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Flat knot 6.818

Min(phi) over symmetries of the knot is: [-3,-3,1,1,2,2,-1,2,3,2,4,1,2,1,3,1,1,0,1,0,-2]
Flat knots (up to 7 crossings) with same phi are :['6.818']
Arrow polynomial of the knot is: -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.235', '6.379', '6.411', '6.547', '6.811', '6.818', '6.823', '6.897', '6.898', '6.1008', '6.1053', '6.1109', '6.1110', '6.1130', '6.1222', '6.1239', '6.1303', '6.1307', '6.1342', '6.1491', '6.1495', '6.1496', '6.1519', '6.1592', '6.1593', '6.1642', '6.1652', '6.1653', '6.1671', '6.1673', '6.1717']
Outer characteristic polynomial of the knot is: t^7+84t^5+116t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.818']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 1024*K1**4*K2 - 2400*K1**4 + 256*K1**3*K2*K3 - 896*K1**3*K3 + 1216*K1**2*K2**3 - 6016*K1**2*K2**2 - 192*K1**2*K2*K4 + 8352*K1**2*K2 - 736*K1**2*K3**2 - 5408*K1**2 + 448*K1*K2**3*K3 - 1536*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 8016*K1*K2*K3 + 976*K1*K3*K4 - 1696*K2**4 - 832*K2**2*K3**2 - 16*K2**2*K4**2 + 1456*K2**2*K4 - 3724*K2**2 + 432*K2*K3*K5 + 16*K2*K4*K6 - 64*K3**4 + 32*K3**2*K6 - 2304*K3**2 - 384*K4**2 - 48*K5**2 - 4*K6**2 + 4350
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.818']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71642', 'vk6.71667', 'vk6.71819', 'vk6.72239', 'vk6.72266', 'vk6.72367', 'vk6.72383', 'vk6.77266', 'vk6.77275', 'vk6.77362', 'vk6.77379', 'vk6.77610', 'vk6.77702', 'vk6.77718', 'vk6.81420', 'vk6.81447', 'vk6.86954', 'vk6.87159', 'vk6.88006', 'vk6.89552']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3O4U1O5U6U4U5U2O6U3
R3 orbit {'O1O2O3O4U1O5U6U4U5U2O6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2O5U3U6U1U5O6U4
Gauss code of K* Same
Gauss code of -K* O1O2O3O4U2O5U3U6U1U5O6U4
Diagrammatic symmetry type +
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 1 2 1 2 -3],[ 3 0 2 3 1 1 1],[-1 -2 0 0 -1 1 -3],[-2 -3 0 0 0 2 -4],[-1 -1 1 0 0 1 -2],[-2 -1 -1 -2 -1 0 -2],[ 3 -1 3 4 2 2 0]]
Primitive based matrix [[ 0 2 2 1 1 -3 -3],[-2 0 2 0 0 -3 -4],[-2 -2 0 -1 -1 -1 -2],[-1 0 1 0 1 -1 -2],[-1 0 1 -1 0 -2 -3],[ 3 3 1 1 2 0 1],[ 3 4 2 2 3 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,-1,3,3,-2,0,0,3,4,1,1,1,2,-1,1,2,2,3,-1]
Phi over symmetry [-3,-3,1,1,2,2,-1,2,3,2,4,1,2,1,3,1,1,0,1,0,-2]
Phi of -K [-3,-3,1,1,2,2,-1,2,3,2,4,1,2,1,3,1,1,0,1,0,-2]
Phi of K* [-2,-2,-1,-1,3,3,-2,0,0,3,4,1,1,1,2,-1,1,2,2,3,-1]
Phi of -K* [-3,-3,1,1,2,2,-1,2,3,2,4,1,2,1,3,1,1,0,1,0,-2]
Symmetry type of based matrix +
u-polynomial 2t^3-2t^2-2t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+56t^4+46t^2+1
Outer characteristic polynomial t^7+84t^5+116t^3+11t
Flat arrow polynomial -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -256*K1**4*K2**2 + 1024*K1**4*K2 - 2400*K1**4 + 256*K1**3*K2*K3 - 896*K1**3*K3 + 1216*K1**2*K2**3 - 6016*K1**2*K2**2 - 192*K1**2*K2*K4 + 8352*K1**2*K2 - 736*K1**2*K3**2 - 5408*K1**2 + 448*K1*K2**3*K3 - 1536*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 8016*K1*K2*K3 + 976*K1*K3*K4 - 1696*K2**4 - 832*K2**2*K3**2 - 16*K2**2*K4**2 + 1456*K2**2*K4 - 3724*K2**2 + 432*K2*K3*K5 + 16*K2*K4*K6 - 64*K3**4 + 32*K3**2*K6 - 2304*K3**2 - 384*K4**2 - 48*K5**2 - 4*K6**2 + 4350
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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