Stephan Klaus: Knots
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Stephan Klaus: Knots
Stephan Klaus is professor of topology and works at the Mathematisches Forschungsinstitut Oberwolfach. He used SURFER to visualize knotted surfaces. He found a method to construct such polynomials by Fourier decomposition and algebraic variable elimination for every knot type.
Formula
- ((a-b)\cdot (x\cdot (x^2+y^2-z^2+1)-2\,yz)-(2a+2b+ab)\cdot (x^2+y^2))^2 -{(x^2+y^2)\cdot ((a+b)\cdot (x^2+y^2+z^2+1)+2\cdot (a-b)\cdot (yz-x))}^2=0
Möbiusband 1
Licence CC BY-NC-SA-3.0
Formula
- ((a-b)\cdot (x\cdot (x^2+y^2-z^2+1)-2\,yz)-(2a+2b+ab)\cdot (x^2+y^2))^2 -{(x^2+y^2)\cdot ((a+b)\cdot (x^2+y^2+z^2+1)+2\cdot (a-b)\cdot (yz-x))^2}=0
Möbiusband 1
Licence CC BY-NC-SA-3.0
Formula
- ((b^2x^2+a^2y^2)\cdot (x^2+y^2)+b^2\cdot (-x+yz)^2+a^2\cdot(y+xz)^2-a^2b^2\cdot (x^2+y^2))^2 -{4\cdot (x^2+y^2)}\cdot (b^2x\cdot(-x+yz)-a^2y\cdot (y+xz))^2=0
Doppeltes Möbiusband
Licence CC BY-NC-SA-3.0
dreifaches Möbiusband 1
Licence CC BY-NC-SA-3.0
dreifaches Möbiusband 2
Licence CC BY-NC-SA-3.0
Formula
- (-8\cdot (x^2+y^2)^2\cdot (x^2+y^2+1+z^2+a^2-b^2) +{4a^2\cdot (x^3-3\,xy^2)\cdot z^2} +{4a^2\cdot (2\cdot (x^2+y^2)^2-(x^3-3\,xy^2)\cdot (x^2+y^2+1))} +{8a^2\cdot (3\,x^2y-y^3)\cdot z})^2 -{(x^2+y^2)\cdot ({2\cdot (x^2+y^2)\cdot (x^2+y^2+1+z^2+a^2-b^2)^2} +{8\cdot (x^2+y^2)^2} +{4a^2\cdot (2\cdot (x^3-3\,xy^2)-(x^2+y^2)\cdot (x^2+y^2+1))} -{8a^2\cdot (3\,x^2y-y^3)\cdot z} -{4\cdot(x^2+y^2)\cdot a^2z^2})^2}=0
Kleeblattknoten 1
Licence CC BY-NC-SA-3.0
Formula
- (-8\cdot (x^2+y^2)^2\cdot (x^2+y^2+1+z^2+a^2-b^2) +{4a^2\cdot (2\cdot(x^2+y^2)^2-(x^3-3\,xy^2)\cdot (x^2+y^2+1))} +{8a^2\cdot (3\,x^2y-y^3)\cdot z} +{4a^2\cdot (x^3-3\,xy^2)\cdot z^2})^2 -{(x^2+y^2)\cdot (2\cdot (x^2+y^2)\cdot (x^2+y^2+1+z^2+a^2-b^2)^2 +{8\cdot (x^2+y^2)^2} +{4a^2\cdot (2\cdot (x^3-3\,xy^2)-(x^2+y^2)\cdot (x^2+y^2+1))} -{8a^2\cdot (3\,x^2y-y^3)\cdot z} -{4\cdot (x^2+y^2)\cdot a^2z^2})^2}=0
Kleeblattknoten 2
Formula
- (-8\cdot (x^2+y^2)^2\cdot (x^2+y^2+1+z^2+a^2-b^2)+4a^2\cdot (2\cdot (x^2+y^2)^2-(x^3-3\,xy^2)\cdot (x^2+y^2+1))+8a^2\cdot (3\,x^2y-y^3)\cdot z+4a^2\cdot (x^3-3\,xy^2)\cdot z^2)^2-(x^2+y^2)\cdot (2\cdot(x^2+y^2)\cdot (x^2+y^2+1+z^2+a^2-b^2)^2+8\cdot (x^2+y^2)^2+4a^2\cdot (2\cdot (x^3-3\,xy^2)-(x^2+y^2)\cdot (x^2+y^2+1))-8a^2\cdot (3\,x^2y-y^3)\cdot z-4\cdot (x^2+y^2)\cdot a^2z^2)^2
Kleeblattknoten 3
Licence CC BY-NC-SA-3.0
Formula
- (-8\cdot (x^2+y^2)^2\cdot (x^2+y^2+1+z^2+a^2-b^2)+4a^2\cdot (2\cdot (x^2+y^2)^2-(x^3-3\,xy^2)\cdot (x^2+y^2+1))+8a^2\cdot (3\,x^2y-y^3)\cdot z+4a^2\cdot (x^3-3\,xy^2)\cdot z^2)^2-(x^2+y^2)\cdot (2\cdot (x^2+y^2)\cdot (x^2+y^2+1+z^2+a^2-b^2)^2+8\cdot (x^2+y^2)^2+4a^2\cdot (2\cdot (x^3-3\,xy^2)-(x^2+y^2)\cdot (x^2+y^2+1))-8a^2\cdot (3\,x^2y-y^3)\cdot z-4\cdot (x^2+y^2)\cdot a^2z^2)^2=0
Kleeblattknoten 4
Licence CC BY-NC-SA-3.0
Formula
- (-8\cdot (x^2+y^2)^2\cdot (x^2+y^2+1+z^2+a^2-b^2)+4a^2\cdot (2\cdot (x^2+y^2)^2-(x^3-3\,xy^2)\cdot (x^2+y^2+1))+8a^2\cdot (3\,x^2y-y^3)\cdot z+4a^2\cdot (x^3-3\,xy^2)\cdot z^2)^2-(x^2+y^2)\cdot (2\cdot (x^2+y^2)\cdot (x^2+y^2+1+z^2+a^2-b^2)^2+8\cdot (x^2+y^2)^2+4a^2\cdot (2\cdot (x^3-3\,xy^2)-(x^2+y^2)\cdot (x^2+y^2+1))-8a^2\cdot (3\,x^2y-y^3)\cdot z-4\cdot (x^2+y^2)\cdot a^2z^2)^2=0
Kleeblattknoten 5
Licence CC BY-NC-SA-3.0
Formula
- (-8\cdot (x^2+y^2)^2\cdot (x^2+y^2+1+z^2+a^2-b^2)+ {4a^2\cdot(2\cdot (x^2+y^2)^2 -(x^3-3\,xy^2)\cdot (x^2+y^2+1))} +{8a^2\cdot(3\,x^2y-y^3)\cdot z} +{4a^2\cdot (x^3-3\,xy^2)\cdot z^2})^2 -(x^2+y^2)\cdot ({2\cdot (x^2+y^2)}\cdot (x^2+y^2+1+z^2+a^2-b^2)^2 +{8\cdot(x^2+y^2)^2} +{4a^2\cdot(2\cdot (x^3-3\,xy^2)-(x^2+y^2)\cdot (x^2+y^2+1))} -{8a^2\cdot(3\,x^2y-y^3)\cdot z} -{4\cdot (x^2+y^2)\cdot a^2z^2})^2=0
Kleeblattknoten 6
Licence CC BY-NC-SA-3.0
Formula
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+0.0001=0
(2,5)-Torusknoten
Licence CC BY-NC-SA-3.0
