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Flat knot 6.1105

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,1,1,2,3,1,0,1,2,0,1,1,-1,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1105']
Arrow polynomial of the knot is: 8*K1**3 - 6*K1**2 - 4*K1*K2 - 4*K1 + 3*K2 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.813', '6.1105', '6.1107', '6.1552', '6.1687']
Outer characteristic polynomial of the knot is: t^7+41t^5+46t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1105']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 640*K1**4*K2**2 + 704*K1**4*K2 - 832*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 + 384*K1**2*K2**5 - 2496*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 3584*K1**2*K2**3 - 7472*K1**2*K2**2 - 352*K1**2*K2*K4 + 5704*K1**2*K2 - 96*K1**2*K3**2 - 96*K1**2*K4**2 - 3368*K1**2 + 2208*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 800*K1*K2**2*K3 - 128*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 4376*K1*K2*K3 + 440*K1*K3*K4 + 104*K1*K4*K5 - 448*K2**6 + 448*K2**4*K4 - 2072*K2**4 - 496*K2**2*K3**2 - 112*K2**2*K4**2 + 1040*K2**2*K4 - 968*K2**2 + 128*K2*K3*K5 - 824*K3**2 - 286*K4**2 - 32*K5**2 + 2396
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1105']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11450', 'vk6.11746', 'vk6.12763', 'vk6.13106', 'vk6.20672', 'vk6.22112', 'vk6.28175', 'vk6.29600', 'vk6.31207', 'vk6.31548', 'vk6.32377', 'vk6.32788', 'vk6.39623', 'vk6.41864', 'vk6.46231', 'vk6.47838', 'vk6.52212', 'vk6.52482', 'vk6.53045', 'vk6.53366', 'vk6.57609', 'vk6.58770', 'vk6.62273', 'vk6.63214', 'vk6.63782', 'vk6.63896', 'vk6.64212', 'vk6.64395', 'vk6.67071', 'vk6.67938', 'vk6.69685', 'vk6.70368']
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 O1O2O3O4U2U3O5U1U4O6U5U6
R3 orbit {'O1O2O3O4U2U3O5U1U4O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6O5U1U4O6U2U3
Gauss code of K* O1O2U3O4O5U6O3O6U4U1U2U5
Gauss code of -K* O1O2U1O3O4U2O5O6U3U5U6U4
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 -2 -2 0 2 1 1],[ 2 0 -1 1 3 2 1],[ 2 1 0 1 2 1 0],[ 0 -1 -1 0 1 1 0],[-2 -3 -2 -1 0 1 1],[-1 -2 -1 -1 -1 0 1],[-1 -1 0 0 -1 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -2 -2],[-2 0 1 1 -1 -2 -3],[-1 -1 0 1 -1 -1 -2],[-1 -1 -1 0 0 0 -1],[ 0 1 1 0 0 -1 -1],[ 2 2 1 0 1 0 1],[ 2 3 2 1 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,2,2,-1,-1,1,2,3,-1,1,1,2,0,0,1,1,1,-1]
Phi over symmetry [-2,-2,0,1,1,2,-1,1,1,2,3,1,0,1,2,0,1,1,-1,-1,-1]
Phi of -K [-2,-2,0,1,1,2,-1,1,2,3,2,1,1,2,1,0,1,1,-1,2,2]
Phi of K* [-2,-1,-1,0,2,2,2,2,1,1,2,-1,1,2,3,0,1,2,1,1,-1]
Phi of -K* [-2,-2,0,1,1,2,-1,1,1,2,3,1,0,1,2,0,1,1,-1,-1,-1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 3z^2+16z+21
Enhanced Jones-Krushkal polynomial -2w^4z^2+5w^3z^2-6w^3z+22w^2z+21w
Inner characteristic polynomial t^6+27t^4+17t^2
Outer characteristic polynomial t^7+41t^5+46t^3+7t
Flat arrow polynomial 8*K1**3 - 6*K1**2 - 4*K1*K2 - 4*K1 + 3*K2 + 4
2-strand cable arrow polynomial 256*K1**4*K2**3 - 640*K1**4*K2**2 + 704*K1**4*K2 - 832*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 + 384*K1**2*K2**5 - 2496*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 3584*K1**2*K2**3 - 7472*K1**2*K2**2 - 352*K1**2*K2*K4 + 5704*K1**2*K2 - 96*K1**2*K3**2 - 96*K1**2*K4**2 - 3368*K1**2 + 2208*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 800*K1*K2**2*K3 - 128*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 4376*K1*K2*K3 + 440*K1*K3*K4 + 104*K1*K4*K5 - 448*K2**6 + 448*K2**4*K4 - 2072*K2**4 - 496*K2**2*K3**2 - 112*K2**2*K4**2 + 1040*K2**2*K4 - 968*K2**2 + 128*K2*K3*K5 - 824*K3**2 - 286*K4**2 - 32*K5**2 + 2396
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}]]
If K is slice False
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