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Two polynomial invariants of a knot diagram, described by J.B. Listing in 1847 [[#References|[a1]]], named by him complexions-symbol (cf. also [[Knot and link diagrams|Knot and link diagrams]]). Listing associated with every corner of a crossing a symbol $\delta$ (for dexiotropic, i.e. right-handed) or $\lambda$ (for leotropic, i.e. left-handed) according to the convention of Fig.a1a. He then coloured regions cut out by the diagram in a checkerboard manner: black and white and such that neighbouring regions have different colours. To each region he associated a $2$-variable monomial $\delta ^ { i } \lambda ^ { j }$, where $i$ is the number of $\delta$ corners and $j$ the number of $\lambda$ corners in the region. The first Listing polynomial, or black Listing polynomial, $P _ { b } ( \delta , \lambda )$ is equal to the sum of the above monomials taken over all black regions. The second Listing polynomial, or white Listing polynomial, $P _ { W } ( \delta , \lambda )$ is defined in a similar manner, summing over white regions.

Listing noted that for an alternating knot diagram each region has corners of the same colour, and that his polynomials are in fact one-variable polynomials. He also stated that the left- and right-handed trefoil knots are not equivalent (cf. also [[Torus knot|Torus knot]]) and that there are knots without an alternating diagram. He noticed further that his polynomials can change when the knot diagram is modified without changing the knot type.

<img src="https://www.encyclopediaofmath.org/legacyimages/common_img/l130090a.gif" style="border:1px solid;"/>

Figure: l130090a

In his 1849 note, Listing showed that the figure-eight knot, Fig.a1b, and its mirror image are equivalent [[#References|[a2]]]. Because of this, this knot is often called the [[Listing knot|Listing knot]].

In modern knot theory, Listing's ideas are widely used: colouring of regions (e.g. to define the Goeritz matrix), labelling of corners (e.g. to define the [[Kauffman bracket polynomial|Kauffman bracket polynomial]]), studying labelling of corners of alternating diagrams (e.g. to proof the Tait conjectures). Listing polynomials can be thought as a precursors of the [[Partition function (statistical physics)|partition function]] of a state model of a physical system in statistical mechanics (cf. also [[Statistical mechanics, mathematical problems in]]).

====References====
<table>
<tr><td valign="top">[a1]</td> <td valign="top">  J.B. Listing,   "Vorstudien zur Topologie"  ''Göttinger Studien (Abt. 1)'' , '''1'''  (1847)  pp. 811–875</td></tr>
<tr><td valign="top">[a2]</td> <td valign="top">  C.N. Little,   "Non-alternate $\pm$ knots"  ''Trans. Royal Soc. Edinburgh'' , '''XXXIX: III''' :  30  (1899)  pp. 771–778  (Read: July 3, 1899) {{ZBL|31.0481.02}}</td></tr>
</table>