# Kmers and super-kmers

## Kmers

A **kmer** is a DNA subsequence of fixed length k. Two constraints govern the choice of k:

- **k ∈ [11, 31]**: the range ensures the kmer is long enough to be specific and short enough to fit in a single machine word.
- **k is odd**: an odd-length sequence cannot equal its own reverse complement (no palindromes). This guarantees that the canonical form `min(kmer, revcomp(kmer))` is always strictly defined — the two orientations are always distinct — which is required for strand-independent counting.

## Super-kmers

A **super-kmer** is a maximal run of consecutive kmers from a DNA read, each overlapping the next by k−1 nucleotides. Each kmer of the run carries the same **canonical minimizer**. The **canonical minimizer** of a kmer is the smallest value of `min(m-mer, revcomp(m-mer))` over all m-mers within the kmer (m < k, m odd).

### Canonical super-kmers

A **canonical super-kmer** is the lexicographic minimum of a super-kmer and its reverse complement:

```
canonical(super-kmer) = min(super-kmer, revcomp(super-kmer))
```

When a read and its reverse-complement are both sequenced, they produce super-kmers that are reverse complements of each other. Both map to the same canonical form: the same genomic region is represented by a single canonical super-kmer regardless of which strand was read.

### Expected length of a super-kmer

For a random minimizer of length m over k-mers of length k, the density of minimizer positions is approximately 2/(k−m+2) [@Zheng2020-ji; @Golan2025-xf], so the expected number of consecutive k-mers per super-kmer is (k−m+2)/2. A run of n k-mers spans n + k − 1 nucleotides, giving:

$$L_{\text{nt}} = \frac{k-m+2}{2} + k - 1$$

For k=31, m=13: expected ≈ 40 nt. In practice super-kmers rarely exceed a few dozen nucleotides.[^superkmer_length]

[^superkmer_length]: The expected length formula and the density approximation 2/(k−m+2) should be verified against the values reported in [@Zheng2020-ji] and [@Golan2025-xf].