# Wikidata:Property proposal/Dowker-Thistlethwaite name

### Dowker-Thistlethwaite name

Under discussion
Description Unambiguous systematic identifier scheme for mathematical knots Dowker–Thistlethwaite name (Q96601703) Mathematical expression knot (Q1188853) (?:0|[1-9][0-9]*)[an]_\{[1-9][0-9]+\} unknot (Q1188344) → ${\displaystyle 0a_{1}}$ trefoil knot (Q168620) → ${\displaystyle 3a_{1}}$ figure-eight knot (Q168697) → ${\displaystyle 4a_{1}}$ (−2,3,7) pretzel knot (Q4544970) → ${\displaystyle 12n_{242}}$ Conway knot (Q94806552) → ${\displaystyle 11n_{34}}$ https://knotinfo.math.indiana.edu/ , en:List of prime knots (see https://knotinfo.math.indiana.edu/descriptions/dt_name.html for description) used as property for bot-generated items for knots with crossing number up to 12 for the use of knot identification, and case-by-case use on any further knots added to the database ~2900 always incomplete (Q21873886) (but complete for the knots of crossing number up to 12)

#### Motivation

This is an unambiguous name for items from a large set of notable mathematical objects. Knot theory is in the awkward postion of having created multiple incompatible and confusing naming schemes for these objects. This naming scheme is algorithmically generated from the Dowker-Thistlethwaite notation for these knots, and lacks the arbitrary and sometimes ambiguous nature of the Alexander-Briggs naming scheme. It can also be extended indefinitely to knots of higher crossing number, for which no Alexander-Briggs name exists, or is ever likely to exist. (See here for how ugly things have become with the traditional naming scheme, for which there are now several competing variants.)

These names are used to identify knots in the academic literature, see for example [1], [2], and [3] as well as being used by w:Morwen Thistlethwaite's program Knotscape (available from [4]). These codes are also used in Knotilus, in a modified form, see for example [5].

The Dowker-Thistlethwaite name is complementary to the Dowker-Thistlethwaite notation, from which the names are derived by combining (a) the crossing number, (b) an indicator ["a" or "n"] of whether the knot is alternating or non-alternating, and (c) a serial number indicating the place in the lexicographical ordering of the normalized Dowker-Thistlethwaite notation for that knot in the listing of all knots sharing that crossing number and alternation property. Unlike the Dowker-Thistlethwaite notation, this notation is compact: for example, the DT notation for 11n34 is [4, 8, 12, 2, -16, -18, 6, -20, -22, -14, -10], but the DT name is just 11n_34.

(Note regarding formatting: these names are usually typeset as LaTeX, so that 11n_{34} is shown as ${\displaystyle 11n_{34}}$. When not typeset, the LaTeX markup is normally skipped, for example as 11n34.)

For discussion: if this is implemented, should the curly brackets be mandatory in the regex, even if there is only a single subscript digit, for example "4a_{1}"? This would have the advantage of making the representation unambiguous.

Update: with this tighter constraint, we should also be able to set a uniqueness constraint on this property, since D-T names should be truly unique identifiers of prime knots (up to reflection). This should help prevent duplicate items from being created, something which is sadly not possible for the A-B naming scheme. -- The Anome (talk) 10:11, 24 June 2020 (UTC)

#### Discussion

This is a useful property that does something that none of the other knot properties we have currently encoded can do: name the for identification purposes in a way this is both compact and unambiguous. It can also serve as a linchpin from which to hang other knot properties, by linking with info in KnotInfo and Knotscape. I intend to fill in these values for all knots up to crossing number 12 using a bot: see Wikidata:Requests for permissions/Bot/The Anomebot 3 2 for the associated bot request.

Note also Wikidata:Property proposal/Dowker-Thistlethwaite notation, in which I propose the similarly named, but distinct, descriptive notation, which is complementary to these names. -- The Anome (talk) 09:53, 23 June 2020 (UTC)

I suggest yes, we should enforce a single consistent format (i.e. require the curly braces). ArthurPSmith (talk) 19:43, 23 June 2020 (UTC)
Thanks. I agree, and I've amended the proposal's regex accordingly, and also made the special case 0a_{1} for the w:unknot possible. With this amendment, we also have the bonus that we can now add a uniqueness constraint to the property, as there is only one way to represent each knot name. -- The Anome (talk) 10:09, 24 June 2020 (UTC)
As an aside, here's a great article on the process of knot enumeration: "The first 1,701,936 Knots" -- The Anome (talk) 20:21, 26 June 2020 (UTC)