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Flat knot 6.1017

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,-1,1,1,3,3,1,0,1,0,0,1,1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1017']
Arrow polynomial of the knot is: 8*K1**3 - 10*K1**2 - 6*K1*K2 - 3*K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.576', '6.581', '6.622', '6.627', '6.983', '6.1017']
Outer characteristic polynomial of the knot is: t^7+42t^5+40t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.611', '6.1017', '6.1020']
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 256*K1**4*K2**2 + 2496*K1**4*K2 - 6192*K1**4 + 480*K1**3*K2*K3 - 1184*K1**3*K3 - 192*K1**2*K2**4 + 1120*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 7296*K1**2*K2**2 - 576*K1**2*K2*K4 + 12944*K1**2*K2 - 944*K1**2*K3**2 - 48*K1**2*K4**2 - 6944*K1**2 + 256*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1248*K1*K2**2*K3 - 64*K1*K2**2*K5 - 160*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8912*K1*K2*K3 + 1864*K1*K3*K4 + 72*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 1032*K2**4 - 256*K2**2*K3**2 - 72*K2**2*K4**2 + 1528*K2**2*K4 - 6042*K2**2 + 296*K2*K3*K5 + 32*K2*K4*K6 - 2928*K3**2 - 946*K4**2 - 96*K5**2 - 6*K6**2 + 6504
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1017']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4816', 'vk6.5159', 'vk6.6378', 'vk6.6809', 'vk6.8343', 'vk6.8777', 'vk6.9717', 'vk6.10020', 'vk6.11638', 'vk6.11991', 'vk6.12984', 'vk6.20470', 'vk6.20725', 'vk6.21825', 'vk6.27858', 'vk6.29368', 'vk6.31437', 'vk6.32615', 'vk6.39292', 'vk6.39765', 'vk6.41472', 'vk6.46325', 'vk6.47595', 'vk6.47902', 'vk6.49053', 'vk6.49883', 'vk6.51315', 'vk6.51532', 'vk6.53233', 'vk6.57329', 'vk6.62019', 'vk6.64314']
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 O1O2O3O4U3U1O5U2O6U5U6U4
R3 orbit {'O1O2O3O4U3U1O5U2O6U5U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U5U6O5U3O6U4U2
Gauss code of K* O1O2O3U4U5U6U3O6O4U1O5U2
Gauss code of -K* O1O2O3U2O4U3O5O6U1U6U4U5
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 -2 -1 -1 3 0 1],[ 2 0 1 0 3 1 0],[ 1 -1 0 0 3 1 1],[ 1 0 0 0 1 0 0],[-3 -3 -3 -1 0 -1 1],[ 0 -1 -1 0 1 0 1],[-1 0 -1 0 -1 -1 0]]
Primitive based matrix [[ 0 3 1 0 -1 -1 -2],[-3 0 1 -1 -1 -3 -3],[-1 -1 0 -1 0 -1 0],[ 0 1 1 0 0 -1 -1],[ 1 1 0 0 0 0 0],[ 1 3 1 1 0 0 -1],[ 2 3 0 1 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,1,1,2,-1,1,1,3,3,1,0,1,0,0,1,1,0,0,1]
Phi over symmetry [-3,-1,0,1,1,2,-1,1,1,3,3,1,0,1,0,0,1,1,0,0,1]
Phi of -K [-2,-1,-1,0,1,3,0,1,1,3,2,0,0,1,1,1,2,3,0,2,3]
Phi of K* [-3,-1,0,1,1,2,3,2,1,3,2,0,1,2,3,0,1,1,0,0,1]
Phi of -K* [-2,-1,-1,0,1,3,0,1,1,0,3,0,0,0,1,1,1,3,1,1,-1]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 3z^2+24z+37
Enhanced Jones-Krushkal polynomial 3w^3z^2+24w^2z+37w
Inner characteristic polynomial t^6+26t^4+15t^2
Outer characteristic polynomial t^7+42t^5+40t^3+4t
Flat arrow polynomial 8*K1**3 - 10*K1**2 - 6*K1*K2 - 3*K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial -256*K1**6 - 256*K1**4*K2**2 + 2496*K1**4*K2 - 6192*K1**4 + 480*K1**3*K2*K3 - 1184*K1**3*K3 - 192*K1**2*K2**4 + 1120*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 7296*K1**2*K2**2 - 576*K1**2*K2*K4 + 12944*K1**2*K2 - 944*K1**2*K3**2 - 48*K1**2*K4**2 - 6944*K1**2 + 256*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1248*K1*K2**2*K3 - 64*K1*K2**2*K5 - 160*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8912*K1*K2*K3 + 1864*K1*K3*K4 + 72*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 1032*K2**4 - 256*K2**2*K3**2 - 72*K2**2*K4**2 + 1528*K2**2*K4 - 6042*K2**2 + 296*K2*K3*K5 + 32*K2*K4*K6 - 2928*K3**2 - 946*K4**2 - 96*K5**2 - 6*K6**2 + 6504
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {5}, {2, 3}, {1}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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