final touches

main.typ

@@ -135,7 +135,8 @@
 
 #show: ams-article.with(
   paper-size: "a4",
-  title: [Implementation of the Kauffman Polynomial in SageMath],
+  title: [Implementation of the \ Kauffman Polynomial in Python],
+  page-title: [Implementation of the Kauffman Polynomial in Python],
   authors: (
     (
       name: "Antonio De Lucreziis",
@@ -150,9 +151,9 @@
   ],
 )
 
-= Introduction
+#pagebreak()
 
-Actually we don't like Python so we will be using Rust and then write bindings for Python that can be used in SageMath.
+= Introduction
 
 == The Kauffman Polynomial
 
@@ -174,6 +175,8 @@ The defining axioms of the Kauffman polynomial are the following, given a link d
 
 We will later be seeing that the Kauffman polynomial can be defined in a more explicit way, using a recursive definition that is the one we will be using to derive our algorithm.
 
+#pagebreak()
+
 = Computational Knot Theory
 
 The first problem in computational knot theory is to find a good representation for knots and links. There are various common representations in the literature, such as:
@@ -295,19 +298,22 @@ $
   epsilon(#skein-generic(direction: (+1, -1))) = -1
 $
 
-*Algorithm*:
+#pagebreak()
 
-```
-Input: An oriented link diagram with starting points on each component
-Output: List<List<(Int, Int)>>
+*Algorithm*: The input is an oriented link diagram with starting points on each component and the output is a list of components where each component is a list of pairs of numbers
+
+1. Label each crossing with a number in order
+
+2. For each component:
+
+  1. Walk along it from the starting point in its orientation
+
+  2. At each crossing with label $i$, write a tuple with components
+
+    - $+i$ or $-i$ if this is an over-crossing or under-crossing
+
+    - $+1$ or $-1$ if this is a left-handed or right-handed
 
-- Label each crossing with a number in order
-- For each component:
-  - Walk along it from the starting point in its orientation
-  - At each crossing, write a tuple with components
-    - +i or -i if this is an over-crossing or under-crossing
-    - +1 or -1 if this is a left-handed or right-handed
-```
 
 Converting one code to the other is not too much work as one just need to first do a labelling step to convert crossing labels and then convert the over/under-strand and left/right-handedness relations between the two notations.
 
@@ -579,8 +585,6 @@ So PD codes are simpler and compact to store (and generate from a diagram) but S
 //   ),
 // $
 
-#pagebreak()
-
 == Link reconstruction from code
 
 We briefly mention that reconstructing a link from a PD or SG code is not trivial and there are various approaches used by various softwares that can be used for this task.
@@ -616,6 +620,8 @@ Another approach used by #link("https://knotfol.io/")[KnotFolio] is based on #li
 
 This condition that every point is the average of its neighbors can be easily expressed as a system of linear equations where some points on a chosen outer face have been fixed. When the graph is planar and 3-vertex-connected the linear system is non degenerate and has a unique solution.
 
+#pagebreak()
+
 = Computing the Polynomial
 
 Let's now recap the main formal algorithm for computing the Kauffman polynomial.

theme.typ

@@ -13,6 +13,7 @@
 #let ams-article(
   // The article's title.
   title: [Paper title],
+  page-title: [Paper title],
   // An array of authors. For each author you can specify a name,
   // department, organization, location, and email. Everything but
   // but the name is optional.
@@ -64,7 +65,7 @@
       if i == 1 { return }
       set text(size: script-size)
       align(center)[
-        #upper(title)
+        #upper(page-title)
       ]
     },