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Flat knot 6.1506

Min(phi) over symmetries of the knot is: [-2,-1,1,1,1,0,0,1,2,1,1,1,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['5.54', '6.1506', '7.22432']
Arrow polynomial of the knot is: 4*K1**3 + 2*K1**2 - 4*K1*K2 - K1 - K2 + K3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.140', '6.569', '6.943', '6.970', '6.1234', '6.1298', '6.1311', '6.1326', '6.1500', '6.1506', '6.1708', '6.1712', '6.1720', '6.1859']
Outer characteristic polynomial of the knot is: t^6+18t^4+16t^2+1
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.54', '6.1506']
2-strand cable arrow polynomial of the knot is: 4608*K1**4*K2 - 7264*K1**4 + 1536*K1**3*K2*K3 - 1472*K1**3*K3 - 384*K1**2*K2**4 + 768*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 8112*K1**2*K2**2 - 736*K1**2*K2*K4 + 6648*K1**2*K2 - 1248*K1**2*K3**2 - 96*K1**2*K4**2 + 904*K1**2 + 416*K1*K2**3*K3 - 544*K1*K2**2*K3 - 256*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 4672*K1*K2*K3 + 816*K1*K3*K4 + 96*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 584*K2**4 - 32*K2**3*K6 - 112*K2**2*K3**2 - 16*K2**2*K4**2 + 496*K2**2*K4 - 1102*K2**2 + 120*K2*K3*K5 + 16*K2*K4*K6 - 556*K3**2 - 158*K4**2 - 28*K5**2 - 2*K6**2 + 1348
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1506']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3254', 'vk6.3280', 'vk6.3298', 'vk6.3382', 'vk6.3413', 'vk6.3431', 'vk6.3471', 'vk6.3517', 'vk6.4627', 'vk6.5918', 'vk6.6037', 'vk6.7957', 'vk6.8074', 'vk6.9389', 'vk6.17838', 'vk6.17855', 'vk6.19057', 'vk6.19868', 'vk6.24351', 'vk6.25673', 'vk6.25690', 'vk6.26313', 'vk6.26756', 'vk6.37777', 'vk6.43776', 'vk6.43793', 'vk6.45050', 'vk6.48114', 'vk6.48127', 'vk6.48152', 'vk6.48203', 'vk6.50663']
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 O1O2O3U1O4U5U4O6O5U6U3U2
R3 orbit {'O1O2O3U1O4U5U4O6O5U6U3U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U1U4O5O4U6U5O6U3
Gauss code of K* O1O2O3U4U3U2O4U5O6O5U1U6
Gauss code of -K* O1O2O3U4U3O5O4U5O6U2U1U6
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 0 -1],[ 2 0 2 1 0 2 0],[-1 -2 0 0 1 -1 -1],[-1 -1 0 0 1 -1 -1],[-1 0 -1 -1 0 -1 -1],[ 0 -2 1 1 1 0 -1],[ 1 0 1 1 1 1 0]]
Primitive based matrix [[ 0 1 1 1 -1 -2],[-1 0 1 0 -1 -1],[-1 -1 0 -1 -1 0],[-1 0 1 0 -1 -2],[ 1 1 1 1 0 0],[ 2 1 0 2 0 0]]
If based matrix primitive False
Phi of primitive based matrix [-1,-1,-1,1,2,-1,0,1,1,1,1,0,1,2,0]
Phi over symmetry [-2,-1,1,1,1,0,0,1,2,1,1,1,-1,-1,0]
Phi of -K [-2,-1,1,1,1,1,1,2,3,1,1,1,0,-1,-1]
Phi of K* [-1,-1,-1,1,2,-1,-1,1,3,0,1,1,1,2,1]
Phi of -K* [-2,-1,1,1,1,0,0,1,2,1,1,1,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 9z^2+30z+25
Enhanced Jones-Krushkal polynomial 9w^3z^2+30w^2z+25w
Inner characteristic polynomial t^5+10t^3+7t
Outer characteristic polynomial t^6+18t^4+16t^2+1
Flat arrow polynomial 4*K1**3 + 2*K1**2 - 4*K1*K2 - K1 - K2 + K3
2-strand cable arrow polynomial 4608*K1**4*K2 - 7264*K1**4 + 1536*K1**3*K2*K3 - 1472*K1**3*K3 - 384*K1**2*K2**4 + 768*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 8112*K1**2*K2**2 - 736*K1**2*K2*K4 + 6648*K1**2*K2 - 1248*K1**2*K3**2 - 96*K1**2*K4**2 + 904*K1**2 + 416*K1*K2**3*K3 - 544*K1*K2**2*K3 - 256*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 4672*K1*K2*K3 + 816*K1*K3*K4 + 96*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 584*K2**4 - 32*K2**3*K6 - 112*K2**2*K3**2 - 16*K2**2*K4**2 + 496*K2**2*K4 - 1102*K2**2 + 120*K2*K3*K5 + 16*K2*K4*K6 - 556*K3**2 - 158*K4**2 - 28*K5**2 - 2*K6**2 + 1348
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {5}, {3, 4}, {2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}]]
If K is slice False
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