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Flat knot 6.479

Min(phi) over symmetries of the knot is: [-4,-2,1,1,2,2,1,1,2,4,4,0,1,2,3,0,-1,1,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.479']
Arrow polynomial of the knot is: 4*K1**2*K2 - 2*K1*K3 - K2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.64', '6.74', '6.106', '6.178', '6.300', '6.397', '6.479', '6.481', '6.500']
Outer characteristic polynomial of the knot is: t^7+87t^5+176t^3+15t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.479']
2-strand cable arrow polynomial of the knot is: -384*K1**2*K2**3*K4 + 768*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 3328*K1**2*K2**2 - 800*K1**2*K2*K4 + 3336*K1**2*K2 - 256*K1**2*K3**2 - 3024*K1**2 + 960*K1*K2**3*K3 + 736*K1*K2**2*K3*K4 - 1312*K1*K2**2*K3 - 352*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 4544*K1*K2*K3 + 1232*K1*K3*K4 + 80*K1*K4*K5 - 288*K2**4*K4**2 + 1120*K2**4*K4 - 2880*K2**4 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 1312*K2**2*K3**2 + 96*K2**2*K4**3 - 1232*K2**2*K4**2 + 2824*K2**2*K4 - 8*K2**2*K6**2 - 1688*K2**2 + 856*K2*K3*K5 + 336*K2*K4*K6 - 1480*K3**2 - 938*K4**2 - 136*K5**2 - 8*K6**2 + 2664
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.479']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73188', 'vk6.73204', 'vk6.73630', 'vk6.74312', 'vk6.74401', 'vk6.74956', 'vk6.75010', 'vk6.75101', 'vk6.75120', 'vk6.75566', 'vk6.75592', 'vk6.76524', 'vk6.76581', 'vk6.76933', 'vk6.78036', 'vk6.78060', 'vk6.78534', 'vk6.78563', 'vk6.79364', 'vk6.79788', 'vk6.79852', 'vk6.79961', 'vk6.80824', 'vk6.80882', 'vk6.83691', 'vk6.84701', 'vk6.84810', 'vk6.85268', 'vk6.85641', 'vk6.87707', 'vk6.88379', 'vk6.89493']
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 O1O2O3O4O5U1U2U5O6U4U3U6
R3 orbit {'O1O2O3O4O5U1U2U5O6U4U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U3U2O6U1U4U5
Gauss code of K* O1O2O3U4O5O6O4U1U2U6U5U3
Gauss code of -K* O1O2O3U1O4O5O6U4U3U2U5U6
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 -4 -2 1 1 2 2],[ 4 0 1 4 3 2 2],[ 2 -1 0 3 2 1 2],[-1 -4 -3 0 0 0 2],[-1 -3 -2 0 0 0 1],[-2 -2 -1 0 0 0 0],[-2 -2 -2 -2 -1 0 0]]
Primitive based matrix [[ 0 2 2 1 1 -2 -4],[-2 0 0 0 0 -1 -2],[-2 0 0 -1 -2 -2 -2],[-1 0 1 0 0 -2 -3],[-1 0 2 0 0 -3 -4],[ 2 1 2 2 3 0 -1],[ 4 2 2 3 4 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,-1,2,4,0,0,0,1,2,1,2,2,2,0,2,3,3,4,1]
Phi over symmetry [-4,-2,1,1,2,2,1,1,2,4,4,0,1,2,3,0,-1,1,0,1,0]
Phi of -K [-4,-2,1,1,2,2,1,1,2,4,4,0,1,2,3,0,-1,1,0,1,0]
Phi of K* [-2,-2,-1,-1,2,4,0,-1,0,2,4,1,1,3,4,0,0,1,1,2,1]
Phi of -K* [-4,-2,1,1,2,2,1,3,4,2,2,2,3,1,2,0,0,1,0,2,0]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 6z^2+19z+15
Enhanced Jones-Krushkal polynomial -4w^4z^2+10w^3z^2-8w^3z+27w^2z+15w
Inner characteristic polynomial t^6+57t^4+41t^2
Outer characteristic polynomial t^7+87t^5+176t^3+15t
Flat arrow polynomial 4*K1**2*K2 - 2*K1*K3 - K2
2-strand cable arrow polynomial -384*K1**2*K2**3*K4 + 768*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 3328*K1**2*K2**2 - 800*K1**2*K2*K4 + 3336*K1**2*K2 - 256*K1**2*K3**2 - 3024*K1**2 + 960*K1*K2**3*K3 + 736*K1*K2**2*K3*K4 - 1312*K1*K2**2*K3 - 352*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 4544*K1*K2*K3 + 1232*K1*K3*K4 + 80*K1*K4*K5 - 288*K2**4*K4**2 + 1120*K2**4*K4 - 2880*K2**4 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 1312*K2**2*K3**2 + 96*K2**2*K4**3 - 1232*K2**2*K4**2 + 2824*K2**2*K4 - 8*K2**2*K6**2 - 1688*K2**2 + 856*K2*K3*K5 + 336*K2*K4*K6 - 1480*K3**2 - 938*K4**2 - 136*K5**2 - 8*K6**2 + 2664
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}]]
If K is slice False
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