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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>
<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.</li>
<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>
<h2 id="super-kmers">Super-kmers</h2>
<p>A <strong>super-kmer</strong> 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 <strong>canonical minimizer</strong>. The <strong>canonical minimizer</strong> of a kmer is the smallest value of <code>min(m-mer, revcomp(m-mer))</code> over all m-mers within the kmer (m &lt; k, m odd).</p>
<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−m+2) (Golan &amp; 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−m+2)/2. A run of n k-mers spans n + k − 1 nucleotides, giving:</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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<hr />
<ol>
<li id="fn:superkmer_length">
<p>The expected length formula and the density approximation 2/(k−m+2) should be verified against the values reported in (Zheng <em>et al.</em> 2020)<sup id="fnref2:Zheng2020-ji"><a class="footnote-ref" href="#fn:Zheng2020-ji">2</a></sup> and (Golan &amp; Shur 2025)<sup id="fnref2:Golan2025-xf"><a class="footnote-ref" href="#fn:Golan2025-xf">3</a></sup>.&#160;<a class="footnote-backref" href="#fnref:superkmer_length" title="Jump back to footnote 1 in the text">&#8617;</a></p>
</li>
<li id="fn:Zheng2020-ji">
<p>Zheng, H., Kingsford, C. &amp; 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.&#160;<a class="footnote-backref" href="#fnref:Zheng2020-ji" title="Jump back to footnote 2 in the text">&#8617;</a><a class="footnote-backref" href="#fnref2:Zheng2020-ji" title="Jump back to footnote 2 in the text">&#8617;</a></p>
</li>
<li id="fn:Golan2025-xf">
<p>Golan, S. &amp; 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.&#160;<a class="footnote-backref" href="#fnref:Golan2025-xf" title="Jump back to footnote 3 in the text">&#8617;</a><a class="footnote-backref" href="#fnref2:Golan2025-xf" title="Jump back to footnote 3 in the text">&#8617;</a></p>
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