(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 69934, 2323] NotebookOptionsPosition[ 51783, 1718] NotebookOutlinePosition[ 56507, 1851] CellTagsIndexPosition[ 55287, 1819] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["EquivalentWords 1.11", "Title"], Cell["documentation notebook", "Subtitle"], Cell[TextData[{ "Eric Rowland\n", ButtonBox["http://math.tulane.edu/~erowland/packages.html", BaseStyle->"Hyperlink", ButtonData:>{ URL["http://math.tulane.edu/~erowland/packages.html"], None}] }], "Subsubtitle"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "EquivalentWords is a package for computing with equivalence classes of \ words in ", Cell[BoxData[ FormBox[ SubscriptBox["F", "2"], TraditionalForm]]], ", the free group on two generators, under ", Cell[BoxData[ FormBox[ RowBox[{"Aut", " ", SubscriptBox["F", "2"]}], TraditionalForm]]], ".\nThis introduction gets you started with a few features of the package; \ the next section provides a complete list of package symbols along with their \ usage messages and further examples." }], "Text"], Cell[TextData[{ "To use ", "EquivalentWords", ", first you will need to load the package by evaluating the following cell. \ (If you need help, see ", ButtonBox["loading a package", BaseStyle->"Hyperlink", ButtonData->{ URL["http://math.tulane.edu/~erowland/packages/loadingapackage.html"], None}, ButtonNote-> "http://math.tulane.edu/~erowland/packages/loadingapackage.html"], ".)" }], "Text"], Cell[BoxData[ RowBox[{"<<", "EquivalentWords`"}]], "Code"], Cell[CellGroupData[{ Cell["Representing words", "Subsection"], Cell[TextData[{ "An element of ", Cell[BoxData[ FormBox[ SubscriptBox["F", "2"], TraditionalForm]]], " is represented as a string. The generators of ", Cell[BoxData[ FormBox[ SubscriptBox["F", "2"], TraditionalForm]]], " are ", Cell[BoxData[ FormBox["\"\\"", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["\"\\"", TraditionalForm]]], ", and their inverses are ", Cell[BoxData[ FormBox["\"\\"", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["\"\\"", TraditionalForm]]], ". The group operation is string concatenation." }], "Text"], Cell[TextData[{ "Here are some elements of ", Cell[BoxData[ FormBox[ SubscriptBox["F", "2"], TraditionalForm]]], ":" }], "Text"], Cell[BoxData[ RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\"", ",", "\"\\""}], "}"}]], "Input"], Cell[TextData[{ "Each element has a canonical representation where there are no adjacent \ inverses. Moreover, it is convenient to take representatives of words modulo \ the inner automorphism group (since eventually we are interested in \ representatives modulo the full automorphism group), meaning that inverses \ can be canceled from the beginning and end of a word. ", StyleBox["CancelInverses", "Input"], " finds this representative." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"CancelInverses", "[", "\"\\"", "]"}]], "Input"], Cell[BoxData["\<\"BaB\"\>"], "Output"] }, Open ]], Cell[TextData[{ "In addition to choosing a representative with respect to inner \ automorphisms (cycles), it is also relatively simple to choose a \ representative with respect to permutations of the generators and their \ inverses. ", StyleBox["CanonicalPermutation", "Input"], " gives the word equivalent appearing first in lexicographic order among all \ the words obtained by permuting the generators." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"CanonicalPermutation", "[", "\"\\"", "]"}]], "Input"], Cell[BoxData["\<\"aba\"\>"], "Output"] }, Open ]], Cell[TextData[{ StyleBox["RCPRepresentative", "Input"], " (\"reduced under cycled permutations\" representative) gives a canonical \ representative with respect to both inner automorphisms and permutations of \ generators:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"RCPRepresentative", "[", "\"\\"", "]"}]], "Input"], Cell[BoxData["\<\"aab\"\>"], "Output"] }, Open ]], Cell[TextData[{ "Finally, we would like to take representatives with respect to word length. \ A word is ", StyleBox["minimal", FontSlant->"Italic"], " if it is of minimum length in its equivalence class under ", Cell[BoxData[ FormBox[ RowBox[{"Aut", " ", SubscriptBox["F", "2"]}], TraditionalForm]]], ". ", StyleBox["MinimalWordQ", "Input"], " implements a simple test for this." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MinimalWordQ", "[", "\"\\"", "]"}]], "Input"], Cell[BoxData["False"], "Output"] }, Open ]], Cell[TextData[{ "In this package we are concerned with computing equivalence classes of ", Cell[BoxData[ FormBox[ SubscriptBox["F", "2"], TraditionalForm]]], " under ", Cell[BoxData[ FormBox[ RowBox[{"Aut", " ", SubscriptBox["F", "2"]}], TraditionalForm]]], ", and we represent an equivalence class by its list of minimal, RCP words. \ ", StyleBox["EquivalenceClass", "Input"], " computes the equivalence class of a word." