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# 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), with the constraint that non-degenerate m-mers are always preferred over degenerate ones. A degenerate m-mer is one composed of a single repeated nucleotide (all-A, all-C, all-G, or all-T); such m-mers are selected only if no non-degenerate candidate exists in the window.

### 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) (Golan & Shur 2025; Zheng et al. 2020)2 3, 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.1

1. The expected length formula and the density approximation 2/(k−m+2) should be verified against the values reported in (Zheng et al. 2020)2 and (Golan & Shur 2025)3. ↩

2. Zheng, H., Kingsford, C. & Marçais, G. (2020). Improved design and analysis of practical minimizers. Bioinformatics (Oxford, England), 36, i119--i127. ↩↩

3. Golan, S. & Shur, A.M. (2025). Expected density of random minimizers. In: Lecture notes in computer science, Lecture notes in computer science. Springer Nature Switzerland, Cham, pp. 347--360. ↩↩