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Flat knot 6.2075

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,-1,1,0,1,1,0,1,0,-1,1,1,1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.2075']
Arrow polynomial of the knot is: 8*K1**3 - 4*K1**2 - 8*K1*K2 - 2*K1 + 2*K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.927', '6.1364', '6.1367', '6.1540', '6.1675', '6.1779', '6.1811', '6.1876', '6.2075']
Outer characteristic polynomial of the knot is: t^7+14t^5+47t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.2075']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 3488*K1**4*K2 - 5664*K1**4 + 672*K1**3*K2*K3 - 1216*K1**3*K3 - 128*K1**2*K2**4 + 1792*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 8992*K1**2*K2**2 - 928*K1**2*K2*K4 + 10176*K1**2*K2 - 544*K1**2*K3**2 - 32*K1**2*K4**2 - 3792*K1**2 + 448*K1*K2**3*K3 - 2048*K1*K2**2*K3 - 288*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 7952*K1*K2*K3 + 1232*K1*K3*K4 + 96*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 1424*K2**4 - 64*K2**3*K6 - 384*K2**2*K3**2 - 128*K2**2*K4**2 + 1976*K2**2*K4 - 4060*K2**2 + 464*K2*K3*K5 + 80*K2*K4*K6 - 1888*K3**2 - 652*K4**2 - 128*K5**2 - 12*K6**2 + 4154
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.2075']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13913', 'vk6.14010', 'vk6.14182', 'vk6.14423', 'vk6.14984', 'vk6.15107', 'vk6.15650', 'vk6.16106', 'vk6.16706', 'vk6.16733', 'vk6.16839', 'vk6.18796', 'vk6.19283', 'vk6.19577', 'vk6.23144', 'vk6.23222', 'vk6.25394', 'vk6.26470', 'vk6.33732', 'vk6.33809', 'vk6.34286', 'vk6.35140', 'vk6.37523', 'vk6.42729', 'vk6.44696', 'vk6.54129', 'vk6.54911', 'vk6.54936', 'vk6.56390', 'vk6.56620', 'vk6.59339', 'vk6.64610']
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 O1O2U1U3O4O5U2U5O3O6U4U6
R3 orbit {'O1O2U1U3O4O5U2U5O3O6U4U6'}
R3 orbit length 1
Gauss code of -K O1O2U3U4O3O5U2U6O4O6U1U5
Gauss code of K* O1O2U3U4O3O5U1U6O4O6U5U2
Gauss code of -K* O1O2U1U3O4O5U4U2O3O6U5U6
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 0 0 -1 1 1],[ 1 0 1 0 1 1 0],[ 0 -1 0 -1 1 1 1],[ 0 0 1 0 -1 1 0],[ 1 -1 -1 1 0 0 1],[-1 -1 -1 -1 0 0 0],[-1 0 -1 0 -1 0 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 0 0 -1 0 -1],[-1 0 0 -1 -1 -1 0],[ 0 0 1 0 1 0 -1],[ 0 1 1 -1 0 -1 1],[ 1 0 1 0 1 0 1],[ 1 1 0 1 -1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,1,1,0,0,1,0,1,1,1,1,0,-1,0,1,1,-1,-1]
Phi over symmetry [-1,-1,0,0,1,1,-1,-1,1,0,1,1,0,1,0,-1,1,1,1,0,0]
Phi of -K [-1,-1,0,0,1,1,-1,0,1,1,2,2,0,2,1,1,0,0,0,1,0]
Phi of K* [-1,-1,0,0,1,1,0,0,0,1,2,0,1,2,1,-1,0,2,1,0,1]
Phi of -K* [-1,-1,0,0,1,1,-1,-1,1,0,1,1,0,1,0,-1,1,1,1,0,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 7z^2+28z+29
Enhanced Jones-Krushkal polynomial 7w^3z^2+28w^2z+29w
Inner characteristic polynomial t^6+10t^4+21t^2+4
Outer characteristic polynomial t^7+14t^5+47t^3+11t
Flat arrow polynomial 8*K1**3 - 4*K1**2 - 8*K1*K2 - 2*K1 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial -256*K1**4*K2**2 + 3488*K1**4*K2 - 5664*K1**4 + 672*K1**3*K2*K3 - 1216*K1**3*K3 - 128*K1**2*K2**4 + 1792*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 8992*K1**2*K2**2 - 928*K1**2*K2*K4 + 10176*K1**2*K2 - 544*K1**2*K3**2 - 32*K1**2*K4**2 - 3792*K1**2 + 448*K1*K2**3*K3 - 2048*K1*K2**2*K3 - 288*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 7952*K1*K2*K3 + 1232*K1*K3*K4 + 96*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 1424*K2**4 - 64*K2**3*K6 - 384*K2**2*K3**2 - 128*K2**2*K4**2 + 1976*K2**2*K4 - 4060*K2**2 + 464*K2*K3*K5 + 80*K2*K4*K6 - 1888*K3**2 - 652*K4**2 - 128*K5**2 - 12*K6**2 + 4154
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
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