $L(p, q)$ is computed as a function of $p$. Using the OEIS, I found a connection to so-called generalized Petersen graphs: For $m\geq 3$ and $1\leq k <m$ relatively prime to $m$, we denote by $P(m, k)$ the graph on vertices $v_1,\dots, v_m, w_1, \dots, w_m$ and with edges $\{v_i, w_i\}, \{v_i, v_{i+1}\}$ and $\{w_i, w_{i+k}\}$ for $1\leq i \leq m$, where the indices are taken modulo $m$.

As was shown by Steimle and Staton, the graphs $P(m, k)$ and $P(m, l)$ are isomorphic if and only if $k\equiv\pm l \text{ mod } m$ or $kl\equiv\pm 1 \text{ mod } m$. The classic result about the homeomorphism classes of 3-dimensional lens spaces states that this condition on $l$ and $k$ is fulfilled if and only if the lens spaces $L(m, k)$ and $L(m, l)$ are homeomorphic. Hence we reach the suggestive conclusion $$P(m, k) \cong P(m, l) \Leftrightarrow L(m, k) \cong L(m, l).$$

On the one hand, I find that the proof for the classification of generalized Petersen graphs given in the paper lacks a conceptual approach. On the other hand the definition of a generalized Petersen graph really captures my intuition on how lens spaces are 'built'. This motivates my

Question: Is there a 'direct' structural correspondence between lens spaces and generalized Petersen graphs which allows to obtain one of the classification results from the other?

Recently I came across this mathoverflow question, in which the number of homeomorphism classes of 3-dimensional lens spaces $L(p, q)$ is computed as a function of $p$. Using the OEIS, I found a connection to so-called generalized Petersen graphs: For $m\geq 3$ and $1\leq k <m$ relatively prime to $m$, we denote by $P(m, k)$ the graph on vertices $v_1,\dots, v_m, w_1, \dots, w_m$ and with edges $\{v_i, w_i\}, \{v_i, v_{i+1}\}$ and $\{w_i, w_{i+k}\}$ for $1\leq i \leq m$, where the indices are taken modulo $m$.

As was shown by Steimle and Staton, the graphs $P(m, k)$ and $P(m, l)$ are isomorphic if and only if $k\equiv\pm l \text{ mod } m$ or $kl\equiv\pm 1 \text{ mod } m$. The classic result about the homeomorphism classes of 3-dimensional lens spaces states that this condition on $l$ and $k$ is fulfilled if and only if the lens spaces $L(m, k)$ and $L(m, l)$ are homeomorphic. Hence we reach the suggestive conclusion $$P(m, k) \cong P(m, l) \Leftrightarrow L(m, k) \cong L(m, l).$$

On the one hand, I find that the proof for the classification of generalized Petersen graphs given in the paper lacks a conceptual approach. On the other hand the definition of a generalized Petersen graph really captures my intuition on how lens spaces are 'built'. This motivates my

Question: Is there a 'direct' structural correspondence between lens spaces and generalized Petersen graphs which allows to obtain one of the classification results from the other?

Recently I came across this mathoverflow question, in which the number of homeomorphism classes of 3-dimensional lens spaces $L(p, q)$ is computed as a function of $p$. Using the OEIS, I found a connection to so-called generalized Petersen graphs: For $m\geq 3$ and $1\leq k <m$ relatively prime to $m$, we denote by $P(m, k)$ the graph on vertices $v_1,\dots, v_m, w_1, \dots, w_m$ and with edges $\{v_i, w_i\}, \{v_i, v_{i+1}\}$ and $\{w_i, v_{i+k}\}$ for $1\leq i \leq m$, where the indices are taken modulo $m$.

As was shown by Steimle and Staton, the graphs $P(m, k)$ and $P(m, l)$ are isomorphic if and only if $k\equiv\pm l \text{ mod } m$ or $kl\equiv\pm 1 \text{ mod } m$. The classic result about the homeomorphism classes of 3-dimensional lens spaces states that this condition on $l$ and $k$ is fulfilled if and only if the lens spaces $L(m, k)$ and $L(m, l)$ are homeomorphic. Hence we reach the suggestive conclusion $$P(m, k) \cong P(m, l) \Leftrightarrow L(m, k) \cong L(m, l).$$

On the one hand, I find that the proof for the classification of generalized Petersen graphs given in the paper lacks a conceptual approach. On the other hand the definition of a generalized Petersen graph really captures my intuition on how lens spaces are 'built'. This motivates my

Question: Is there a 'direct' structural correspondence between lens spaces and generalized Petersen graphs which allows to obtain one of the classification results from the other?

