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Flat knot 6.1928

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,1,1,2,3,0,1,1,2,0,1,1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1928']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.209', '6.231', '6.391', '6.419', '6.600', '6.661', '6.744', '6.812', '6.826', '6.1114', '6.1125', '6.1202', '6.1275', '6.1292', '6.1305', '6.1322', '6.1365', '6.1481', '6.1483', '6.1497', '6.1543', '6.1549', '6.1572', '6.1577', '6.1580', '6.1594', '6.1641', '6.1658', '6.1683', '6.1753', '6.1830', '6.1907', '6.1928']
Outer characteristic polynomial of the knot is: t^7+27t^5+55t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1928']
2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 224*K1**4*K2 - 1008*K1**4 + 96*K1**3*K2*K3 - 32*K1**3*K3 + 256*K1**2*K2**3 - 2288*K1**2*K2**2 - 256*K1**2*K2*K4 + 4056*K1**2*K2 - 144*K1**2*K3**2 - 3220*K1**2 + 288*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 416*K1*K2**2*K3 - 320*K1*K2*K3*K4 + 3176*K1*K2*K3 - 224*K1*K2*K4*K5 + 1008*K1*K3*K4 + 216*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 576*K2**4 - 512*K2**2*K3**2 - 400*K2**2*K4**2 + 1464*K2**2*K4 - 2918*K2**2 - 64*K2*K3**2*K4 + 656*K2*K3*K5 + 336*K2*K4*K6 - 1404*K3**2 - 956*K4**2 - 248*K5**2 - 42*K6**2 + 3058
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1928']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11294', 'vk6.11372', 'vk6.12559', 'vk6.12670', 'vk6.18353', 'vk6.18693', 'vk6.24799', 'vk6.25258', 'vk6.30978', 'vk6.31105', 'vk6.32162', 'vk6.32281', 'vk6.36977', 'vk6.37433', 'vk6.44163', 'vk6.44485', 'vk6.52050', 'vk6.52133', 'vk6.52893', 'vk6.52956', 'vk6.56127', 'vk6.56353', 'vk6.60648', 'vk6.60991', 'vk6.63667', 'vk6.63712', 'vk6.64099', 'vk6.64144', 'vk6.65777', 'vk6.66038', 'vk6.68782', 'vk6.68992']
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 O1O2O3U1U4U3O5O4O6U2U6U5
R3 orbit {'O1O2O3U1U4U3O5O4O6U2U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U2U5O4O6O5U3U1U6
Gauss code of K* O1O2O3U4U1U5O4O6O5U3U6U2
Gauss code of -K* O1O2O3U1U4U3O5O6O4U6U2U5
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 2 0 0 1],[ 2 0 2 1 1 1 1],[ 1 -2 0 1 0 1 1],[-2 -1 -1 0 -2 -1 0],[ 0 -1 0 2 0 0 1],[ 0 -1 -1 1 0 0 0],[-1 -1 -1 0 -1 0 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 0 -1 -2 -1 -1],[-1 0 0 0 -1 -1 -1],[ 0 1 0 0 0 -1 -1],[ 0 2 1 0 0 0 -1],[ 1 1 1 1 0 0 -2],[ 2 1 1 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,0,1,2,1,1,0,1,1,1,0,1,1,0,1,2]
Phi over symmetry [-2,-1,0,0,1,2,-1,1,1,2,3,0,1,1,2,0,1,1,0,0,1]
Phi of -K [-2,-1,0,0,1,2,-1,1,1,2,3,0,1,1,2,0,1,1,0,0,1]
Phi of K* [-2,-1,0,0,1,2,1,0,1,2,3,0,1,1,2,0,1,1,0,1,-1]
Phi of -K* [-2,-1,0,0,1,2,2,1,1,1,1,0,1,1,1,0,1,2,0,1,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 3z^2+20z+29
Enhanced Jones-Krushkal polynomial 3w^3z^2+20w^2z+29w
Inner characteristic polynomial t^6+17t^4+27t^2+1
Outer characteristic polynomial t^7+27t^5+55t^3+5t
Flat arrow polynomial 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -64*K1**4*K2**2 + 224*K1**4*K2 - 1008*K1**4 + 96*K1**3*K2*K3 - 32*K1**3*K3 + 256*K1**2*K2**3 - 2288*K1**2*K2**2 - 256*K1**2*K2*K4 + 4056*K1**2*K2 - 144*K1**2*K3**2 - 3220*K1**2 + 288*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 416*K1*K2**2*K3 - 320*K1*K2*K3*K4 + 3176*K1*K2*K3 - 224*K1*K2*K4*K5 + 1008*K1*K3*K4 + 216*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 576*K2**4 - 512*K2**2*K3**2 - 400*K2**2*K4**2 + 1464*K2**2*K4 - 2918*K2**2 - 64*K2*K3**2*K4 + 656*K2*K3*K5 + 336*K2*K4*K6 - 1404*K3**2 - 956*K4**2 - 248*K5**2 - 42*K6**2 + 3058
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {4, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {3, 5}, {2, 4}, {1}]]
If K is slice False
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