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Flat knot 6.1687

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,0,1,1,1,1,1,1,1,1,2,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1687']
Arrow polynomial of the knot is: 8*K1**3 - 6*K1**2 - 4*K1*K2 - 4*K1 + 3*K2 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.813', '6.1105', '6.1107', '6.1552', '6.1687']
Outer characteristic polynomial of the knot is: t^7+26t^5+35t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1687']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 512*K1**4*K2**3 - 1664*K1**4*K2**2 + 3136*K1**4*K2 - 3760*K1**4 - 512*K1**3*K2**2*K3 + 736*K1**3*K2*K3 - 1088*K1**3*K3 - 960*K1**2*K2**4 + 4768*K1**2*K2**3 - 11520*K1**2*K2**2 - 768*K1**2*K2*K4 + 10664*K1**2*K2 - 16*K1**2*K3**2 - 5004*K1**2 + 1696*K1*K2**3*K3 - 1824*K1*K2**2*K3 - 96*K1*K2**2*K5 + 7360*K1*K2*K3 + 184*K1*K3*K4 - 64*K2**6 + 128*K2**4*K4 - 2776*K2**4 - 496*K2**2*K3**2 - 48*K2**2*K4**2 + 1544*K2**2*K4 - 2504*K2**2 + 24*K2*K3*K5 - 1244*K3**2 - 154*K4**2 + 3872
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1687']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19929', 'vk6.19984', 'vk6.21163', 'vk6.21251', 'vk6.26878', 'vk6.27007', 'vk6.28640', 'vk6.28730', 'vk6.38309', 'vk6.38417', 'vk6.40440', 'vk6.40596', 'vk6.45179', 'vk6.45303', 'vk6.47013', 'vk6.47082', 'vk6.56715', 'vk6.56798', 'vk6.57802', 'vk6.57928', 'vk6.61131', 'vk6.61302', 'vk6.62380', 'vk6.62485', 'vk6.66412', 'vk6.66510', 'vk6.67176', 'vk6.67294', 'vk6.69064', 'vk6.69156', 'vk6.69850', 'vk6.69910']
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 O1O2O3U4O5U2U3U1O4O6U5U6
R3 orbit {'O1O2O3U4O5U2U3U1O4O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O4O6U3U1U2O5U6
Gauss code of K* O1O2O3U4U5O6O5U3U1U2O4U6
Gauss code of -K* O1O2O3U4O5U2U3U1O6O4U6U5
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 0 -1 1 -2 1 1],[ 0 0 -1 1 -2 2 1],[ 1 1 0 1 0 1 1],[-1 -1 -1 0 -1 0 1],[ 2 2 0 1 0 1 0],[-1 -2 -1 0 -1 0 1],[-1 -1 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 -1 -1],[-1 -1 0 -1 -1 -1 0],[-1 0 1 0 -2 -1 -1],[ 0 1 1 2 0 -1 -2],[ 1 1 1 1 1 0 0],[ 2 1 0 1 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,-1,0,1,1,1,1,1,1,0,2,1,1,1,2,0]
Phi over symmetry [-2,-1,0,1,1,1,0,2,0,1,1,1,1,1,1,1,1,2,-1,-1,0]
Phi of -K [-2,-1,0,1,1,1,1,0,2,2,3,0,1,1,1,-1,0,0,0,-1,-1]
Phi of K* [-1,-1,-1,0,1,2,-1,-1,0,1,3,0,-1,1,2,0,1,2,0,0,1]
Phi of -K* [-2,-1,0,1,1,1,0,2,0,1,1,1,1,1,1,1,1,2,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+18t^4+18t^2+1
Outer characteristic polynomial t^7+26t^5+35t^3+8t
Flat arrow polynomial 8*K1**3 - 6*K1**2 - 4*K1*K2 - 4*K1 + 3*K2 + 4
2-strand cable arrow polynomial -128*K1**6 + 512*K1**4*K2**3 - 1664*K1**4*K2**2 + 3136*K1**4*K2 - 3760*K1**4 - 512*K1**3*K2**2*K3 + 736*K1**3*K2*K3 - 1088*K1**3*K3 - 960*K1**2*K2**4 + 4768*K1**2*K2**3 - 11520*K1**2*K2**2 - 768*K1**2*K2*K4 + 10664*K1**2*K2 - 16*K1**2*K3**2 - 5004*K1**2 + 1696*K1*K2**3*K3 - 1824*K1*K2**2*K3 - 96*K1*K2**2*K5 + 7360*K1*K2*K3 + 184*K1*K3*K4 - 64*K2**6 + 128*K2**4*K4 - 2776*K2**4 - 496*K2**2*K3**2 - 48*K2**2*K4**2 + 1544*K2**2*K4 - 2504*K2**2 + 24*K2*K3*K5 - 1244*K3**2 - 154*K4**2 + 3872
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}]]
If K is slice False
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