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Flat knot 6.366

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,-1,1,1,3,3,0,1,2,2,0,0,0,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.366']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 6*K1*K2 - 2*K1*K3 - 2*K2**2 + 2*K2 + 2*K3 + 2*K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.366']
Outer characteristic polynomial of the knot is: t^7+52t^5+52t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.366', '7.35701']
2-strand cable arrow polynomial of the knot is: 1248*K1**4*K2 - 3280*K1**4 + 1152*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1664*K1**3*K3 - 128*K1**2*K2**4 + 416*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 4640*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 896*K1**2*K2*K4 + 8472*K1**2*K2 - 1744*K1**2*K3**2 - 96*K1**2*K3*K5 - 224*K1**2*K4**2 - 96*K1**2*K4*K6 - 5572*K1**2 + 320*K1*K2**3*K3 - 1536*K1*K2**2*K3 - 160*K1*K2**2*K5 + 32*K1*K2*K3**3 - 960*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8312*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2984*K1*K3*K4 + 776*K1*K4*K5 + 168*K1*K5*K6 + 16*K1*K6*K7 - 32*K2**6 + 64*K2**4*K4 - 744*K2**4 - 32*K2**3*K6 - 672*K2**2*K3**2 - 32*K2**2*K3*K7 - 88*K2**2*K4**2 + 1928*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 4816*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1000*K2*K3*K5 - 32*K2*K4**2*K6 + 296*K2*K4*K6 + 80*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 - 16*K3**2*K4**2 + 112*K3**2*K6 - 3056*K3**2 + 48*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1528*K4**2 - 456*K5**2 - 184*K6**2 - 36*K7**2 - 4*K8**2 + 5154
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.366']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4357', 'vk6.4390', 'vk6.5675', 'vk6.5708', 'vk6.7740', 'vk6.7773', 'vk6.9218', 'vk6.9251', 'vk6.10501', 'vk6.10551', 'vk6.10646', 'vk6.10722', 'vk6.10755', 'vk6.10833', 'vk6.14618', 'vk6.15297', 'vk6.15422', 'vk6.16237', 'vk6.17994', 'vk6.24434', 'vk6.30180', 'vk6.30230', 'vk6.30325', 'vk6.30452', 'vk6.33943', 'vk6.34344', 'vk6.34399', 'vk6.43861', 'vk6.50427', 'vk6.50459', 'vk6.54221', 'vk6.63436']
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 O1O2O3O4O5U3U5O6U1U6U4U2
R3 orbit {'O1O2O3O4O5U3U5O6U1U6U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U2U6U5O6U1U3
Gauss code of K* O1O2O3O4U1U4U5U3U6O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U5U2U6U1U4
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 -3 1 -2 2 1 1],[ 3 0 3 -1 3 1 1],[-1 -3 0 -2 1 1 0],[ 2 1 2 0 2 1 0],[-2 -3 -1 -2 0 0 0],[-1 -1 -1 -1 0 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 1 1 1 -2 -3],[-2 0 0 0 -1 -2 -3],[-1 0 0 0 0 0 -1],[-1 0 0 0 -1 -1 -1],[-1 1 0 1 0 -2 -3],[ 2 2 0 1 2 0 1],[ 3 3 1 1 3 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,-1,2,3,0,0,1,2,3,0,0,0,1,1,1,1,2,3,-1]
Phi over symmetry [-3,-2,1,1,1,2,-1,1,1,3,3,0,1,2,2,0,0,0,-1,0,1]
Phi of -K [-3,-2,1,1,1,2,2,1,3,3,2,1,2,3,2,-1,0,0,0,1,1]
Phi of K* [-2,-1,-1,-1,2,3,0,1,1,2,2,0,1,1,1,0,3,3,2,3,2]
Phi of -K* [-3,-2,1,1,1,2,-1,1,1,3,3,0,1,2,2,0,0,0,-1,0,1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 6z^2+26z+29
Enhanced Jones-Krushkal polynomial 6w^3z^2+26w^2z+29w
Inner characteristic polynomial t^6+32t^4+22t^2+1
Outer characteristic polynomial t^7+52t^5+52t^3+7t
Flat arrow polynomial 4*K1**3 - 2*K1**2 - 6*K1*K2 - 2*K1*K3 - 2*K2**2 + 2*K2 + 2*K3 + 2*K4 + 3
2-strand cable arrow polynomial 1248*K1**4*K2 - 3280*K1**4 + 1152*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1664*K1**3*K3 - 128*K1**2*K2**4 + 416*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 4640*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 896*K1**2*K2*K4 + 8472*K1**2*K2 - 1744*K1**2*K3**2 - 96*K1**2*K3*K5 - 224*K1**2*K4**2 - 96*K1**2*K4*K6 - 5572*K1**2 + 320*K1*K2**3*K3 - 1536*K1*K2**2*K3 - 160*K1*K2**2*K5 + 32*K1*K2*K3**3 - 960*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8312*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2984*K1*K3*K4 + 776*K1*K4*K5 + 168*K1*K5*K6 + 16*K1*K6*K7 - 32*K2**6 + 64*K2**4*K4 - 744*K2**4 - 32*K2**3*K6 - 672*K2**2*K3**2 - 32*K2**2*K3*K7 - 88*K2**2*K4**2 + 1928*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 4816*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1000*K2*K3*K5 - 32*K2*K4**2*K6 + 296*K2*K4*K6 + 80*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 - 16*K3**2*K4**2 + 112*K3**2*K6 - 3056*K3**2 + 48*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1528*K4**2 - 456*K5**2 - 184*K6**2 - 36*K7**2 - 4*K8**2 + 5154
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {5}, {3, 4}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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