Encoded diagram size and exact output

Let $D$ be a nonempty finite regular oriented link diagram: a smooth immersion of a finite union of oriented circles into the plane with finitely many transverse double points, no triple points, and one overpassing branch specified at each double point. Let $c(D)$ be its number of crossings and $m(D)$ its number of components. An arc is a connected strand segment cut at each undercrossing; each component with no undercrossings contributes one closed arc. Write $a(D)$ for the number of arcs. If $q(D)$ components have no undercrossings, then $a(D)=c(D)+q(D)$ and $0\le q(D)\le m(D)$.

A supplied finite code for $D$ has code length $L$, the total number of bits in that literal code, including its tags, signs, labels, and separators. The length depends on the chosen encoding. In particular, for one of the tagged Gauss, signed Dowker–Thistlethwaite, or oriented planar-diagram formats, $L$ is the bit length of the supplied valid code in that format. The procedures in this page take typed mathematical diagrams or valid codes as stated; no string parser or malformed-code behavior is included in any operation count.

For an integer $z$, define its signed-magnitude bit length by ${\mathrm{\ell }}_{\mathbb{Z}}(0)=1$ and ${\mathrm{\ell }}_{\mathbb{Z}}(z)=1+\lceil {\log }_2(\mathrm{\mid }z\mathrm{\mid }+1)\rceil$ for $z\mathrm{\ne }0$. For a Laurent polynomial $P(A)={\sum }_{j\in \mathbb{Z}}{z}_j{A}^j$ with finite support, define its coefficient-bit size as $0$ if $P=0$, and otherwise as ${\max }_{j:z_j\mathrm{\ne }0}{\mathrm{\ell }}_{\mathbb{Z}}(z_j)$. Its absolute Laurent degree is ${\max }_{j:z_j\mathrm{\ne }0}\mathrm{\mid }j\mathrm{\mid }$, with value $0$ for $P=0$. For a prime $p$, elements of ${\mathbb{F}}_p$ are represented exactly by residues $0,\dots ,p-1$, and their representation uses $\lceil {\log }_{2}p\rceil$ bits.

An exact output is the complete integer, finite list, or Laurent polynomial with its integer coefficients and exponents. A floating-point approximation is not an exact output. A field-operation count treats each addition, subtraction, multiplication, division by a nonzero element, and equality test in ${\mathbb{F}}_p$ as one operation; this convention does not assert that these operations have constant bit cost as $p$ varies.

Regular oriented link diagrams

Quandles and knot-diagram colorings

Kauffman bracket smoothing and loop value

Planar-diagram, Gauss, and signed Dowker–Thistlethwaite codes

Calegari, Introduction to Knot Theory (2025), §§1.2–1.3 and 4.1

Roberts, Knots Knotes (1999), §§3.3–3.4 and 4.3