Analysis

Each crossing will be shown visually as a pair of arrows, one crossing over the other. One arrow will be a solid colour, and will run horizontally from left to right; this is the part of the link that you are traversing. The other arrow will be grey, and will run vertically (either up or down); this is the part of the link that you are passing over or under. The crossing number will be written in the top-right corner beside the arrows.

For each crossing, the arrow that you are traversing will be coloured blue for a positive crossing, or yellow for a negative crossing. (This is just a visual aid, and is not strictly necessary; you can already tell the sign of the crossing from the arrangement of the arrows.)

Crossings are visited in order from left to right along each row of the display, one row at a time (i.e., the same way that you would read English text in a book).

Each crossing will be shown as a number with a plus or minus sign attached. The number indicates which crossing you are passing through, and the sign indicates whether it is a positive or negative crossing. The sign will appear as a superscript if you are passing over the crossing, or a subscript if you are passing under.

As before, crossings are visited in order from left to right along each row, one row at a time. Also as before, positive crossings will be coloured blue and negative crossings will be coloured yellow (but again the colour is just a visual aid, since you already have this information from the plus/minus sign).

The Alexander polynomial is a single-variable polynomial invariant. It is relatively weak, but (unlike many of the other invariants listed here) it can be computed in polynomial time.

Regina will only compute Alexander polynomials for knots (not multiple-component links).

The Jones polynomial is a single-variable polynomial in the variable t, whose exponents are integers for classical knots, but whose exponents may be half-integers in other cases (depending on the number of link components and/or whether the link is virtual). You may therefore see this presented either as a polynomial in t or as a polynomial in √t.

If you have opted not to use unicode symbols, then you will see this as a polynomial in either t or sqrt_t.

Regina can compute Jones polynomials for link diagrams with any number of components.

The HOMFLY-PT polynomial is a two-variable polynomial with two common formulations: one in the variables (α, z) and one in the variables (l, m). These formulations give different polynomials, which are related via a simple substitution. Regina offers a pair of buttons beside the HOMFLY-PT polynomial for you to choose which formulation is displayed.

Regina can compute HOMFLY-PT polynomials for link diagrams with any number of components.

The Kauffman bracket is a property of the link diagram, not an invariant of the link, and is related to the Jones polynomial by a simple substitution. See Adams [Ada94] for a simple combinatorial introduction to the Kauffman bracket and the Jones polynomial.

Regina can compute Kauffman brackets for link diagrams with any number of components.

The affine index polynomial is a single-variable polynomial invariant that is relatively weak, but which nevertheless provides some useful information for virtual knots. Its main advantage is that it is extremey fast to compute even for large knot diagrams (the computation runs in very small polynomial time).

Regina will only compute affine index polynomials for virtual knots (not multiple-component virtual links).

The affine index polynomial is not shown at all for classical link diagrams, since in the classical case the affine index polynomial is always zero.

The arrow polynomial enhances the Kauffman bracket in a way that makes it more powerful for virtual links, and that makes it a true link invariant. Whereas the Kauffman bracket is a polynomial in the single variable A, the arrow polynomial is a polynomial in many variables (A, K₁, K₂, K₃, …).

Regina can compute arrow polynomials for virtual link diagrams with any number of components.

The arrow polynomial is not shown at all for classical link diagrams, since in the classical case the arrow polynomial is just the Kauffman bracket with some normalisation applied.

This is the link group as constructed from the Wirtinger presentation (though Regina will typically simplify the presentation before showing it here). Each relation is some variant of the form xy=yz, where y corresponds to the upper strand at some crossing, and x and z correspond to the two sides of the lower strand at that same crossing.

This is the extended group of the link, as defined by Silver and Williams [SW06]. This invariant is particularly useful for virtual links, where the ordinary link group is a fairly weak invariant. However, it also yields more complex group presentations.

Like the ordinary link group, the extended group of a virtual link may change depending upon whether you view the link from above or below the diagram, and so for virtual link diagrams Regina will display both variants.

This displays the classical, oriented and signed Gauss codes. Classical Gauss codes are common in the literature, but they suffer from ambiguities related to orientation; for composite knots they may not even uniquely define the topology. Oriented and signed Gauss codes solve this problem by adding a little more information.

For a more detailed description of these formats, see the section on creating knots from Gauss codes.

Currently Regina only displays Gauss codes for knots (not multiple component links). If your knot diagram is virtual (not classical), then Regina will still display all three Gauss codes; however, only the oriented and signed variants contain enough informtion to reconstruct the diagram.

Like classical Gauss codes, Dowker-Thistlethwaite notation is also common in the literature, and also suffers from ambiguities related to orientation and composite knots. Dowker-Thistlethwaite notation comes in two forms—numerical and alphabetical—and Regina displays both forms here.

For a more detailed discussion of Dowker-Thistlethwaite notation, see the section on creating knots from Dowker-Thistlethwaite notation.

Currently Regina only displays Dowker-Thistlethwaite notation for knots (not multiple component links), and only for classical (not virtual) knot diagrams. Be aware that alphabetical Dowker-Thistlethwaite notation is case-sensitive (i.e., upper-case and lower-case matters).

Knot/link signatures are Reginas own text-based format, and were designed to have a combinatorial uniqueness property: two link diagrams have the same signature if and only if they are combinatorially isomorphic (which includes relabelling crossings and/or components, reflecting the entire diagram, rotating the entire diagram, and/or reversing individual link components).

For a more detailed discussion of knot/link signatures, see the section on creating knots from knot/link signatures.

These signatures are supported for all links with fewer than 64 components, and they support both classical and virtual link diagrams. Be aware that knot/link signatures are case-sensitive (i.e., upper-case and lower-case matters).

This displays the planar diagram code for the link. Planar diagram codes support multiple-component links, as well as both classical and virtual link diagrams. Their main limitations are that they cannot represent zero-crossing unknot components, and they cannot fix the orientation of a link component that consists entirely of over-crossings.

For a detailed description of planar diagram codes, see the section on creating links from planar diagram codes.

This is a multiline text code described by Bob Jenkins, and used in his HOMFLY-PT polynomial software. Regina can happily display the Jenkins format for multiple-component links, and for both classical and virtual link diagrams.

In this format, a link is described by a sequence of integers separated by whitespace. We assume that there are n crossings in the link, labelled arbitrarily as 0,1,2,…. The sequence of integers will contain, in order:

A tree decomposition (illustrated below) models the diagram graph using a rooted tree; you can read more about tree decompositions in the triangulations chapter. Regina uses these tree decompositions in its fixed-parameter tractable algorithms for the Jones and HOMFLY-PT polynomials [Bur18].

Regina will display both the width and the number of bags above the tree decomposition itself.

As with triangulations, Regina does not guarantee to find a tree decomposition of the smallest possible width (i.e., it does not compute the precise treewidth of the diagram graph). Instead it uses fast heuristics that are found to produce low-width tree decompositions in practice.

This is a variant of the tree decomposition that is less concise, but more useful in practice for algorithms since it imposes very tight constraints on the relationships between parent and child bags. Again, you can read more about these in the triangulations chapter.

As with tree decompositions, Regina will display both the width of the nice tree decomposition and the number of bags that it contains. Typically the width will be the same as for the (plain) tree decomposition above, but the number of bags may be significantly larger (though still linear in the overall number of crossings).