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< h1 id = "kmers-and-super-kmers" > Kmers and super-kmers</ h1 >
< h2 id = "kmers" > Kmers</ h2 >
< p > A < strong > kmer</ strong > is a DNA subsequence of fixed length k. Two constraints govern the choice of k:</ p >
< ul >
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< li >< strong > k ∈ [11, 31]</ strong > : the range ensures the kmer is long enough to be specific and short enough to fit in a single machine word (u64 at 2 bits/base requires k ≤ 32; k < 11 yields insufficient specificity).</ li >
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< li >< strong > k is odd</ strong > : an odd-length sequence cannot equal its own reverse complement (no palindromes). This guarantees that the canonical form < code > min(kmer, revcomp(kmer))</ code > is always strictly defined — the two orientations are always distinct — which is required for strand-independent counting.</ li >
</ ul >
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< p > Both constraints are < strong > enforced at CLI entry</ strong > by < code > CommonArgs::validate()</ code > in < code > superkmer</ code > and < code > index</ code > . Passing an invalid k exits immediately with an error message.</ p >
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< h2 id = "super-kmers" > Super-kmers</ h2 >
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< p > A < strong > super-kmer</ strong > is a maximal run of consecutive kmers from a DNA read, each overlapping the next by k− < strong > canonical minimizer</ strong > . The < strong > canonical minimizer</ strong > of a kmer is the m-mer (m < k) whose canonical hash < code > hash_kmer(min(m-mer, revcomp(m-mer)))</ code > is smallest over all m-mers in the kmer window. The hash function is a < code > mix64</ code > -based bijection; selection is purely hash-ordered with no degeneracy filter. A super-kmer is capped at 256 nucleotides; a longer run is split at that boundary.</ p >
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< h3 id = "canonical-super-kmers" > Canonical super-kmers</ h3 >
< p > A < strong > canonical super-kmer</ strong > is the lexicographic minimum of a super-kmer and its reverse complement:</ p >
< div class = "highlight" >< pre >< span ></ span >< code > canonical(super-kmer) = min(super-kmer, revcomp(super-kmer))
</ code ></ pre ></ div >
< p > 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.</ p >
< h3 id = "expected-length-of-a-super-kmer" > Expected length of a super-kmer</ h3 >
< p > For a random minimizer of length m over k-mers of length k, the density of minimizer positions is approximately 2/(k− & Shur 2025; Zheng < em > et al.</ em > 2020)< sup id = "fnref:Zheng2020-ji" >< a class = "footnote-ref" href = "#fn:Zheng2020-ji" > 2</ a ></ sup > < sup id = "fnref:Golan2025-xf" >< a class = "footnote-ref" href = "#fn:Golan2025-xf" > 3</ a ></ sup > , so the expected number of consecutive k-mers per super-kmer is (k− − </ p >
< div class = "arithmatex" > \[L_{\text{nt}} = \frac{k-m+2}{2} + k - 1\]</ div >
< p > For k=31, m=13: expected ≈ 40 nt. In practice super-kmers rarely exceed a few dozen nucleotides.< sup id = "fnref:superkmer_length" >< a class = "footnote-ref" href = "#fn:superkmer_length" > 1</ a ></ sup ></ p >
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< li id = "fn:superkmer_length" >
< p > The expected length formula and the density approximation 2/(k− < em > et al.</ em > 2020)< sup id = "fnref2:Zheng2020-ji" >< a class = "footnote-ref" href = "#fn:Zheng2020-ji" > 2</ a ></ sup > and (Golan & Shur 2025)< sup id = "fnref2:Golan2025-xf" >< a class = "footnote-ref" href = "#fn:Golan2025-xf" > 3</ a ></ sup > .  < a class = "footnote-backref" href = "#fnref:superkmer_length" title = "Jump back to footnote 1 in the text" > ↩ </ a ></ p >
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< li id = "fn:Zheng2020-ji" >
< p > Zheng, H., Kingsford, C. & Marçais, G. (2020). < a href = "https://doi.org/10.1093/bioinformatics/btaa472" > Improved design and analysis of practical minimizers</ a > . < em > Bioinformatics (Oxford, England)</ em > , 36, i119--i127.  < a class = "footnote-backref" href = "#fnref:Zheng2020-ji" title = "Jump back to footnote 2 in the text" > ↩ </ a >< a class = "footnote-backref" href = "#fnref2:Zheng2020-ji" title = "Jump back to footnote 2 in the text" > ↩ </ a ></ p >
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< li id = "fn:Golan2025-xf" >
< p > Golan, S. & Shur, A.M. (2025). < a href = "https://doi.org/10.1007/978-3-031-82670-2\_25" > Expected density of random minimizers</ a > . In: < em > Lecture notes in computer science</ em > , Lecture notes in computer science. Springer Nature Switzerland, Cham, pp. 347--360.  < a class = "footnote-backref" href = "#fnref:Golan2025-xf" title = "Jump back to footnote 3 in the text" > ↩ </ a >< a class = "footnote-backref" href = "#fnref2:Golan2025-xf" title = "Jump back to footnote 3 in the text" > ↩ </ a ></ p >
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