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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 418346<Chapter><Heading> Knots and Quandles </Heading>12<Table Align="|l|" >34<Row >5<Item> Knots </Item>6</Row>78<Row>9<Item>10<Index> PresentationKnotQuandle</Index>11<C>PresentationKnotQuandle(gaussCode) </C>12<P/>1314Inputs a Gauss Code of a knot (with the orientations; see <M>GaussCodeOfPureCubicalKnot</M> in HAP package) and outputs the generators and relators of the knot quandle associated (in the form of a record).15</Item>16</Row>1718<Row>19<Item>20<Index> PD2GC</Index>21<C>PD2GC(PD) </C>22<P/>2324Inputs a Planar Diagram of a knot; outputs the Gauss Code associated (with the orientations).25</Item>26</Row>2728<Row>29<Item>30<Index>PlanarDiagramKnot</Index>31<C>PlanarDiagramKnot(n,k) </C>32<P/>3334Returns a Planar Diagram for the <M>k</M>-th knot with <M>n</M> crossings (<M>n</M>≤<M>12</M>) if it exists; fail otherwise.35</Item>36</Row>3738<Row>39<Item>40<Index> GaussCodeKnot</Index>41<C>GaussCodeKnot(n,k) </C>42<P/>4344Returns a Gauss Code (with orientations) for the <M>k</M>-th knot with <M>n</M> crossings (<M>n</M>≤<M>12</M>) if it exists; fail otherwise.45</Item>46</Row>4748<Row>49<Item>50<Index> PresentationKnotQuandleKnot</Index>51<C>PresentationKnotQuandleKnot(n,k) </C>52<P/>5354Returns generators and relators (in the form of a record) for the <M>k</M>-th knot with <M>n</M> crossings (<M>n</M>≤<M>12</M>) if it exists; fail otherwise.55</Item>56</Row>5758<Row>59<Item>60<Index> NumberOfHomomorphisms</Index>61<C>NumberOfHomomorphisms(genRelQ,finiteQ) </C>62<P/>6364Inputs generators and relators <M>genRelQ</M> of a knot quandle (in the form of a record, see above) and a finite quandle <M>finiteQ</M>; outputs the number of homomorphisms from the former to the latter.65</Item>66</Row>6768<Row>69<Item>70<Index> PartitionedNumberOfHomomorphisms</Index>71<C>PartitionedNumberOfHomomorphisms(genRelQ,finiteQ) </C>72<P/>7374Inputs generators and relators <M>genRelQ</M> of a knot quandle (in the form of a record, see above) and a finite connected quandle <M>finiteQ</M>; outputs a partition of the number of homomorphisms from the former to the latter.75</Item>76</Row>7778<Row >79<Item> Quandles </Item>80</Row>8182<Row>83<Item>84<Index> ConjugationQuandle</Index>85<C>ConjugationQuandle(G,n) </C>86<P/>8788Inputs a finite group <M>G</M> and an integer <M>n</M>; outputs the associated <M>n</M>-fold conjugation quandle.89</Item>90</Row>9192<Row>93<Item>94<Index> QuandleAxiomIsSatisfied</Index>95<C>FirstQuandleAxiomIsSatisfied(M) </C>96<Br/>97<C>SecondQuandleAxiomIsSatisfied(M) </C>98<Br/>99<C>ThirdQuandleAxiomIsSatisfied(M) </C>100<P/>101102Inputs a finite magma <M>M</M>; returns true if <M>M</M> satisfy the first/second/third axiom of a quandle, false otherwise.103</Item>104</Row>105106<Row>107<Item>108<Index> IsQuandle</Index>109<C> IsQuandle(M) </C>110<P/>111112Inputs a finite magma <M>M</M>; returns true if <M>M</M> is a quandle, false otherwise.113</Item>114</Row>115116<Row>117<Item>118<Index> Quandles</Index>119<C>Quandles(n) </C>120<P/>121122Returns a list of all quandles of size <M>n</M>, <M>n</M>≤<M>6</M>. If <M>n</M>≥<M>7</M>, it returns fail.123</Item>124</Row>125126<Row>127<Item>128<Index> Quandle</Index>129<C> Quandle(n,k) </C>130<P/>131132Returns the <M>k</M>-th quandle of size <M>n</M> (<M>n</M>≤<M>6</M>) if such a quandle exists, fail otherwise.133</Item>134</Row>135136<Row>137<Item>138<Index> IdQuandle</Index>139<C>IdQuandle(Q) </C>140<P/>141142Inputs a quandle <M>Q</M>; and outputs a