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Flat knot 6.541

Min(phi) over symmetries of the knot is: [-2,-2,-1,1,2,2,0,-1,0,2,3,-1,1,3,3,0,1,1,1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.541']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 2*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 - 2*K2**2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.541', '6.1257']
Outer characteristic polynomial of the knot is: t^7+59t^5+100t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.541']
2-strand cable arrow polynomial of the knot is: -64*K1**4 + 96*K1**3*K2*K3 - 64*K1**3*K3 - 192*K1**2*K2**4 + 832*K1**2*K2**3 + 160*K1**2*K2**2*K4 - 3664*K1**2*K2**2 - 384*K1**2*K2*K4 + 4904*K1**2*K2 - 64*K1**2*K3**2 - 128*K1**2*K4**2 - 4584*K1**2 + 480*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1536*K1*K2**2*K3 - 224*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 5176*K1*K2*K3 - 224*K1*K2*K4*K5 + 1632*K1*K3*K4 + 384*K1*K4*K5 + 96*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1208*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 624*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 400*K2**2*K4**2 - 32*K2**2*K4*K8 + 2240*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 3674*K2**2 - 192*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 912*K2*K3*K5 + 384*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 + 32*K3**2*K6 - 2136*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1284*K4**2 - 376*K5**2 - 86*K6**2 - 8*K7**2 - 2*K8**2 + 3900
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.541']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4742', 'vk6.5070', 'vk6.6276', 'vk6.6717', 'vk6.8239', 'vk6.8688', 'vk6.9628', 'vk6.9946', 'vk6.20652', 'vk6.22083', 'vk6.28138', 'vk6.29567', 'vk6.39576', 'vk6.41807', 'vk6.46191', 'vk6.47809', 'vk6.48774', 'vk6.48984', 'vk6.49580', 'vk6.49787', 'vk6.50784', 'vk6.50997', 'vk6.51268', 'vk6.51468', 'vk6.57560', 'vk6.58730', 'vk6.62234', 'vk6.63180', 'vk6.67038', 'vk6.67911', 'vk6.69663', 'vk6.70344']
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 O1O2O3O4O5U3U2U5O6U4U1U6
R3 orbit {'O1O2O3O4O5U3U2U5O6U4U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U5U2O6U1U4U3
Gauss code of K* O1O2O3U4O5O6O4U6U2U1U5U3
Gauss code of -K* O1O2O3U1O4O5O6U4U3U6U5U2
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 -1 -2 -2 1 2 2],[ 1 0 -2 -2 2 2 2],[ 2 2 0 0 3 2 1],[ 2 2 0 0 2 1 1],[-1 -2 -3 -2 0 0 1],[-2 -2 -2 -1 0 0 0],[-2 -2 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 2 2 1 -1 -2 -2],[-2 0 0 0 -2 -1 -2],[-2 0 0 -1 -2 -1 -1],[-1 0 1 0 -2 -2 -3],[ 1 2 2 2 0 -2 -2],[ 2 1 1 2 2 0 0],[ 2 2 1 3 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,2,2,0,0,2,1,2,1,2,1,1,2,2,3,2,2,0]
Phi over symmetry [-2,-2,-1,1,2,2,0,-1,0,2,3,-1,1,3,3,0,1,1,1,0,0]
Phi of -K [-2,-2,-1,1,2,2,0,-1,0,2,3,-1,1,3,3,0,1,1,1,0,0]
Phi of K* [-2,-2,-1,1,2,2,0,0,1,3,3,1,1,2,3,0,0,1,-1,-1,0]
Phi of -K* [-2,-2,-1,1,2,2,0,2,2,1,1,2,3,1,2,2,2,2,1,0,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 8z^2+27z+23
Enhanced Jones-Krushkal polynomial 8w^3z^2+27w^2z+23w
Inner characteristic polynomial t^6+41t^4+36t^2
Outer characteristic polynomial t^7+59t^5+100t^3+5t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 2*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 - 2*K2**2 + K3 + K4 + 2
2-strand cable arrow polynomial -64*K1**4 + 96*K1**3*K2*K3 - 64*K1**3*K3 - 192*K1**2*K2**4 + 832*K1**2*K2**3 + 160*K1**2*K2**2*K4 - 3664*K1**2*K2**2 - 384*K1**2*K2*K4 + 4904*K1**2*K2 - 64*K1**2*K3**2 - 128*K1**2*K4**2 - 4584*K1**2 + 480*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1536*K1*K2**2*K3 - 224*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 5176*K1*K2*K3 - 224*K1*K2*K4*K5 + 1632*K1*K3*K4 + 384*K1*K4*K5 + 96*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1208*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 624*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 400*K2**2*K4**2 - 32*K2**2*K4*K8 + 2240*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 3674*K2**2 - 192*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 912*K2*K3*K5 + 384*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 + 32*K3**2*K6 - 2136*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1284*K4**2 - 376*K5**2 - 86*K6**2 - 8*K7**2 - 2*K8**2 + 3900
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}]]
If K is slice False
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