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Flat knot 6.881

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,1,1,2,3,3,0,0,0,0,1,0,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.881']
Arrow polynomial of the knot is: -6*K1*K2 + 3*K1 - 4*K2**2 + 3*K3 + 2*K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.881', '6.967', '6.1179']
Outer characteristic polynomial of the knot is: t^7+38t^5+37t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.881']
2-strand cable arrow polynomial of the knot is: -2928*K1**4 + 320*K1**3*K2*K3 + 64*K1**3*K3*K4 - 768*K1**3*K3 + 32*K1**3*K4*K5 + 96*K1**2*K2**2*K4 - 1216*K1**2*K2**2 - 1184*K1**2*K2*K4 + 5400*K1**2*K2 - 1344*K1**2*K3**2 - 256*K1**2*K3*K5 - 1008*K1**2*K4**2 - 32*K1**2*K4*K6 - 64*K1**2*K5**2 - 4452*K1**2 - 96*K1*K2**2*K3 - 320*K1*K2**2*K5 - 480*K1*K2*K3*K4 + 5016*K1*K2*K3 - 96*K1*K2*K4*K5 + 4280*K1*K3*K4 + 1760*K1*K4*K5 + 112*K1*K5*K6 - 32*K2**4 - 56*K2**2*K4**2 + 968*K2**2*K4 - 3350*K2**2 + 696*K2*K3*K5 + 112*K2*K4*K6 + 8*K2*K5*K7 - 48*K3**4 - 112*K3**2*K4**2 + 32*K3**2*K6 - 2748*K3**2 + 88*K3*K4*K7 - 48*K4**4 + 32*K4**2*K8 - 2060*K4**2 - 732*K5**2 - 58*K6**2 - 20*K7**2 - 4*K8**2 + 4494
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.881']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4872', 'vk6.5215', 'vk6.6458', 'vk6.6877', 'vk6.8423', 'vk6.8842', 'vk6.9771', 'vk6.10062', 'vk6.11687', 'vk6.12038', 'vk6.13033', 'vk6.20492', 'vk6.20761', 'vk6.21851', 'vk6.27896', 'vk6.29400', 'vk6.29726', 'vk6.32676', 'vk6.33017', 'vk6.39333', 'vk6.39793', 'vk6.46353', 'vk6.47603', 'vk6.47930', 'vk6.48838', 'vk6.49107', 'vk6.51361', 'vk6.51572', 'vk6.53286', 'vk6.57353', 'vk6.64347', 'vk6.66910']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is c.
The reverse -K is
The mirror image K* is
The reversed mirror image -K* is
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

invariant value
Gauss code O1O2O3O4U1U5O6O5U4U6U3U2
R3 orbit {'O1O2O3O4U1U5O6O5U4U6U3U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U2U5U1O6O5U6U4
Gauss code of K* O1O2O3O4U5U4U3U1O5O6U2U6
Gauss code of -K* O1O2O3O4U5U3O5O6U4U2U1U6
Diagrammatic symmetry type c
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 1 1 0 1 0],[ 3 0 3 2 1 3 1],[-1 -3 0 0 -1 0 0],[-1 -2 0 0 -1 0 0],[ 0 -1 1 1 0 0 0],[-1 -3 0 0 0 0 0],[ 0 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 1 1 1 0 0 -3],[-1 0 0 0 0 0 -3],[-1 0 0 0 0 -1 -2],[-1 0 0 0 0 -1 -3],[ 0 0 0 0 0 0 -1],[ 0 0 1 1 0 0 -1],[ 3 3 2 3 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,0,3,0,0,0,0,3,0,0,1,2,0,1,3,0,1,1]
Phi over symmetry [-3,0,0,1,1,1,1,1,2,3,3,0,0,0,0,1,0,1,0,0,0]
Phi of -K [-3,0,0,1,1,1,2,2,1,1,2,0,0,1,0,1,1,1,0,0,0]
Phi of K* [-1,-1,-1,0,0,3,0,0,0,1,1,0,0,1,2,1,1,1,0,2,2]
Phi of -K* [-3,0,0,1,1,1,1,1,2,3,3,0,0,0,0,1,0,1,0,0,0]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+26t^4+21t^2
Outer characteristic polynomial t^7+38t^5+37t^3+3t
Flat arrow polynomial -6*K1*K2 + 3*K1 - 4*K2**2 + 3*K3 + 2*K4 + 3
2-strand cable arrow polynomial -2928*K1**4 + 320*K1**3*K2*K3 + 64*K1**3*K3*K4 - 768*K1**3*K3 + 32*K1**3*K4*K5 + 96*K1**2*K2**2*K4 - 1216*K1**2*K2**2 - 1184*K1**2*K2*K4 + 5400*K1**2*K2 - 1344*K1**2*K3**2 - 256*K1**2*K3*K5 - 1008*K1**2*K4**2 - 32*K1**2*K4*K6 - 64*K1**2*K5**2 - 4452*K1**2 - 96*K1*K2**2*K3 - 320*K1*K2**2*K5 - 480*K1*K2*K3*K4 + 5016*K1*K2*K3 - 96*K1*K2*K4*K5 + 4280*K1*K3*K4 + 1760*K1*K4*K5 + 112*K1*K5*K6 - 32*K2**4 - 56*K2**2*K4**2 + 968*K2**2*K4 - 3350*K2**2 + 696*K2*K3*K5 + 112*K2*K4*K6 + 8*K2*K5*K7 - 48*K3**4 - 112*K3**2*K4**2 + 32*K3**2*K6 - 2748*K3**2 + 88*K3*K4*K7 - 48*K4**4 + 32*K4**2*K8 - 2060*K4**2 - 732*K5**2 - 58*K6**2 - 20*K7**2 - 4*K8**2 + 4494
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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