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Flat knot 6.532

Min(phi) over symmetries of the knot is: [-3,-2,-1,2,2,2,0,0,2,3,4,0,1,2,2,0,0,1,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.532']
Arrow polynomial of the knot is: 4*K1**2*K2 - 2*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.128', '6.408', '6.452', '6.532', '6.867', '6.917', '6.938', '6.1164', '6.1173', '6.1174']
Outer characteristic polynomial of the knot is: t^7+87t^5+79t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.532']
2-strand cable arrow polynomial of the knot is: -64*K1**4 + 96*K1**3*K2*K3 - 64*K1**3*K3 + 640*K1**2*K2**3 + 224*K1**2*K2**2*K4 - 2784*K1**2*K2**2 - 1312*K1**2*K2*K4 + 4448*K1**2*K2 - 64*K1**2*K3**2 - 256*K1**2*K4**2 - 4696*K1**2 + 160*K1*K2**2*K3*K4 - 1120*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 384*K1*K2**2*K5 - 768*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 5040*K1*K2*K3 - 160*K1*K2*K4*K5 - 64*K1*K2*K4*K7 + 1968*K1*K3*K4 + 872*K1*K4*K5 + 88*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 760*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 192*K2**2*K3**2 + 32*K2**2*K4**3 - 424*K2**2*K4**2 + 2464*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 3994*K2**2 - 64*K2*K3*K4*K5 + 944*K2*K3*K5 - 32*K2*K4**2*K6 + 376*K2*K4*K6 + 72*K2*K5*K7 + 8*K2*K6*K8 + 32*K3**2*K6 - 2032*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1572*K4**2 - 536*K5**2 - 110*K6**2 - 8*K7**2 - 2*K8**2 + 3924
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.532']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4235', 'vk6.4316', 'vk6.5506', 'vk6.5626', 'vk6.7615', 'vk6.7704', 'vk6.9104', 'vk6.9185', 'vk6.18369', 'vk6.18707', 'vk6.24818', 'vk6.25275', 'vk6.37006', 'vk6.37454', 'vk6.44183', 'vk6.44502', 'vk6.48547', 'vk6.48604', 'vk6.49250', 'vk6.49368', 'vk6.50340', 'vk6.50399', 'vk6.51077', 'vk6.51110', 'vk6.56150', 'vk6.56377', 'vk6.60675', 'vk6.61024', 'vk6.65818', 'vk6.66070', 'vk6.68811', 'vk6.69019']
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 O1O2O3O4O5U2U6U5O6U1U4U3
R3 orbit {'O1O2O3O4O5U2U6U5O6U1U4U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U2U5O6U1U6U4
Gauss code of K* O1O2O3U2O4O5O6U4U1U6U5U3
Gauss code of -K* O1O2O3U4O5O4O6U5U2U1U6U3
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 -3 2 2 2 -1],[ 2 0 -1 3 2 2 1],[ 3 1 0 3 2 1 2],[-2 -3 -3 0 0 1 -3],[-2 -2 -2 0 0 1 -3],[-2 -2 -1 -1 -1 0 -2],[ 1 -1 -2 3 3 2 0]]
Primitive based matrix [[ 0 2 2 2 -1 -2 -3],[-2 0 1 0 -3 -2 -2],[-2 -1 0 -1 -2 -2 -1],[-2 0 1 0 -3 -3 -3],[ 1 3 2 3 0 -1 -2],[ 2 2 2 3 1 0 -1],[ 3 2 1 3 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-2,1,2,3,-1,0,3,2,2,1,2,2,1,3,3,3,1,2,1]
Phi over symmetry [-3,-2,-1,2,2,2,0,0,2,3,4,0,1,2,2,0,0,1,0,-1,-1]
Phi of -K [-3,-2,-1,2,2,2,0,0,2,3,4,0,1,2,2,0,0,1,0,-1,-1]
Phi of K* [-2,-2,-2,1,2,3,-1,-1,1,2,4,0,0,1,2,0,2,3,0,0,0]
Phi of -K* [-3,-2,-1,2,2,2,1,2,1,2,3,1,2,2,3,2,3,3,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 8z^2+27z+23
Enhanced Jones-Krushkal polynomial 8w^3z^2-2w^3z+29w^2z+23w
Inner characteristic polynomial t^6+61t^4+39t^2+4
Outer characteristic polynomial t^7+87t^5+79t^3+13t
Flat arrow polynomial 4*K1**2*K2 - 2*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + K3 + K4 + 2
2-strand cable arrow polynomial -64*K1**4 + 96*K1**3*K2*K3 - 64*K1**3*K3 + 640*K1**2*K2**3 + 224*K1**2*K2**2*K4 - 2784*K1**2*K2**2 - 1312*K1**2*K2*K4 + 4448*K1**2*K2 - 64*K1**2*K3**2 - 256*K1**2*K4**2 - 4696*K1**2 + 160*K1*K2**2*K3*K4 - 1120*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 384*K1*K2**2*K5 - 768*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 5040*K1*K2*K3 - 160*K1*K2*K4*K5 - 64*K1*K2*K4*K7 + 1968*K1*K3*K4 + 872*K1*K4*K5 + 88*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 760*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 192*K2**2*K3**2 + 32*K2**2*K4**3 - 424*K2**2*K4**2 + 2464*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 3994*K2**2 - 64*K2*K3*K4*K5 + 944*K2*K3*K5 - 32*K2*K4**2*K6 + 376*K2*K4*K6 + 72*K2*K5*K7 + 8*K2*K6*K8 + 32*K3**2*K6 - 2032*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1572*K4**2 - 536*K5**2 - 110*K6**2 - 8*K7**2 - 2*K8**2 + 3924
Genus of based matrix 1
Fillings of based matrix [[{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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