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

Let $P$ be the projection plane in an oriented $3$-sphere and let $n$ be the unit viewing normal pointing toward the viewer. Orient $P$ so that $(u,v)$ is a positive basis exactly when $(n,u,v)$ is a positive ambient basis. A crossing code has a finite set $C$ of crossing labels and a finite set $D$ of darts, where a dart is one directed half-edge incident to a crossing. A fixed-point-free involution $\alpha :D\to D$ pairs the two ends of each edge: ${\alpha }^2=\mathrm{i}\mathrm{d}$ and $\alpha (d)\mathrm{\ne }d$. A vertex map $v:D\to C$ records the crossing incident to each dart. A rotation system is the permutation $\sigma$ whose cycles give the clockwise order of darts at each crossing. Each crossing has four darts; the two over-strand darts and the two under-strand darts are opposite in that cyclic order. An arrow on each dart records whether its edge is directed into or out of its incident crossing. The pairing $\alpha$ must join one incoming end to one outgoing end. At a crossing, each opposite pair has one incoming and one outgoing dart, so following an edge and then the opposite dart defines the oriented link components. A code is connected when the graph with vertices $C$ and edges the $\alpha$-pairs is connected.

The sign of an oriented crossing is $+1$ or $-1$ according as $\det (v_{\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}},v_{\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}})$ is positive or negative, where the two tangent vectors follow the oriented over- and under-strands. This item fixes the local cyclic-order convention as follows. Starting with the outgoing over-dart and reading counterclockwise, a positive crossing has order $(O_{\mathrm{o}\mathrm{u}\mathrm{t}},U_{\mathrm{o}\mathrm{u}\mathrm{t}},O_{\mathrm{i}\mathrm{n}},U_{\mathrm{i}\mathrm{n}})$; a negative crossing has order $(O_{\mathrm{o}\mathrm{u}\mathrm{t}},U_{\mathrm{i}\mathrm{n}},O_{\mathrm{i}\mathrm{n}},U_{\mathrm{o}\mathrm{u}\mathrm{t}})$. The stored crossing sign and cyclic order must agree. This is the page's explicit sign convention. It agrees with the current KT-03 supplier: reversing its positive counterclockwise order from outgoing over gives the clockwise record $(O_{\mathrm{o}\mathrm{u}\mathrm{t}},U_{\mathrm{i}\mathrm{n}},O_{\mathrm{i}\mathrm{n}},U_{\mathrm{o}\mathrm{u}\mathrm{t}})$.

For a nonempty connected crossing graph, let $V=\mathrm{\mid }C\mathrm{\mid }$, $E=\mathrm{\mid }D\mathrm{\mid }\mathrm{/}2$, and let $F$ be the number of cycles of the face permutation $\varphi =\alpha \circ \sigma$. A spherical code is a code satisfying $V-E+F=2$. A spherical code becomes a plane code by distinguishing one face as the outer face and deleting a point in that face from the sphere. The crossingless one-component code is a separate case: its Gauss data are one empty component word, its DT pairing and crossing-sign vector are empty at $c=0$, and its PD data are the single loop Loop[1]; every format also carries an orientation marker, ccw or cw, for the unit circle $S^1=\{(x,y)\in {\mathbb{R}}^2:x^2+y^2=1\}$ in the oriented plane, and the outer-face marker is outside. Thus the crossingless code records which direction orients its circle. It has no crossing graph or face-permutation test. Disconnected crossing graphs and disconnected crossingless links are outside this code class.

