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Flat knot 6.278

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,0,1,4,2,4,1,3,2,2,1,1,1,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.278']
Arrow polynomial of the knot is: -8*K1**4 + 4*K1**3 + 4*K1**2*K2 + 4*K1**2 - 2*K1*K2 - 2*K1 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.179', '6.278', '6.415']
Outer characteristic polynomial of the knot is: t^7+93t^5+128t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.278']
2-strand cable arrow polynomial of the knot is: -736*K1**4 - 192*K1**3*K3 - 768*K1**2*K2**6 + 1152*K1**2*K2**5 - 3264*K1**2*K2**4 - 512*K1**2*K2**3*K4 + 3232*K1**2*K2**3 - 3584*K1**2*K2**2 - 192*K1**2*K2*K4 + 3240*K1**2*K2 - 64*K1**2*K3**2 - 2128*K1**2 + 1408*K1*K2**5*K3 + 384*K1*K2**4*K3*K4 - 1024*K1*K2**4*K3 - 128*K1*K2**4*K5 + 2432*K1*K2**3*K3 - 256*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 2272*K1*K2*K3 + 216*K1*K3*K4 + 40*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1312*K2**6 - 704*K2**4*K3**2 - 288*K2**4*K4**2 + 992*K2**4*K4 - 784*K2**4 + 64*K2**3*K3*K5 - 224*K2**2*K3**2 - 8*K2**2*K4**2 + 408*K2**2*K4 - 560*K2**2 + 72*K2*K3*K5 - 632*K3**2 - 156*K4**2 - 40*K5**2 + 1578
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.278']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16898', 'vk6.17142', 'vk6.20205', 'vk6.21487', 'vk6.23282', 'vk6.23584', 'vk6.27383', 'vk6.29013', 'vk6.35284', 'vk6.35726', 'vk6.38806', 'vk6.40981', 'vk6.42799', 'vk6.43084', 'vk6.45557', 'vk6.47342', 'vk6.55039', 'vk6.55286', 'vk6.57046', 'vk6.58146', 'vk6.59419', 'vk6.59714', 'vk6.61545', 'vk6.62725', 'vk6.64882', 'vk6.65099', 'vk6.66662', 'vk6.67489', 'vk6.68195', 'vk6.68342', 'vk6.69307', 'vk6.70069']
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 O1O2O3O4O5U1U5O6U2U3U4U6
R3 orbit {'O1O2O3O4O5U1U5O6U2U3U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U2U3U4O6U1U5
Gauss code of K* O1O2O3O4U5U1U2U3U6O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U5U2U3U4U6
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 -4 -2 0 2 1 3],[ 4 0 2 3 4 1 3],[ 2 -2 0 1 2 0 3],[ 0 -3 -1 0 1 0 2],[-2 -4 -2 -1 0 0 1],[-1 -1 0 0 0 0 0],[-3 -3 -3 -2 -1 0 0]]
Primitive based matrix [[ 0 3 2 1 0 -2 -4],[-3 0 -1 0 -2 -3 -3],[-2 1 0 0 -1 -2 -4],[-1 0 0 0 0 0 -1],[ 0 2 1 0 0 -1 -3],[ 2 3 2 0 1 0 -2],[ 4 3 4 1 3 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,0,2,4,1,0,2,3,3,0,1,2,4,0,0,1,1,3,2]
Phi over symmetry [-4,-2,0,1,2,3,0,1,4,2,4,1,3,2,2,1,1,1,1,2,0]
Phi of -K [-4,-2,0,1,2,3,0,1,4,2,4,1,3,2,2,1,1,1,1,2,0]
Phi of K* [-3,-2,-1,0,2,4,0,2,1,2,4,1,1,2,2,1,3,4,1,1,0]
Phi of -K* [-4,-2,0,1,2,3,2,3,1,4,3,1,0,2,3,0,1,2,0,0,1]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t
Normalized Jones-Krushkal polynomial 5z+11
Enhanced Jones-Krushkal polynomial -6w^4z^2+6w^3z^2-10w^3z+15w^2z+11w
Inner characteristic polynomial t^6+59t^4+44t^2
Outer characteristic polynomial t^7+93t^5+128t^3+6t
Flat arrow polynomial -8*K1**4 + 4*K1**3 + 4*K1**2*K2 + 4*K1**2 - 2*K1*K2 - 2*K1 + 1
2-strand cable arrow polynomial -736*K1**4 - 192*K1**3*K3 - 768*K1**2*K2**6 + 1152*K1**2*K2**5 - 3264*K1**2*K2**4 - 512*K1**2*K2**3*K4 + 3232*K1**2*K2**3 - 3584*K1**2*K2**2 - 192*K1**2*K2*K4 + 3240*K1**2*K2 - 64*K1**2*K3**2 - 2128*K1**2 + 1408*K1*K2**5*K3 + 384*K1*K2**4*K3*K4 - 1024*K1*K2**4*K3 - 128*K1*K2**4*K5 + 2432*K1*K2**3*K3 - 256*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 2272*K1*K2*K3 + 216*K1*K3*K4 + 40*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1312*K2**6 - 704*K2**4*K3**2 - 288*K2**4*K4**2 + 992*K2**4*K4 - 784*K2**4 + 64*K2**3*K3*K5 - 224*K2**2*K3**2 - 8*K2**2*K4**2 + 408*K2**2*K4 - 560*K2**2 + 72*K2*K3*K5 - 632*K3**2 - 156*K4**2 - 40*K5**2 + 1578
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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