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Flat knot 6.39

Min(phi) over symmetries of the knot is: [-5,-1,0,0,2,4,1,3,4,2,5,1,1,1,2,0,1,2,1,3,0]
Flat knots (up to 7 crossings) with same phi are :['6.39']
Arrow polynomial of the knot is: 4*K1**2*K2 - 2*K1**2 + K1 - 2*K2**2 - 2*K2*K3 - K2 + K5 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.39']
Outer characteristic polynomial of the knot is: t^7+123t^5+48t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.39']
2-strand cable arrow polynomial of the knot is: -384*K1**2*K2**2 - 640*K1**2*K2*K4 + 1784*K1**2*K2 - 448*K1**2*K3**2 - 224*K1**2*K4**2 - 3104*K1**2 + 160*K1*K2*K3*K4**2 - 416*K1*K2*K3*K4 + 2480*K1*K2*K3 - 96*K1*K2*K4*K5 - 64*K1*K2*K4*K7 + 96*K1*K3**3*K4 - 32*K1*K3*K4*K6 + 2072*K1*K3*K4 + 608*K1*K4*K5 + 56*K1*K5*K6 + 48*K1*K6*K7 - 2*K10**2 + 8*K10*K4*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 488*K2**4 - 96*K2**2*K3**2 + 32*K2**2*K4**3 - 432*K2**2*K4**2 + 1608*K2**2*K4 - 2614*K2**2 - 192*K2*K3**2*K4 + 32*K2*K3*K4**2*K5 - 64*K2*K3*K4*K5 + 576*K2*K3*K5 + 384*K2*K4*K6 + 40*K2*K5*K7 - 192*K3**4 - 192*K3**2*K4**2 + 248*K3**2*K6 - 1712*K3**2 + 176*K3*K4*K7 - 8*K4**4 - 32*K4**2*K5**2 - 8*K4**2*K6**2 - 1518*K4**2 + 8*K4*K5*K9 - 416*K5**2 - 168*K6**2 - 80*K7**2 + 3108
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.39']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71578', 'vk6.71696', 'vk6.72113', 'vk6.72315', 'vk6.73475', 'vk6.74116', 'vk6.74138', 'vk6.74686', 'vk6.74707', 'vk6.75234', 'vk6.75483', 'vk6.76152', 'vk6.76185', 'vk6.77194', 'vk6.77294', 'vk6.77497', 'vk6.77652', 'vk6.78444', 'vk6.79117', 'vk6.79141', 'vk6.80032', 'vk6.80180', 'vk6.80622', 'vk6.80639', 'vk6.83726', 'vk6.83854', 'vk6.85058', 'vk6.85327', 'vk6.86668', 'vk6.86974', 'vk6.87408', 'vk6.89527']
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 O1O2O3O4O5O6U1U4U6U3U2U5
R3 orbit {'O1O2O3O4O5O6U1U4U6U3U2U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U2U5U4U1U3U6
Gauss code of K* O1O2O3O4O5O6U1U5U4U2U6U3
Gauss code of -K* O1O2O3O4O5O6U4U1U5U3U2U6
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 -5 0 0 -1 4 2],[ 5 0 4 3 1 5 2],[ 0 -4 0 0 -1 3 1],[ 0 -3 0 0 -1 2 1],[ 1 -1 1 1 0 2 1],[-4 -5 -3 -2 -2 0 0],[-2 -2 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 4 2 0 0 -1 -5],[-4 0 0 -2 -3 -2 -5],[-2 0 0 -1 -1 -1 -2],[ 0 2 1 0 0 -1 -3],[ 0 3 1 0 0 -1 -4],[ 1 2 1 1 1 0 -1],[ 5 5 2 3 4 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,0,0,1,5,0,2,3,2,5,1,1,1,2,0,1,3,1,4,1]
Phi over symmetry [-5,-1,0,0,2,4,1,3,4,2,5,1,1,1,2,0,1,2,1,3,0]
Phi of -K [-5,-1,0,0,2,4,3,1,2,5,4,0,0,2,3,0,1,1,1,2,2]
Phi of K* [-4,-2,0,0,1,5,2,1,2,3,4,1,1,2,5,0,0,1,0,2,3]
Phi of -K* [-5,-1,0,0,2,4,1,3,4,2,5,1,1,1,2,0,1,2,1,3,0]
Symmetry type of based matrix c
u-polynomial t^5-t^4-t^2+t
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial 6w^3z^2+23w^2z+23w
Inner characteristic polynomial t^6+77t^4+11t^2
Outer characteristic polynomial t^7+123t^5+48t^3+4t
Flat arrow polynomial 4*K1**2*K2 - 2*K1**2 + K1 - 2*K2**2 - 2*K2*K3 - K2 + K5 + 2
2-strand cable arrow polynomial -384*K1**2*K2**2 - 640*K1**2*K2*K4 + 1784*K1**2*K2 - 448*K1**2*K3**2 - 224*K1**2*K4**2 - 3104*K1**2 + 160*K1*K2*K3*K4**2 - 416*K1*K2*K3*K4 + 2480*K1*K2*K3 - 96*K1*K2*K4*K5 - 64*K1*K2*K4*K7 + 96*K1*K3**3*K4 - 32*K1*K3*K4*K6 + 2072*K1*K3*K4 + 608*K1*K4*K5 + 56*K1*K5*K6 + 48*K1*K6*K7 - 2*K10**2 + 8*K10*K4*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 488*K2**4 - 96*K2**2*K3**2 + 32*K2**2*K4**3 - 432*K2**2*K4**2 + 1608*K2**2*K4 - 2614*K2**2 - 192*K2*K3**2*K4 + 32*K2*K3*K4**2*K5 - 64*K2*K3*K4*K5 + 576*K2*K3*K5 + 384*K2*K4*K6 + 40*K2*K5*K7 - 192*K3**4 - 192*K3**2*K4**2 + 248*K3**2*K6 - 1712*K3**2 + 176*K3*K4*K7 - 8*K4**4 - 32*K4**2*K5**2 - 8*K4**2*K6**2 - 1518*K4**2 + 8*K4*K5*K9 - 416*K5**2 - 168*K6**2 - 80*K7**2 + 3108
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {5}, {2, 4}, {1}]]
If K is slice False
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