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Flat knot 6.177

Min(phi) over symmetries of the knot is: [-4,0,0,0,2,2,1,2,3,2,4,0,1,1,1,1,2,1,2,1,-2]
Flat knots (up to 7 crossings) with same phi are :['6.177']
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+76t^5+131t^3+25t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.177']
2-strand cable arrow polynomial of the knot is: -128*K2**8 + 128*K2**6*K4 - 1088*K2**6 - 32*K2**4*K4**2 + 1216*K2**4*K4 - 5088*K2**4 - 128*K2**3*K6 - 336*K2**2*K4**2 + 3888*K2**2*K4 + 1504*K2**2 + 64*K2*K4*K6 - 632*K4**2 + 630
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.177']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70474', 'vk6.70489', 'vk6.70530', 'vk6.70608', 'vk6.70652', 'vk6.70678', 'vk6.70760', 'vk6.70845', 'vk6.70934', 'vk6.70961', 'vk6.71007', 'vk6.71113', 'vk6.71164', 'vk6.71179', 'vk6.71243', 'vk6.71301', 'vk6.72386', 'vk6.72401', 'vk6.72752', 'vk6.73064', 'vk6.73617', 'vk6.74396', 'vk6.74933', 'vk6.75402', 'vk6.76498', 'vk6.76695', 'vk6.77731', 'vk6.78365', 'vk6.79436', 'vk6.79953', 'vk6.87173', 'vk6.90135']
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 O1O2O3O4O5U1O6U5U4U6U2U3
R3 orbit {'O1O2O3O4O5U1O6U5U4U6U2U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U4U6U2U1O6U5
Gauss code of K* O1O2O3O4O5U6U4U5U2U1O6U3
Gauss code of -K* O1O2O3O4O5U3O6U5U4U1U2U6
Diagrammatic symmetry type c
Flat genus of the diagram 3
If K is checkerboard colorable True
If K is almost classical False
Based matrix from Gauss code [[ 0 -4 0 2 0 0 2],[ 4 0 3 4 2 1 2],[ 0 -3 0 1 -1 -1 2],[-2 -4 -1 0 -1 -1 2],[ 0 -2 1 1 0 0 2],[ 0 -1 1 1 0 0 1],[-2 -2 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 2 2 0 0 0 -4],[-2 0 2 -1 -1 -1 -4],[-2 -2 0 -1 -2 -2 -2],[ 0 1 1 0 1 0 -1],[ 0 1 2 -1 0 -1 -3],[ 0 1 2 0 1 0 -2],[ 4 4 2 1 3 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,0,0,4,-2,1,1,1,4,1,2,2,2,-1,0,1,1,3,2]
Phi over symmetry [-4,0,0,0,2,2,1,2,3,2,4,0,1,1,1,1,2,1,2,1,-2]
Phi of -K [-4,0,0,0,2,2,1,2,3,2,4,1,1,1,0,0,1,0,1,1,-2]
Phi of K* [-2,-2,0,0,0,4,-2,0,0,1,4,1,1,1,2,-1,-1,1,0,2,3]
Phi of -K* [-4,0,0,0,2,2,1,2,3,2,4,0,1,1,1,1,2,1,2,1,-2]
Symmetry type of based matrix c
u-polynomial t^4-2t^2
Normalized Jones-Krushkal polynomial 3z^3+17z^2+29z+15
Enhanced Jones-Krushkal polynomial 3w^4z^3+17w^3z^2+29w^2z+15
Inner characteristic polynomial t^6+52t^4+27t^2+1
Outer characteristic polynomial t^7+76t^5+131t^3+25t
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 - 1088*K2**6 - 32*K2**4*K4**2 + 1216*K2**4*K4 - 5088*K2**4 - 128*K2**3*K6 - 336*K2**2*K4**2 + 3888*K2**2*K4 + 1504*K2**2 + 64*K2*K4*K6 - 632*K4**2 + 630
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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