Algorithm for finding symplectic fillings of links of quotient surface singularities of Type  $(3,1)$

As mentioned above, corresponding to each link of a quotient surface singularity of Type  $(3,1)$ there is a standard compactifying divisor $K$ consisting of a cuspidal rational curve $D$ , a string of rational curves $C_{1},\ldots ,C_{k}$ and two more rational curves $A$ and $B$ . For our purposes it is sufficient to record the self-intersection numbers $d=D\cdot D,-c_{1}=C_{1}\cdot C_{1},\ldots ,-c_{k}=C_{k}\cdot C_{k}$ . We will write $Q=Q_{K}=(d,-c_{1},\ldots ,-c_{k})$ . We handle Case I and Case II symplectic fillings of the singularity link associated with the data $Q$ separately. For Case I fillings, by Proposition 4.8, it is sufficient to list the numerical data associated to all preadmissible configurations $A^{\prime }\cup B^{\prime }\cup D^{\prime }\cup C_{1}^{\prime }\cup \cdots \cup C_{k}^{\prime }$ such that $D^{\prime }\cdot D^{\prime }=D\cdot D$ and $C_{i}^{\prime }\cdot C_{i}^{\prime }\geqslant C_{i}\cdot C_{i}$ for all $i$ . For Case II fillings, by assumption and Proposition 4.10, first we have to identify the curves $C_{i}$ and $C_{j}$ such that there are exceptional curves $E$ and $F$ intersecting $B$ and $C_{i}$ and $D$ and $C_{j}$ , respectively, in some rational surface $R$ containing the compactifying divisor $K$ . Blowing down the curves $E$ and $F$ and denoting the resulting configuration $A^{\prime }\cup B^{\prime }\cup D^{\prime }\cup C_{1}^{\prime }\cup \cdots \cup C_{k}^{\prime }$ , by Proposition 4.10, it is then sufficient to find all preadmissible configurations $A^{\prime \prime }\cup B^{\prime \prime }\cup D^{\prime \prime }\cup C_{1}^{\prime \prime }\cup \cdots \cup C_{k}^{\prime \prime }$ such that $D^{\prime \prime }\cdot D^{\prime \prime }=D^{\prime }\cdot D^{\prime }$ and $C_{i}^{\prime \prime }\cdot C_{i}^{\prime \prime }\geqslant C_{i}^{\prime }\cdot C_{i}^{\prime }$ for all $i$ . (Note that in [Reference Bhupal and Ono2], we did not define preadmissible configurations for Case II symplectic fillings of links of tetrahedral, octahedral or icosahedral singularities of Type  $(3,1)$ , however a definition similar to the other cases can be given.)

In the algorithm above, we assume that the compactifying divisor $K$ associated with the data $Q$ sits in some rational surface $R$ . Collections of exceptional curves $\{E_{\unicode[STIX]{x1D6FC}}\}$ in $R$ that are disjoint from $D\cup A\cup B$ and such that $E_{\unicode[STIX]{x1D6FC}}\cdot \bigcup C_{i}=1$ for each $\unicode[STIX]{x1D6FC}$ will be recorded by the data $S=(e_{1},\ldots ,e_{k})$ , where $e_{i}$ denotes the number of exceptional curves intersecting $C_{i}$ . Thus to list all possible Case I fillings of the singularity link associated to $Q$ , it is sufficient to enumerate all possible $k$ -tuples $S$ such that on blowing down a collection of exceptional curves represented by $S$ , $K$ is transformed into a preadmissible configuration. Similarly, to list all possible Case II fillings of the singularity link associated to $Q$ , it is sufficient to list all possible numerical data $(i,j)$ and $S$ such that on blowing down $E$ and $F$ and a collection of exceptional curves represented by $S$ , $K$ is transformed into a preadmissible configuration.