Recently I came across this mathoverflow question, in which the number of homeomorphism classes of 3-dimensional lens spaces $L(p, q)$ is computed as a function of $p$. Using the OEIS, I found a connection to so-called generalized Petersen graphs: For $m\geq 3$ and $1\leq k <m$ relatively prime to $m$, we denote by $P(m, k)$ the graph on vertices $v_1,\dots, v_m, w_1, \dots, w_m$ and with edges $\{v_i, w_i\}, \{v_i, v_{i+1}\}$ and $\{w_i, w_{i+k}\}$ for $1\leq i \leq m$, where the indices are taken modulo $m$.

As was shown by Steimle and Staton, the graphs $P(m, k)$ and $P(m, l)$ are isomorphic if and only if $k\equiv\pm l \text{ mod } m$ or $kl\equiv\pm 1 \text{ mod } m$. The classic result about the homeomorphism classes of 3-dimensional lens spaces states that this condition on $l$ and $k$ is fulfilled if and only if the lens spaces $L(m, k)$ and $L(m, l)$ are homeomorphic. Hence we reach the suggestive conclusion $$P(m, k) \cong P(m, l) \Leftrightarrow L(m, k) \cong L(m, l).$$

On the one hand, I find that the proof for the classification of generalized Petersen graphs given in the paper lacks a conceptual approach. On the other hand the definition of a generalized Petersen graph really captures my intuition on how lens spaces are 'built'. This motivates my

Question: Is there a 'direct' structural correspondence between lens spaces and generalized Petersen graphs which allows to obtain one of the classification results from the other?

Recently I came across this mathoverflow question, in which the number of homeomorphism classes of 3-dimensional lens spaces $L(p, q)$ is computed as a function of $p$. Using the OEIS, I found a connection to so-called generalized Petersen graphs: For $m\geq 3$ and $1\leq k <m$ relatively prime to $m$, we denote by $P(m, k)$ the graph on vertices $v_1,\dots, v_m, w_1, \dots, w_m$ and with edges $\{v_i, w_i\}, \{v_i, v_{i+1}\}$ and $\{w_i, v_{i+k}\}$ for $1\leq i \leq m$, where the indices are taken modulo $m$. As was shown by Steimle and Staton, the graphs $P(m, k)$ and $P(m, l)$ are isomorphic if and only if $k\equiv\pm l \text{ mod } m$ or $kl\equiv\pm 1 \text{ mod } m$. The classic result about the homeomorphism classes of 3-dimensional lens spaces states that this condition on $l$ and $k$ is fulfilled if and only if the lens spaces $L(m, k)$ and $L(m, l)$ are homeomorphic. Hence we reach the suggestive conclusion $$P(m, k) \cong P(m, l) \Leftrightarrow L(m, k) \cong L(m, l).$$

On the one hand, I find that the proof for the classification of generalized Petersen graphs given in the paper lacks a conceptual approach. On the other hand the definition of a generalized Petersen graph really captures my intuition on how lens spaces are 'built'. This motivates my

Question: Is there a 'direct' structural correspondence between lens spaces and generalized Petersen graphs which allows to obtain one of the classification results from the other?

Recently I came across this mathoverflow question, in which the number of homeomorphism classes of 3-dimensional lens spaces $L(p, q)$ is computed as a function of $p$. Using the OEIS, I found a connection to so-called generalized Petersen graphs: For $m\geq 3$ and $1\leq k <m$ relatively prime to $m$, we denote by $P(m, k)$ the graph on vertices $v_1,\dots, v_m, w_1, \dots, w_m$ and with edges $\{v_i, w_i\}, \{v_i, v_{i+1}\}$ and $\{w_i, v_{i+k}\}$ for $1\leq i \leq m$, where the indices are taken modulo $m$. 

As was shown by Steimle and Staton, the graphs $P(m, k)$ and $P(m, l)$ are isomorphic if and only if $k\equiv\pm l \text{ mod } m$ or $kl\equiv\pm 1 \text{ mod } m$. The classic result about the homeomorphism classes of 3-dimensional lens spaces states that this condition on $l$ and $k$ is fulfilled if and only if the lens spaces $L(m, k)$ and $L(m, l)$ are homeomorphic. Hence we reach the suggestive conclusion $$P(m, k) \cong P(m, l) \Leftrightarrow L(m, k) \cong L(m, l).$$

On the one hand, I find that the proof for the classification of generalized Petersen graphs given in the paper lacks a conceptual approach. On the other hand the definition of a generalized Petersen graph really captures my intuition on how lens spaces are 'built'. This motivates my

Question: Is there a 'direct' structural correspondence between lens spaces and generalized Petersen graphs which allows to obtain one of the classification results from the other?