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Flat knot 6.1914

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,2,3,0,1,2,1,1,2,2,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1914']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 8*K1*K2 + K1 + 4*K2 + 3*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1209', '6.1245', '6.1509', '6.1541', '6.1704', '6.1778', '6.1914']
Outer characteristic polynomial of the knot is: t^7+32t^5+58t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1914']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 480*K1**4*K2 - 1920*K1**4 + 320*K1**3*K2*K3 - 512*K1**3*K3 + 256*K1**2*K2**3 - 2784*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 96*K1**2*K2*K4 + 5600*K1**2*K2 - 1280*K1**2*K3**2 - 48*K1**2*K4**2 - 3988*K1**2 + 224*K1*K2**3*K3 - 992*K1*K2**2*K3 - 224*K1*K2**2*K5 + 160*K1*K2*K3**3 - 224*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 5384*K1*K2*K3 + 1680*K1*K3*K4 + 216*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 544*K2**4 - 32*K2**3*K6 - 672*K2**2*K3**2 - 32*K2**2*K4**2 + 928*K2**2*K4 - 3282*K2**2 - 128*K2*K3**2*K4 + 744*K2*K3*K5 + 120*K2*K4*K6 - 160*K3**4 + 208*K3**2*K6 - 1984*K3**2 - 676*K4**2 - 244*K5**2 - 94*K6**2 + 3578
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1914']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4064', 'vk6.4095', 'vk6.5306', 'vk6.5337', 'vk6.7436', 'vk6.7463', 'vk6.8937', 'vk6.8968', 'vk6.10122', 'vk6.10289', 'vk6.10312', 'vk6.14556', 'vk6.15279', 'vk6.15406', 'vk6.15780', 'vk6.16195', 'vk6.29872', 'vk6.29903', 'vk6.33913', 'vk6.33996', 'vk6.34231', 'vk6.34381', 'vk6.48451', 'vk6.49153', 'vk6.50204', 'vk6.50229', 'vk6.51598', 'vk6.53956', 'vk6.54019', 'vk6.54180', 'vk6.54461', 'vk6.63317']
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 O1O2O3U1U3U4O5O4O6U2U5U6
R3 orbit {'O1O2O3U1U3U4O5O4O6U2U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U4U5O4O6O5U1U3U6
Gauss code of K* O1O2O3U4U1U5O4O5O6U2U6U3
Gauss code of -K* O1O2O3U1U4U2O4O5O6U5U3U6
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 -1 2],[ 2 0 2 1 2 1 1],[ 1 -2 0 0 1 0 2],[-1 -1 0 0 -1 0 0],[-1 -2 -1 1 0 -1 1],[ 1 -1 0 0 1 0 1],[-2 -1 -2 0 -1 -1 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -1 -1 -2 -1],[-1 0 0 -1 0 0 -1],[-1 1 1 0 -1 -1 -2],[ 1 1 0 1 0 0 -1],[ 1 2 0 1 0 0 -2],[ 2 1 1 2 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,0,1,1,2,1,1,0,0,1,1,1,2,0,1,2]
Phi over symmetry [-2,-1,-1,1,1,2,-1,0,1,2,3,0,1,2,1,1,2,2,-1,0,1]
Phi of -K [-2,-1,-1,1,1,2,-1,0,1,2,3,0,1,2,1,1,2,2,-1,0,1]
Phi of K* [-2,-1,-1,1,1,2,0,1,1,2,3,1,1,1,1,2,2,2,0,-1,0]
Phi of -K* [-2,-1,-1,1,1,2,1,2,1,2,1,0,0,1,1,0,1,2,-1,0,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial z^2+18z+33
Enhanced Jones-Krushkal polynomial w^3z^2+18w^2z+33w
Inner characteristic polynomial t^6+20t^4+26t^2+1
Outer characteristic polynomial t^7+32t^5+58t^3+5t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 8*K1*K2 + K1 + 4*K2 + 3*K3 + 5
2-strand cable arrow polynomial -64*K1**6 + 480*K1**4*K2 - 1920*K1**4 + 320*K1**3*K2*K3 - 512*K1**3*K3 + 256*K1**2*K2**3 - 2784*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 96*K1**2*K2*K4 + 5600*K1**2*K2 - 1280*K1**2*K3**2 - 48*K1**2*K4**2 - 3988*K1**2 + 224*K1*K2**3*K3 - 992*K1*K2**2*K3 - 224*K1*K2**2*K5 + 160*K1*K2*K3**3 - 224*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 5384*K1*K2*K3 + 1680*K1*K3*K4 + 216*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 544*K2**4 - 32*K2**3*K6 - 672*K2**2*K3**2 - 32*K2**2*K4**2 + 928*K2**2*K4 - 3282*K2**2 - 128*K2*K3**2*K4 + 744*K2*K3*K5 + 120*K2*K4*K6 - 160*K3**4 + 208*K3**2*K6 - 1984*K3**2 - 676*K4**2 - 244*K5**2 - 94*K6**2 + 3578
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}]]
If K is slice False
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