R-Matrix Invariants: Difference between revisions
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KnotTheory` can compute knot and link invariants associated with matrix solutions of the Yang-Baxter equation, using the program <code>REngine</code>: |
KnotTheory` can compute knot and link invariants associated with matrix solutions of the Yang-Baxter equation, using the program <code>REngine</code>: |
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{{Startup Note}} |
{{Startup Note}} |
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<!--$$?REngine$$--> |
<!--$$?REngine$$--> |
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<!--END--> |
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In more detail, let <math>V</math> be a free module of finite rank over a commutative ring, and let <math> R, \bar{R} \in End(V \otimes V) </math> be invertible. Further, let there be invertible endomorphisms <math>M_\leftarrow, M_\rightarrow, M^\leftarrow, M^\rightarrow \in End(V) </math>; these correspond to McupL, McupR, McapL, and McapR respectively. Fixing a basis of V, we can regard all of these endomorphisms as matrices. |
In more detail, let <math>V</math> be a free module of finite rank over a commutative ring, and let <math> R, \bar{R} \in End(V \otimes V) </math> be invertible. Further, let there be invertible endomorphisms <math>M_\leftarrow, M_\rightarrow, M^\leftarrow, M^\rightarrow \in End(V) </math>; these correspond to McupL, McupR, McapL, and McapR respectively. Fixing a basis of V, we can regard all of these endomorphisms as matrices. |
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The following notation is useful: for <math>n = dim(V)</math>, <math>R_{ab}^{cd}</math> refers to the element in row <math>n(c-1)+d</math> row and column <math>n(a-1) + b</math> column. |
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For an oriented knot or link <math>K</math>, REngine returns the product <math> \tau(K) </math>, which is computed as follows: |
For an oriented knot or link <math>K</math>, REngine returns the product <math> \tau(K) </math>, which is computed as follows: |
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#Find a [[MorseLink Presentations| Morse link presentation]] of <math>K</math> such that it is composed only of crossings of the <code>X[k, Over/Under, Down, Down]</code> variety; this is |
#Find a [[MorseLink Presentations| Morse link presentation]] of <math>K</math> such that it is composed only of crossings of the <code>X[k, Over/Under, Down, Down]</code> variety; this is accomplished by adding caps and cups where necessary. |
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#Label the edges in between events in the Morse link presentation of K. |
#Label the edges in between events in the Morse link presentation of <math>K</math>. |
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#Associate each event with one of the matrices as shown: |
#Associate each event with one of the matrices as shown, with the indices given by the labelling: |
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#* |
#*<code>X[k, Over, Down, Down]</code>:[[Image:REposcr.png|Positive crossing]]<math>\Rightarrow R_{ab}^{cd}</math> |
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#*<code>X[k, Under, Down, Down]</code>:[[Image:REnegcr.png|Negative crossing]] <math>\Rightarrow\bar{R}_{ab}^{cd}</math> |
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#Define <math>\tau(K)</math> as the result of taking the product of the matrices associated with the elements of ''K'', and summing over repeated indices; each sum runs from 1 to <math>n=dim(V)</math>. |
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</table> |
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As an example, let ''K'' be the left-handed trefoil, as shown. Then |
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<math>\tau(K) = \sum {M_\leftarrow}_{ab} </math> |
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Revision as of 15:05, 7 September 2005
KnotTheory` can compute knot and link invariants associated with matrix solutions of the Yang-Baxter equation, using the program REngine:
(For In[1] see Setup)
In more detail, let [math]\displaystyle{ V }[/math] be a free module of finite rank over a commutative ring, and let [math]\displaystyle{ R, \bar{R} \in End(V \otimes V) }[/math] be invertible. Further, let there be invertible endomorphisms [math]\displaystyle{ M_\leftarrow, M_\rightarrow, M^\leftarrow, M^\rightarrow \in End(V) }[/math]; these correspond to McupL, McupR, McapL, and McapR respectively. Fixing a basis of V, we can regard all of these endomorphisms as matrices.
The following notation is useful: for [math]\displaystyle{ n = dim(V) }[/math], [math]\displaystyle{ R_{ab}^{cd} }[/math] refers to the element in row [math]\displaystyle{ n(c-1)+d }[/math] row and column [math]\displaystyle{ n(a-1) + b }[/math] column.
For an oriented knot or link [math]\displaystyle{ K }[/math], REngine returns the product [math]\displaystyle{ \tau(K) }[/math], which is computed as follows:
- Find a Morse link presentation of [math]\displaystyle{ K }[/math] such that it is composed only of crossings of the
X[k, Over/Under, Down, Down]variety; this is accomplished by adding caps and cups where necessary. - Label the edges in between events in the Morse link presentation of [math]\displaystyle{ K }[/math].
- Associate each event with one of the matrices as shown, with the indices given by the labelling:
X[k, Over, Down, Down]:
[math]\displaystyle{ \Rightarrow R_{ab}^{cd} }[/math]X[k, Under, Down, Down]:
[math]\displaystyle{ \Rightarrow\bar{R}_{ab}^{cd} }[/math]
- Define [math]\displaystyle{ \tau(K) }[/math] as the result of taking the product of the matrices associated with the elements of K, and summing over repeated indices; each sum runs from 1 to [math]\displaystyle{ n=dim(V) }[/math].
As an example, let K be the left-handed trefoil, as shown. Then
[math]\displaystyle{ \tau(K) = \sum {M_\leftarrow}_{ab} }[/math]
