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Flat knot 6.266

Min(phi) over symmetries of the knot is: [-4,-1,-1,1,2,3,0,2,3,4,3,1,1,1,1,1,2,2,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.266']
Arrow polynomial of the knot is: 4*K1**2*K2 - 8*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.266']
Outer characteristic polynomial of the knot is: t^7+90t^5+37t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.266']
2-strand cable arrow polynomial of the knot is: -80*K1**4 + 128*K1**3*K2*K3 + 32*K1**3*K3*K4 - 128*K1**3*K3 + 64*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 2080*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 384*K1**2*K2*K4 + 2760*K1**2*K2 - 176*K1**2*K3**2 - 80*K1**2*K4**2 - 2384*K1**2 + 544*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 544*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 224*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 3016*K1*K2*K3 - 32*K1*K2*K4*K5 + 504*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**4*K4**2 + 128*K2**4*K4 - 992*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 416*K2**2*K3**2 - 184*K2**2*K4**2 + 1088*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 1482*K2**2 + 248*K2*K3*K5 + 64*K2*K4*K6 + 8*K2*K5*K7 - 876*K3**2 - 318*K4**2 - 44*K5**2 - 6*K6**2 + 1740
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.266']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71571', 'vk6.71684', 'vk6.72102', 'vk6.72310', 'vk6.73749', 'vk6.73888', 'vk6.74592', 'vk6.75694', 'vk6.75711', 'vk6.75902', 'vk6.76070', 'vk6.76784', 'vk6.77191', 'vk6.77494', 'vk6.78684', 'vk6.78692', 'vk6.78890', 'vk6.79016', 'vk6.79594', 'vk6.80309', 'vk6.80317', 'vk6.80440', 'vk6.80555', 'vk6.81007', 'vk6.81100', 'vk6.81140', 'vk6.81210', 'vk6.81314', 'vk6.81701', 'vk6.82204', 'vk6.82458', 'vk6.83969', 'vk6.84437', 'vk6.86319', 'vk6.87106', 'vk6.87780', 'vk6.88014', 'vk6.88113', 'vk6.88321', 'vk6.88394']
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 O1O2O3O4O5U1U3O6U5U2U6U4
R3 orbit {'O1O2O3O4O5U1U3U4O6U2U5U6', 'O1O2O3O4O5U1U3O6U5U2U6U4'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5U2U6U4U1O6U3U5
Gauss code of K* O1O2O3O4U5U2U6U4U1O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U4U1U5U3U6
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 -1 -1 3 1 2],[ 4 0 3 1 4 2 2],[ 1 -3 0 -1 3 1 2],[ 1 -1 1 0 2 1 1],[-3 -4 -3 -2 0 -1 1],[-1 -2 -1 -1 1 0 1],[-2 -2 -2 -1 -1 -1 0]]
Primitive based matrix [[ 0 3 2 1 -1 -1 -4],[-3 0 1 -1 -2 -3 -4],[-2 -1 0 -1 -1 -2 -2],[-1 1 1 0 -1 -1 -2],[ 1 2 1 1 0 1 -1],[ 1 3 2 1 -1 0 -3],[ 4 4 2 2 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,1,1,4,-1,1,2,3,4,1,1,2,2,1,1,2,-1,1,3]
Phi over symmetry [-4,-1,-1,1,2,3,0,2,3,4,3,1,1,1,1,1,2,2,0,1,2]
Phi of -K [-4,-1,-1,1,2,3,0,2,3,4,3,1,1,1,1,1,2,2,0,1,2]
Phi of K* [-3,-2,-1,1,1,4,2,1,1,2,3,0,1,2,4,1,1,3,-1,0,2]
Phi of -K* [-4,-1,-1,1,2,3,1,3,2,2,4,1,1,1,2,1,2,3,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial 4z^2+17z+19
Enhanced Jones-Krushkal polynomial 4w^3z^2+17w^2z+19w
Inner characteristic polynomial t^6+58t^4+8t^2
Outer characteristic polynomial t^7+90t^5+37t^3+3t
Flat arrow polynomial 4*K1**2*K2 - 8*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial -80*K1**4 + 128*K1**3*K2*K3 + 32*K1**3*K3*K4 - 128*K1**3*K3 + 64*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 2080*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 384*K1**2*K2*K4 + 2760*K1**2*K2 - 176*K1**2*K3**2 - 80*K1**2*K4**2 - 2384*K1**2 + 544*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 544*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 224*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 3016*K1*K2*K3 - 32*K1*K2*K4*K5 + 504*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**4*K4**2 + 128*K2**4*K4 - 992*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 416*K2**2*K3**2 - 184*K2**2*K4**2 + 1088*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 1482*K2**2 + 248*K2*K3*K5 + 64*K2*K4*K6 + 8*K2*K5*K7 - 876*K3**2 - 318*K4**2 - 44*K5**2 - 6*K6**2 + 1740
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {4}, {2, 3}, {1}]]
If K is slice False
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