{{Short description|Figure formed by knights moves on a grid}} thumb|right|The 35 free tetraknightsA '''polyknight''' is a plane geometric figure formed by selecting cells in a square lattice that could represent the path of a chess knight in which doubling back is allowed. It is a polyform with square cells which are not necessarily connected, comparable to the polyking. Alternatively, it can be interpreted as a connected subset of the vertices of a knight's graph, a graph formed by connecting pairs of lattice squares that are a knight's move apart.<ref>{{citation | last1 = Aleksandrowicz | first1 = Gadi | last2 = Barequet | first2 = Gill | editor1-last = Atallah | editor1-first = Mikhail J. | editor2-last = Li | editor2-first = Xiang-Yang | editor3-last = Zhu | editor3-first = Binhai | contribution = Parallel enumeration of lattice animals | doi = 10.1007/978-3-642-21204-8_13 | pages = 90–99 | publisher = Springer | series = Lecture Notes in Computer Science | title = Frontiers in Algorithmics and Algorithmic Aspects in Information and Management - Joint International Conference, FAW-AAIM 2011, Jinhua, China, May 28-31, 2011. Proceedings | volume = 6681 | year = 2011| isbn = 978-3-642-21203-1 }}.</ref>
==Enumeration of polyknights== === Free, one-sided, and fixed polyknights === Three common ways of distinguishing polyominoes for enumeration<ref>{{citation |last=Redelmeier |first=D. Hugh |year=1981 |title=Counting polyominoes: yet another attack |journal=Discrete Mathematics |volume=36 |issue=2 |pages=191–203 |doi=10.1016/0012-365X(81)90237-5|doi-access=free }}</ref> can also be extended to polyknights: *''free'' polyknights are distinct when none is a rigid transformation (translation, rotation, reflection or glide reflection) of another (pieces that can be picked up and flipped over). *''one-sided'' polyknights are distinct when none is a translation or rotation of another (pieces that cannot be flipped over). *''fixed'' polyknights are distinct when none is a translation of another (pieces that can be neither flipped nor rotated).
The following table shows the numbers of polyknights of various types with ''n'' cells. {| class=wikitable ! ''n'' !! free !! one-sided !! fixed |- align=right | 1 || 1 || 1 || 1 |- align=right | 2 || 1 || 2 || 4 |- align=right | 3 || 6 || 8 || 28 |- align=right | 4 || 35 || 68 || 234 |- align=right | 5 || 290 || 550 || 2,162 |- align=right | 6 || 2,680 || 5,328 || 20,972 |- align=right | 7 || 26,379 || 52,484 || 209,608 |- align=right | 8 || 267,598 || 534,793 || 2,135,572 |- align=right | 9 || 2,758,016 || 5,513,338 || 22,049,959 |- align=right | 10 || 28,749,456 || 57,494,308 || 229,939,414 |- align=right | OEIS || {{OEIS link|id=A030446}} || {{OEIS link|id=A030445}} || {{OEIS link|id=A030444}} |}
{{Gallery |title=Free polyknights |File:Pentaknights.png|The 290 free pentaknights. |File:Hexaknights.png|The 2,680 free hexaknights. }}
==Notes== {{reflist}}
{{Polyforms}}
Category:Polyforms