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Flat knot 6.1237

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,0,1,2,2,0,1,1,1,0,0,1,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1237', '7.41479']
Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.374', '6.446', '6.527', '6.1218', '6.1237', '6.1276', '6.1498', '6.1523', '6.1595', '6.1703', '6.1751', '6.1766', '6.1849', '6.1926']
Outer characteristic polynomial of the knot is: t^7+24t^5+34t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1237', '7.41479']
2-strand cable arrow polynomial of the knot is: -768*K1**6 - 1024*K1**4*K2**2 + 3040*K1**4*K2 - 6784*K1**4 + 1824*K1**3*K2*K3 + 256*K1**3*K3*K4 - 1248*K1**3*K3 - 448*K1**2*K2**4 + 2400*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 11824*K1**2*K2**2 - 1440*K1**2*K2*K4 + 12048*K1**2*K2 - 1888*K1**2*K3**2 - 32*K1**2*K3*K5 - 496*K1**2*K4**2 - 2516*K1**2 + 1696*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 1952*K1*K2**2*K3 - 480*K1*K2**2*K5 - 480*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 10088*K1*K2*K3 - 96*K1*K2*K4*K5 + 2128*K1*K3*K4 + 360*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 2632*K2**4 - 32*K2**3*K6 - 1760*K2**2*K3**2 - 384*K2**2*K4**2 + 2264*K2**2*K4 - 2650*K2**2 - 64*K2*K3**2*K4 + 984*K2*K3*K5 + 240*K2*K4*K6 - 1868*K3**2 - 622*K4**2 - 104*K5**2 - 14*K6**2 + 3764
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1237']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.22', 'vk6.34', 'vk6.36', 'vk6.153', 'vk6.159', 'vk6.168', 'vk6.174', 'vk6.1204', 'vk6.1210', 'vk6.1300', 'vk6.1312', 'vk6.1314', 'vk6.2355', 'vk6.2387', 'vk6.2393', 'vk6.2957', 'vk6.3533', 'vk6.3541', 'vk6.6917', 'vk6.6925', 'vk6.6948', 'vk6.6956', 'vk6.15385', 'vk6.15391', 'vk6.15503', 'vk6.33434', 'vk6.33451', 'vk6.33491', 'vk6.33506', 'vk6.33603', 'vk6.49930', 'vk6.53745']
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 O1O2O3O4U3U4U5O6O5U1U6U2
R3 orbit {'O1O2O3O4U3U4U5O6O5U1U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5U4O6O5U6U1U2
Gauss code of K* O1O2O3U4U3O5O4O6U5U6U1U2
Gauss code of -K* O1O2O3U4U2O4O5O6U5U6U1U3
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 0],[ 2 0 2 -1 1 2 0],[-1 -2 0 -1 1 0 -1],[ 1 1 1 0 1 1 0],[-1 -1 -1 -1 0 -1 0],[-1 -2 0 -1 1 0 0],[ 0 0 1 0 0 0 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 0 -1 -2],[-1 -1 0 -1 0 -1 -1],[-1 0 1 0 -1 -1 -2],[ 0 0 0 1 0 0 0],[ 1 1 1 1 0 0 1],[ 2 2 1 2 0 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,-1,0,0,1,2,1,0,1,1,1,1,2,0,0,-1]
Phi over symmetry [-2,-1,0,1,1,1,-1,0,1,2,2,0,1,1,1,0,0,1,-1,-1,0]
Phi of -K [-2,-1,0,1,1,1,2,2,1,1,2,1,1,1,1,0,1,1,0,-1,-1]
Phi of K* [-1,-1,-1,0,1,2,-1,-1,1,1,2,0,0,1,1,1,1,1,1,2,2]
Phi of -K* [-2,-1,0,1,1,1,-1,0,1,2,2,0,1,1,1,0,0,1,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+16t^4+17t^2+4
Outer characteristic polynomial t^7+24t^5+34t^3+11t
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6
2-strand cable arrow polynomial -768*K1**6 - 1024*K1**4*K2**2 + 3040*K1**4*K2 - 6784*K1**4 + 1824*K1**3*K2*K3 + 256*K1**3*K3*K4 - 1248*K1**3*K3 - 448*K1**2*K2**4 + 2400*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 11824*K1**2*K2**2 - 1440*K1**2*K2*K4 + 12048*K1**2*K2 - 1888*K1**2*K3**2 - 32*K1**2*K3*K5 - 496*K1**2*K4**2 - 2516*K1**2 + 1696*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 1952*K1*K2**2*K3 - 480*K1*K2**2*K5 - 480*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 10088*K1*K2*K3 - 96*K1*K2*K4*K5 + 2128*K1*K3*K4 + 360*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 2632*K2**4 - 32*K2**3*K6 - 1760*K2**2*K3**2 - 384*K2**2*K4**2 + 2264*K2**2*K4 - 2650*K2**2 - 64*K2*K3**2*K4 + 984*K2*K3*K5 + 240*K2*K4*K6 - 1868*K3**2 - 622*K4**2 - 104*K5**2 - 14*K6**2 + 3764
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {5}, {3, 4}, {1}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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