I am running a cell-based screen by introducing a library of mutants into a reporter cell line and then sorting them via FACS. I have two conditions--control and treatment and cells from each condition are sorted into "top" and "bottom" pools based on the reporter activity. I have triplicates for each condition so 12 samples in total. Now I want to compare the difference in the ratio of top/bottom between the two conditions. Since top and bottom pools are derived from the same cell population for each replicate, I thought of adding a batch interaction term for pair-wise comparison.

coldata <- data.frame(
  c("sample1", "sample2", "sample3","sample4","sample5","sample6",
    "sample7", "sample8", "sample9","sample10","sample11","sample12"),
  batch = rep(c("1","2","3"),3),
  sort=rep(c("top","bottom"),6),
  condition=c(rep("control",6), rep("treatment",6))

  dds <- DESeqDataSetFromMatrix(countData = E,
                              colData = coldata,
                              design = ~ batch + sort + condition + sort:condition)

dds$condition <- relevel(dds$condition, ref="control")
dds$sort <- relevel(dds$sort, ref="bottom")

dds <- DESeq(dds)
resultsNames(dds)

The resultNames returns

[1] "Intercept" "batch_2_vs_1" "batch_3_vs_1" "sort_top_vs_bottom"
[5] "condition_treatment_vs_control" "sorttop.conditiontreatment"

Is my design correct and if so is sorttop.conditiontreatment what I am ultimately looking for?

Check the vignette section on interactions. If you need further guidance on how to interpret results columns, I'd recommend consulting with a local statistician or someone familiar with linear models in R.