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"EquivalenceClass", "[", "\"\\"", "]"}]], "Input"], Cell[BoxData[ RowBox[{"{", "\<\"a\"\>", "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"EquivalenceClass", "[", "\"\\"", "]"}]], "Input"], Cell[BoxData[ RowBox[{"{", "\<\"a\"\>", "}"}]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Manipulating words", "Subsection"], Cell[TextData[{ StyleBox["InverseGenerator", "Input"], " gives the inverse of a generator." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"InverseGenerator", "/@", RowBox[{"{", RowBox[{ "\"\\"", ",", "\"\\"", ",", "\"\\"", ",", "\"\\""}], "}"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"A\"\>", ",", "\<\"B\"\>", ",", "\<\"a\"\>", ",", "\<\"b\"\>"}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ StyleBox["SyllableCount", "Input"], " gives the number of occurrences of a subword and its inverse in a word. \ For example, this counts the number of ", Cell[BoxData[ FormBox[ SuperscriptBox["b", RowBox[{"\[PlusMinus]", "1"}]], TraditionalForm]]], "'s:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SyllableCount", "[", RowBox[{"\"\\"", ",", "\"\\""}], "]"}]], "Input"], Cell[BoxData["6"], "Output"] }, Open ]], Cell[TextData[{ StyleBox["ApplyAutomorphism", "Input"], " applies a Whitehead type II automorphism to a word." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ApplyAutomorphism", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", "\"\\"", "}"}], ",", "\"\\""}], "}"}], ",", "\"\\""}], "]"}]], "Input"], Cell[BoxData["\<\"abababbbaabbba\"\>"], "Output"] }, Open ]], Cell[TextData[{ "In ", Cell[BoxData[ FormBox[ SubscriptBox["F", "2"], TraditionalForm]]], ", it suffices to consider automorphisms where the set ", Cell[BoxData[ FormBox["A", TraditionalForm]]], " consists of a single generator.\n", StyleBox["$AllAutomorphisms", "Input"], " contains eight one-letter automorphisms:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData["$AllAutomorphisms"], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", "\<\"a\"\>", "}"}], ",", "\<\"b\"\>"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", "\<\"a\"\>", "}"}], ",", "\<\"B\"\>"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", "\<\"b\"\>", "}"}], ",", "\<\"a\"\>"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", "\<\"b\"\>", "}"}], ",", "\<\"A\"\>"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", "\<\"A\"\>", "}"}], ",", "\<\"b\"\>"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", "\<\"A\"\>", "}"}], ",", "\<\"B\"\>"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", "\<\"B\"\>", "}"}], ",", "\<\"a\"\>"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", "\<\"B\"\>", "}"}], ",", "\<\"A\"\>"}], "}"}]}], "}"}]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Computing all equivalence classes", "Subsection"], Cell[TextData[{ "To compute all equivalence classes of words of length ", Cell[BoxData[ FormBox["n", TraditionalForm]]], ", we start by computing a set of RCP candidate words of that length." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"RCPCandidates", "[", "4", "]"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"aaaa\"\>", ",", "\<\"aaab\"\>", ",", "\<\"aabb\"\>", ",", "\<\"abab\"\>", ",", "\<\"abaB\"\>", ",", "\<\"abAb\"\>", ",", "\<\"abAB\"\>"}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ StyleBox["RCPWords", "Input"], " calls ", StyleBox["RCPCandidates", "Input"], " and then removes the nonminimal words and equivalent duplicates in the \ resulting list to obtain the set of minimal RCP words of length ", Cell[BoxData[ FormBox["n", TraditionalForm]]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"RCPWords", "[", "4", "]"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"aaaa\"\>", ",", "\<\"aabb\"\>", ",", "\<\"abaB\"\>", ",", "\<\"abAB\"\>"}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ StyleBox["ClassifyWords", "Input"], " partitions this set into equivalence classes. Thus these are the \ equivalence classes of length-", Cell[BoxData[ FormBox["4", TraditionalForm]]], " words:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ClassifyWords", "[", RowBox[{"{", RowBox[{ "\"\\"", ",", "\"\\"", ",", "\"\\"", ",", "\"\\""}], "}"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", "\<\"aaaa\"\>", "}"}], ",", RowBox[{"{", "\<\"abAB\"\>", "}"}], ",", RowBox[{"{", RowBox[{"\<\"aabb\"\>", ",", "\<\"abaB\"\>"}], "}"}]}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ StyleBox["WordEquivalenceClasses", "Input"], " does this all behind the scenes." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"WordEquivalenceClasses", "[", "4", "]"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", "\<\"aaaa\"\>", "}"}], ",", RowBox[{"{", "\<\"abAB\"\>", "}"}], ",", RowBox[{"{", RowBox[{"\<\"aabb\"\>", ",", "\<\"abaB\"\>"}], "}"}]}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ "Root words are a special class of minimal words. 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