On genus 2 Heegaard diagrams for the 3-sphere
Takeshi Kaneto · Transactions of the American Mathematical Society · 1983
Let D D be any genus 2 2 Heegaard diagram for the 3 3 -sphere and ⟨ a 1 , a 2 ; r ~ 1 , r ~ 2 ⟩ \left \langle {{a_1}, {a_2}; {{\tilde r}_1}, {{\tilde r}_2}} \right \rangle be the cyclically reduced presentation associated with D D . We shall show that r ~ 1 {{\tilde {r}}_1} contains r ~ 2 {{\tilde {r}}_2} or r ~ 2 − 1 {\tilde {r}}_2^{-1} as a subword in cyclic sense if { r ~ 1 , r ~ 2 } ≠ { a 1 ± 1 , a 2 ± 1 } \left \{{\tilde r}_1, {\tilde r}_2 \right \} e \left \{{a_1}^{\pm 1}, {a_2}^{\pm 1} \right \} holds, and that, using this property, ⟨ a 1 , a 2 ; r 1 , r 2 ⟩ \left \langle {a_1}, {a_2};{r_1}, {r_2} \right \rangle can be transformed to the trivial one ⟨ a 1 , a 2 ; a 1 ± 1 , a