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Flat knot 6.319

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,0,1,3,5,3,0,1,2,1,0,1,1,1,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.319']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + 2*K2 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.155', '6.319']
Outer characteristic polynomial of the knot is: t^7+95t^5+75t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.319']
2-strand cable arrow polynomial of the knot is: -400*K1**4 + 96*K1**3*K2*K3 + 64*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 784*K1**2*K2**2 + 1016*K1**2*K2 - 416*K1**2*K3**2 - 1212*K1**2 + 224*K1*K2**3*K3 + 96*K1*K2*K3**3 + 2008*K1*K2*K3 + 520*K1*K3*K4 + 24*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 328*K2**4 + 32*K2**3*K3*K5 - 480*K2**2*K3**2 - 8*K2**2*K4**2 + 176*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 892*K2**2 + 328*K2*K3*K5 + 16*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 80*K3**4 + 64*K3**2*K6 - 932*K3**2 - 232*K4**2 - 96*K5**2 - 36*K6**2 - 2*K8**2 + 1304
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.319']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71375', 'vk6.71436', 'vk6.71897', 'vk6.71958', 'vk6.72463', 'vk6.72615', 'vk6.72732', 'vk6.72826', 'vk6.72889', 'vk6.73044', 'vk6.74224', 'vk6.74370', 'vk6.74418', 'vk6.74853', 'vk6.75033', 'vk6.76607', 'vk6.76902', 'vk6.77032', 'vk6.77402', 'vk6.77765', 'vk6.77816', 'vk6.79270', 'vk6.79416', 'vk6.79744', 'vk6.79832', 'vk6.79875', 'vk6.80862', 'vk6.80906', 'vk6.81370', 'vk6.85521', 'vk6.87218', 'vk6.89262']
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 O1O2O3O4O5U2U4O6U3U1U6U5
R3 orbit {'O1O2O3O4O5U2U4O6U3U1U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U6U5U3O6U2U4
Gauss code of K* O1O2O3O4U2U5U1U6U4O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U1U5U4U6U3
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 -3 -1 0 4 2],[ 2 0 -2 1 1 5 2],[ 3 2 0 2 1 3 1],[ 1 -1 -2 0 0 3 1],[ 0 -1 -1 0 0 1 0],[-4 -5 -3 -3 -1 0 0],[-2 -2 -1 -1 0 0 0]]
Primitive based matrix [[ 0 4 2 0 -1 -2 -3],[-4 0 0 -1 -3 -5 -3],[-2 0 0 0 -1 -2 -1],[ 0 1 0 0 0 -1 -1],[ 1 3 1 0 0 -1 -2],[ 2 5 2 1 1 0 -2],[ 3 3 1 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,0,1,2,3,0,1,3,5,3,0,1,2,1,0,1,1,1,2,2]
Phi over symmetry [-4,-2,0,1,2,3,0,1,3,5,3,0,1,2,1,0,1,1,1,2,2]
Phi of -K [-3,-2,-1,0,2,4,-1,0,2,4,4,0,1,2,1,1,2,2,2,3,2]
Phi of K* [-4,-2,0,1,2,3,2,3,2,1,4,2,2,2,4,1,1,2,0,0,-1]
Phi of -K* [-3,-2,-1,0,2,4,2,2,1,1,3,1,1,2,5,0,1,3,0,1,0]
Symmetry type of based matrix c
u-polynomial -t^4+t^3+t
Normalized Jones-Krushkal polynomial 11z+23
Enhanced Jones-Krushkal polynomial -2w^3z+13w^2z+23w
Inner characteristic polynomial t^6+61t^4+19t^2
Outer characteristic polynomial t^7+95t^5+75t^3
Flat arrow polynomial 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + 2*K2 + K4 + 2
2-strand cable arrow polynomial -400*K1**4 + 96*K1**3*K2*K3 + 64*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 784*K1**2*K2**2 + 1016*K1**2*K2 - 416*K1**2*K3**2 - 1212*K1**2 + 224*K1*K2**3*K3 + 96*K1*K2*K3**3 + 2008*K1*K2*K3 + 520*K1*K3*K4 + 24*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 328*K2**4 + 32*K2**3*K3*K5 - 480*K2**2*K3**2 - 8*K2**2*K4**2 + 176*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 892*K2**2 + 328*K2*K3*K5 + 16*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 80*K3**4 + 64*K3**2*K6 - 932*K3**2 - 232*K4**2 - 96*K5**2 - 36*K6**2 - 2*K8**2 + 1304
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {5}, {1, 4}, {2}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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