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Flat knot 6.254

Min(phi) over symmetries of the knot is: [-4,-2,0,2,2,2,1,2,2,3,4,1,1,2,2,0,1,0,0,-2,-1]
Flat knots (up to 7 crossings) with same phi are :['6.254']
Arrow polynomial of the knot is: -8*K1**4 + 4*K1**2*K2 + 4*K1**2 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.61', '6.177', '6.254', '6.357', '6.477']
Outer characteristic polynomial of the knot is: t^7+98t^5+113t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.254']
2-strand cable arrow polynomial of the knot is: -128*K2**8 + 128*K2**6*K4 - 832*K2**6 - 32*K2**4*K4**2 + 640*K2**4*K4 - 704*K2**4 - 144*K2**2*K4**2 + 752*K2**2*K4 + 304*K2**2 + 16*K2*K4*K6 - 136*K4**2 + 134
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.254']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73181', 'vk6.73194', 'vk6.73199', 'vk6.74152', 'vk6.74302', 'vk6.74305', 'vk6.74740', 'vk6.74942', 'vk6.74947', 'vk6.75086', 'vk6.75096', 'vk6.75115', 'vk6.76265', 'vk6.76273', 'vk6.76509', 'vk6.76516', 'vk6.76814', 'vk6.76926', 'vk6.78026', 'vk6.78030', 'vk6.78053', 'vk6.79172', 'vk6.79176', 'vk6.79352', 'vk6.79636', 'vk6.79768', 'vk6.79773', 'vk6.80654', 'vk6.80658', 'vk6.80810', 'vk6.85557', 'vk6.85572', 'vk6.85782', 'vk6.85857', 'vk6.87639', 'vk6.87647', 'vk6.87840', 'vk6.89380', 'vk6.90177', 'vk6.90203']
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 O1O2O3O4O5U1U2O6U4U5U6U3
R3 orbit {'O1O2O3O4O5U1U2U3O6U5U4U6', 'O1O2O3O4O5U1U2O6U4U5U6U3'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5U3U6U1U2O6U4U5
Gauss code of K* O1O2O3O4U5U6U4U1U2O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U3U4U1U5U6
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable True
If K is almost classical False
Based matrix from Gauss code [[ 0 -4 -2 2 0 2 2],[ 4 0 1 4 2 3 2],[ 2 -1 0 3 1 2 2],[-2 -4 -3 0 -2 0 2],[ 0 -2 -1 2 0 1 2],[-2 -3 -2 0 -1 0 1],[-2 -2 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 2 2 2 0 -2 -4],[-2 0 2 0 -2 -3 -4],[-2 -2 0 -1 -2 -2 -2],[-2 0 1 0 -1 -2 -3],[ 0 2 2 1 0 -1 -2],[ 2 3 2 2 1 0 -1],[ 4 4 2 3 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-2,0,2,4,-2,0,2,3,4,1,2,2,2,1,2,3,1,2,1]
Phi over symmetry [-4,-2,0,2,2,2,1,2,2,3,4,1,1,2,2,0,1,0,0,-2,-1]
Phi of -K [-4,-2,0,2,2,2,1,2,2,3,4,1,1,2,2,0,1,0,0,-2,-1]
Phi of K* [-2,-2,-2,0,2,4,-2,-1,0,2,4,0,0,1,2,1,2,3,1,2,1]
Phi of -K* [-4,-2,0,2,2,2,1,2,2,3,4,1,2,2,3,2,1,2,-1,-2,0]
Symmetry type of based matrix c
u-polynomial t^4-2t^2
Normalized Jones-Krushkal polynomial 3z^2+8z+5
Enhanced Jones-Krushkal polynomial -4w^4z^2+7w^3z^2+8w^2z+5
Inner characteristic polynomial t^6+66t^4+17t^2
Outer characteristic polynomial t^7+98t^5+113t^3
Flat arrow polynomial -8*K1**4 + 4*K1**2*K2 + 4*K1**2 + 1
2-strand cable arrow polynomial -128*K2**8 + 128*K2**6*K4 - 832*K2**6 - 32*K2**4*K4**2 + 640*K2**4*K4 - 704*K2**4 - 144*K2**2*K4**2 + 752*K2**2*K4 + 304*K2**2 + 16*K2*K4*K6 - 136*K4**2 + 134
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}]]
If K is slice False
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