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Flat knot 6.1346

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,-1,1,2,4,0,0,1,1,0,1,2,0,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1346']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1*K2 - K1 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.395', '6.430', '6.440', '6.548', '6.551', '6.774', '6.832', '6.887', '6.908', '6.911', '6.1205', '6.1332', '6.1339', '6.1341', '6.1346', '6.1382', '6.1488', '6.1651', '6.1655', '6.1686']
Outer characteristic polynomial of the knot is: t^7+44t^5+50t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1346']
2-strand cable arrow polynomial of the knot is: -1152*K1**4*K2**2 + 4864*K1**4*K2 - 6048*K1**4 - 384*K1**3*K2**2*K3 + 1536*K1**3*K2*K3 - 1088*K1**3*K3 - 128*K1**2*K2**4 + 2624*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 10224*K1**2*K2**2 + 384*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 7976*K1**2*K2 - 1056*K1**2*K3**2 - 32*K1**2*K4**2 - 1288*K1**2 + 576*K1*K2**3*K3 - 1664*K1*K2**2*K3 - 256*K1*K2**2*K5 - 352*K1*K2*K3*K4 + 6624*K1*K2*K3 + 824*K1*K3*K4 + 56*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1472*K2**4 - 32*K2**3*K6 - 464*K2**2*K3**2 - 16*K2**2*K4**2 + 1216*K2**2*K4 - 1998*K2**2 + 312*K2*K3*K5 + 16*K2*K4*K6 - 1036*K3**2 - 212*K4**2 - 28*K5**2 - 2*K6**2 + 2466
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1346']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3259', 'vk6.3289', 'vk6.3294', 'vk6.3395', 'vk6.3420', 'vk6.3425', 'vk6.3465', 'vk6.3520', 'vk6.4630', 'vk6.5915', 'vk6.6036', 'vk6.7958', 'vk6.8077', 'vk6.9386', 'vk6.17847', 'vk6.17862', 'vk6.19054', 'vk6.19866', 'vk6.24364', 'vk6.25668', 'vk6.25683', 'vk6.26307', 'vk6.26750', 'vk6.37774', 'vk6.43785', 'vk6.43800', 'vk6.45052', 'vk6.48111', 'vk6.48119', 'vk6.48144', 'vk6.48202', 'vk6.50666']
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 O1O2O3U1O4O5U4U6U5O6U3U2
R3 orbit {'O1O2O3U1O4O5U4U6U5O6U3U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U1O4U5U4U6O5O6U3
Gauss code of K* O1O2O3U2O4O5U6U5U4O6U1U3
Gauss code of -K* O1O2O3U1U3O4U5U6U4O6O5U2
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 -1 2 -1],[ 2 0 2 1 0 0 2],[-1 -2 0 0 -1 2 -2],[-1 -1 0 0 -1 2 -2],[ 1 0 1 1 0 1 0],[-2 0 -2 -2 -1 0 -2],[ 1 -2 2 2 0 2 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 -2 -2 -1 -2 0],[-1 2 0 0 -1 -2 -1],[-1 2 0 0 -1 -2 -2],[ 1 1 1 1 0 0 0],[ 1 2 2 2 0 0 -2],[ 2 0 1 2 0 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,2,2,1,2,0,0,1,2,1,1,2,2,0,0,2]
Phi over symmetry [-2,-1,-1,1,1,2,-1,-1,1,2,4,0,0,1,1,0,1,2,0,-1,1]
Phi of -K [-2,-1,-1,1,1,2,-1,1,1,2,4,0,0,0,1,1,1,2,0,-1,-1]
Phi of K* [-2,-1,-1,1,1,2,-1,-1,1,2,4,0,0,1,1,0,1,2,0,-1,1]
Phi of -K* [-2,-1,-1,1,1,2,0,2,1,2,0,0,1,1,1,2,2,2,0,2,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 9z^2+30z+25
Enhanced Jones-Krushkal polynomial 9w^3z^2+30w^2z+25w
Inner characteristic polynomial t^6+32t^4+30t^2
Outer characteristic polynomial t^7+44t^5+50t^3+3t
Flat arrow polynomial 4*K1**3 - 4*K1*K2 - K1 + K3 + 1
2-strand cable arrow polynomial -1152*K1**4*K2**2 + 4864*K1**4*K2 - 6048*K1**4 - 384*K1**3*K2**2*K3 + 1536*K1**3*K2*K3 - 1088*K1**3*K3 - 128*K1**2*K2**4 + 2624*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 10224*K1**2*K2**2 + 384*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 7976*K1**2*K2 - 1056*K1**2*K3**2 - 32*K1**2*K4**2 - 1288*K1**2 + 576*K1*K2**3*K3 - 1664*K1*K2**2*K3 - 256*K1*K2**2*K5 - 352*K1*K2*K3*K4 + 6624*K1*K2*K3 + 824*K1*K3*K4 + 56*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1472*K2**4 - 32*K2**3*K6 - 464*K2**2*K3**2 - 16*K2**2*K4**2 + 1216*K2**2*K4 - 1998*K2**2 + 312*K2*K3*K5 + 16*K2*K4*K6 - 1036*K3**2 - 212*K4**2 - 28*K5**2 - 2*K6**2 + 2466
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {4}, {1, 3}, {2}]]
If K is slice False
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