User:Leo algknt: Difference between revisions
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'''Question 1.''' |
'''Question 1.''' |
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A. Prove that the set of all 3-colourings of a knot diagram is a vector space over {\mathbb F}_3. Hence \lambda(K) is always a power of 3 |
A. Prove that the set of all 3-colourings of a knot diagram is a vector space over <math>{\mathbb F}_3</math>. Hence <math>\lambda(K)</math> is always a power of 3 |
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Attempt: Let <math>D</math> be a knot diagram with <math>n</math> crossings. There are <math>n</math> arcs. Let <math>a_1, a_1, \ldots, a_n \in \mathbb{Z}/3\mathbb{Z}</math> represent the arcs. Now let <math>a,b,c \in \mathbb{Z}/3\mathbb{Z}</math> , with |
Attempt: Let <math>D</math> be a knot diagram with <math>n</math> crossings. There are <math>n</math> arcs. Let <math>a_1, a_1, \ldots, a_n \in \mathbb{Z}/3\mathbb{Z}</math> represent the arcs. Now let <math>a,b,c \in \mathbb{Z}/3\mathbb{Z}</math> , with |
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$ |
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a\wedge b = |
<math>a\wedge b = |
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\left\{ |
\left\{ |
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c, & a\not= b |
c, & a\not= b |
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\end{array} |
\end{array} |
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\right. |
\right.</math> |
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$ |
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At each crossing we have a linear equation x |
At each crossing we have a linear equation x |
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Let |
Let |
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B. Prove that \lambda(K) is computable in polynomial time in the number of crossings of K. |
B. Prove that <math>\lambda(K)</math> is computable in polynomial time in the number of crossings of K. |
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Revision as of 00:13, 24 May 2018
Home Work 1
Question 1.
A. Prove that the set of all 3-colourings of a knot diagram is a vector space over . Hence is always a power of 3
Attempt: Let be a knot diagram with crossings. There are arcs. Let represent the arcs. Now let , with
At each crossing we have a linear equation x
Let
B. Prove that is computable in polynomial time in the number of crossings of K.
