# Polyknight

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{{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](/source/knight_(chess)) in which doubling back is allowed. It is a [polyform](/source/polyform) with square cells which are not necessarily connected, comparable to the [polyking](/source/polyking). Alternatively, it can be interpreted as a connected subset of the vertices of a [knight's graph](/source/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](/source/translation_(geometry)), [rotation](/source/rotation), [reflection](/source/reflection_(mathematics)) or [glide reflection](/source/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](/source/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

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Adapted from the Wikipedia article [Polyknight](https://en.wikipedia.org/wiki/Polyknight) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Polyknight?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.