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Flat knot 6.334

Min(phi) over symmetries of the knot is: [-3,-1,0,0,1,3,-1,0,1,3,4,0,0,1,1,0,1,1,1,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.334']
Arrow polynomial of the knot is: 4*K1**2*K2 - 2*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + 2*K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.110', '6.328', '6.334', '6.842']
Outer characteristic polynomial of the knot is: t^7+64t^5+71t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.334']
2-strand cable arrow polynomial of the knot is: -1312*K1**4 + 736*K1**3*K2*K3 + 96*K1**3*K3*K4 - 672*K1**3*K3 + 160*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 2000*K1**2*K2**2 + 32*K1**2*K2*K4**2 - 832*K1**2*K2*K4 + 5184*K1**2*K2 - 864*K1**2*K3**2 - 32*K1**2*K3*K5 - 480*K1**2*K4**2 - 4708*K1**2 - 928*K1*K2**2*K3 - 256*K1*K2**2*K5 + 64*K1*K2*K3**3 + 128*K1*K2*K3*K4**2 - 576*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5360*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2680*K1*K3*K4 + 760*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**4*K4**2 + 96*K2**4*K4 - 280*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 256*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 368*K2**2*K4**2 - 32*K2**2*K4*K8 + 1760*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 4184*K2**2 - 160*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 744*K2*K3*K5 + 440*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 32*K3**4 - 64*K3**2*K4**2 + 40*K3**2*K6 - 2488*K3**2 + 40*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1584*K4**2 - 376*K5**2 - 88*K6**2 - 4*K7**2 - 2*K8**2 + 4272
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.334']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11648', 'vk6.11657', 'vk6.12001', 'vk6.12008', 'vk6.12990', 'vk6.12999', 'vk6.13265', 'vk6.20443', 'vk6.20447', 'vk6.21802', 'vk6.27819', 'vk6.27827', 'vk6.29327', 'vk6.29335', 'vk6.31450', 'vk6.32625', 'vk6.32642', 'vk6.32971', 'vk6.32984', 'vk6.39243', 'vk6.39251', 'vk6.47566', 'vk6.52365', 'vk6.53247', 'vk6.53264', 'vk6.57304', 'vk6.57316', 'vk6.61989', 'vk6.64320', 'vk6.64329', 'vk6.64502', 'vk6.66891']
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 O1O2O3O4O5U2U5O6U4U3U1U6
R3 orbit {'O1O2O3O4O5U2U5O6U4U3U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U5U3U2O6U1U4
Gauss code of K* O1O2O3O4U3U5U2U1U6O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U5U4U3U6U2
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 -1 -3 0 0 1 3],[ 1 0 -3 1 1 1 3],[ 3 3 0 3 2 1 2],[ 0 -1 -3 0 0 0 2],[ 0 -1 -2 0 0 0 1],[-1 -1 -1 0 0 0 0],[-3 -3 -2 -2 -1 0 0]]
Primitive based matrix [[ 0 3 1 0 0 -1 -3],[-3 0 0 -1 -2 -3 -2],[-1 0 0 0 0 -1 -1],[ 0 1 0 0 0 -1 -2],[ 0 2 0 0 0 -1 -3],[ 1 3 1 1 1 0 -3],[ 3 2 1 2 3 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,0,1,3,0,1,2,3,2,0,0,1,1,0,1,2,1,3,3]
Phi over symmetry [-3,-1,0,0,1,3,-1,0,1,3,4,0,0,1,1,0,1,1,1,2,2]
Phi of -K [-3,-1,0,0,1,3,-1,0,1,3,4,0,0,1,1,0,1,1,1,2,2]
Phi of K* [-3,-1,0,0,1,3,2,1,2,1,4,1,1,1,3,0,0,0,0,1,-1]
Phi of -K* [-3,-1,0,0,1,3,3,2,3,1,2,1,1,1,3,0,0,1,0,2,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 6z^2+25z+27
Enhanced Jones-Krushkal polynomial 6w^3z^2+25w^2z+27w
Inner characteristic polynomial t^6+44t^4+21t^2
Outer characteristic polynomial t^7+64t^5+71t^3+5t
Flat arrow polynomial 4*K1**2*K2 - 2*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + 2*K3 + K4 + 2
2-strand cable arrow polynomial -1312*K1**4 + 736*K1**3*K2*K3 + 96*K1**3*K3*K4 - 672*K1**3*K3 + 160*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 2000*K1**2*K2**2 + 32*K1**2*K2*K4**2 - 832*K1**2*K2*K4 + 5184*K1**2*K2 - 864*K1**2*K3**2 - 32*K1**2*K3*K5 - 480*K1**2*K4**2 - 4708*K1**2 - 928*K1*K2**2*K3 - 256*K1*K2**2*K5 + 64*K1*K2*K3**3 + 128*K1*K2*K3*K4**2 - 576*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5360*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2680*K1*K3*K4 + 760*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**4*K4**2 + 96*K2**4*K4 - 280*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 256*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 368*K2**2*K4**2 - 32*K2**2*K4*K8 + 1760*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 4184*K2**2 - 160*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 744*K2*K3*K5 + 440*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 32*K3**4 - 64*K3**2*K4**2 + 40*K3**2*K6 - 2488*K3**2 + 40*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1584*K4**2 - 376*K5**2 - 88*K6**2 - 4*K7**2 - 2*K8**2 + 4272
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}]]
If K is slice False
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