list of integers [<M>n</M>,<M>k</M>] such that <M>Q</M> is isomorphic to <C>Quandle(n,k)</C>. If <M>n</M>≥<M>7</M>, then it returns [<M>n</M>,fail] (where <M>n</M> is the size of <M>Q</M>).143</Item>144</Row>145146<Row>147<Item>148<Index> IsLatin</Index>149<C>IsLatin(Q) </C>150<P/>151152Inputs a finite quandle <M>Q</M>; returns true if <M>Q</M> is latin, false otherwise.153</Item>154</Row>155156<Row>157<Item>158<Index> IsConnectedQuandle</Index>159<C>IsConnectedQuandle(Q) </C>160<P/>161162Inputs a finite quandle <M>Q</M>; returns true if <M>Q</M> is connected, false otherwise.163</Item>164</Row>165166<Row>167<Item>168<Index> ConnectedQuandles</Index>169<C>ConnectedQuandles(n) </C>170<P/>171172Returns a list of all connected quandles of size <M>n</M>.173</Item>174</Row>175176<Row>177<Item>178<Index> ConnectedQuandle</Index>179<C>ConnectedQuandle(n,k) </C>180<P/>181182Returns the <M>k</M>-th quandle of size <M>n</M> if such a quandle exists, fail otherwise.183</Item>184</Row>185186<Row>187<Item>188<Index> IdConnectedQuandle</Index>189<C>IdConnectedQuandle(Q) </C>190<P/>191192Inputs a connected quandle <M>Q</M>; and outputs a list of integers [<M>n</M>,<M>k</M>] such that <M>Q</M> is isomorphic to <C>ConnectedQuandle(n,k)</C>.193</Item>194</Row>195196<Row>197<Item>198<Index> IsQuandleEnvelope</Index>199<C>IsQuandleEnvelope(Q,G,e,stigma) </C>200<P/>201202Inputs a set <M>Q</M>, a permutation group <M>G</M>, an element <M>e</M> ∈ <M>Q</M> and an element <M>stigma </M> ∈ <M>G</M>; returns true if this structure describes a quandle envelope, false otherwise.203</Item>204</Row>205206<Row>207<Item>208<Index> QuandleQuandleEnveloppe</Index>209<C> QuandleQuandleEnveloppe(Q,G,e,stigma) </C>210<P/>211212Inputs a set <M>Q</M>, a permutation group <M>G</M>, an element <M>e</M> ∈ <M>Q</M> and an element <M>stigma</M> ∈ <M>G</M>.213If this structure describes a quandle envelope, the function returns the quandle from this quandle envelope; and fail otherwise.214Nb: this quandle is a connected quandle.215</Item>216</Row>217218<Row>219<Item>220<Index> KnotInvariantCedric</Index>221<C> KnotInvariantCedric(genRelQ,n,m) </C>222<P/>223224Inputs generators and relators of a knot quandle (in the form of a record, see above) and two integers <M>n</M> and <M>m</M>; outputs a list [<M>n</M>1,<M>n</M>2,...,<M>n</M>k] where <M>n</M>j is a partition of the number of homomorphisms from the considered knot quandle to the <M>j</M>-th connected quandle of size <M>n</M>≤<M>i</M>≤<M>m</M>.225</Item>226</Row>227228<Row>229<Item>230<Index> RightMultiplicationGroupAsPerm</Index>231<C>RightMultiplicationGroupAsPerm(Q) </C>232<P/>233234Inputs a connected quandle <M>Q</M>; output its right multiplication group whose elements are permutations.235</Item>236</Row>237238<Row>239<Item>240<Index> RightMultiplicationGroup</Index>241<C>RightMultiplicationGroup(Q) </C>242<P/>243244Inputs a connected quandle <M>Q</M>; output its right multiplication group whose elements are mappings from <M>Q</M> to <M>Q</M>.245</Item>246</Row>247248<Row>249<Item>250<Index> AutomorphismGroupQuandleAsPerm</Index>251<C>AutomorphismGroupQuandleAsPerm(Q) </C>252<P/>253254Inputs a connected quandle <M>Q</M>; outputs its automorphism group whose elements are permutations.255</Item>256</Row>257258<Row>259<Item>260<Index> AutomorphismGroupQuandle</Index>261<C>AutomorphismGroupQuandle(Q)</C>262<P/>263264Inputs a connected quandle <M>Q</M>; outputs its automorphism group whose elements are mappings from <M>Q</M> to <M>Q</M>.265</Item>266</Row>267268269</Table>270</Chapter>271272273