A normalized planar-diagram code with $c>0$ has crossing labels $C=\{1,\dots ,c\}$ and four dart positions $0,1,2,3$ at each crossing, in clockwise order beginning with the outgoing over-dart. The opposite pair $(0,2)$ is over and $(1,3)$ is under. For sign $+1$, the clockwise record is $(O_{\mathrm{o}\mathrm{u}\mathrm{t}},U_{\mathrm{i}\mathrm{n}},O_{\mathrm{i}\mathrm{n}},U_{\mathrm{o}\mathrm{u}\mathrm{t}})$; for sign $-1$ it is $(O_{\mathrm{o}\mathrm{u}\mathrm{t}},U_{\mathrm{o}\mathrm{u}\mathrm{t}},O_{\mathrm{i}\mathrm{n}},U_{\mathrm{i}\mathrm{n}})$. Edge labels are assigned by scanning crossings in increasing order and their four dart positions in order, giving the next positive integer to the pair containing the first not-yet-labelled dart. The outer face is recorded by its least dart in its face cycle. These rules remove arbitrary edge-name and face-name choices; different crossing labelings may still represent the same diagram. At $c=0$ a normalized code is the crossingless record with one of the two orientation markers above.

A Gauss code for an oriented diagram records, in traversal order on each oriented component, a signed crossing label: $+j$ means the component passes over at crossing $j$, and $-j$ means it passes under. For a knot there is one cyclic word; for a link there is one cyclic word per component. Each crossing label must occur exactly twice in total, once positively and once negatively. When $c>0$, every listed component word is nonempty; a separate crossingless component is not part of this code class. The zero-crossing case is only the one-component code specified above. The code also carries a separate algebraic crossing-sign vector ${\epsilon }_j\in \{+1,-1\}$, one entry for each label. Choosing a basepoint turns the knot's cyclic word into a based word; changing the basepoint cyclically shifts the word, and relabeling crossings changes only their names. The visit order, over/under signs, and crossing-sign vector determine the crossing rotation system; it is realizable in the sphere only when the resulting code is spherical. A plane Gauss code also carries an identifier of one cycle of its face permutation to designate the outer face.

A signed Dowker–Thistlethwaite code, in this page's convention, encodes a based oriented knot with $c$ crossings by numbering its $2c$ visits $1,\dots ,2c$ in traversal order. For each odd visit $2j-1$, record the even visit $2k$ paired with it as $d_j=+2k$ when the odd visit is over and $d_j=-2k$ when it is under; also record the algebraic crossing sign ${\epsilon }_j\in \{+1,-1\}$ for crossing label $j$, the pair containing odd visit $2j-1$. Its parity conditions are that $\mathrm{\mid }{d}_1\mathrm{\mid },\dots ,\mathrm{\mid }{d}_c\mathrm{\mid }$ are exactly the even numbers $2,4,\dots ,2c$, each once, and each paired crossing has one over-visit and one under-visit. The separate ${\epsilon }_j$ records the crossing sign; signed DT conventions elsewhere may encode signs differently. The associated cyclic traversal and crossing signs determine a rotation system, which must pass the same spherical realizability test. A plane DT code also carries an identifier of one cycle of its face permutation to designate the outer face.

An oriented planar-diagram code labels each edge segment by an integer and lists its two incident darts at crossing records. Each crossing record lists its four directed edge ends in clockwise order, beginning with the outgoing over-dart, and records which opposite pair is over, the orientation of each strand, and the crossing sign. Every edge label occurs exactly twice, once as an incoming end and once as an outgoing end. The code must satisfy the local crossing-order rule above; its crossing graph must be connected when nonempty; and its rotation system must satisfy $V-E+F=2$. The record also carries an identifier of one cycle of its face permutation as the outer face. Thus an arbitrary word in labels, even when each label occurs twice, is not declared realizable until all incidence, orientation, crossing, connectedness, and face-count tests pass.

Oriented crossing signs, diagram writhe, and blackboard framing — determinant convention in the oriented projection plane, calibrated so a positive Artin half twist has sign $+1$.

Regular oriented link diagrams — regular diagrams have finite transverse double points, four-valent crossing vertices, oriented strands, and over/under data.

Hass–Lagarias–Pippenger, §§1, 3 — finite diagram representation and four-valent crossing data.

Erickson, “Planar Graphs” — dart, rotation-system, face-cycle, and outer-face conventions.

Knot Atlas, Gauss Codes — signed-visit convention and trefoil code.

Knot Atlas, Planar Diagrams — PD edge and crossing records.

Knot Atlas, DT Codes — odd/even DT pairing and signed-code examples.