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Flat knot 6.505

Min(phi) over symmetries of the knot is: [-4,-1,0,0,2,3,0,1,3,4,3,0,1,1,1,0,1,1,2,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.505']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 6*K1*K2 - 2*K2**2 + 3*K2 + 2*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.101', '6.505']
Outer characteristic polynomial of the knot is: t^7+89t^5+49t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.505']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 448*K1**4*K2 - 2800*K1**4 + 256*K1**3*K2*K3 + 32*K1**3*K3*K4 - 992*K1**3*K3 - 128*K1**2*K2**4 + 320*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3184*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 768*K1**2*K2*K4 + 7264*K1**2*K2 - 1104*K1**2*K3**2 - 96*K1**2*K3*K5 - 256*K1**2*K4**2 - 4716*K1**2 + 256*K1*K2**3*K3 - 480*K1*K2**2*K3 - 64*K1*K2**2*K5 + 64*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 352*K1*K2*K3*K4 + 6248*K1*K2*K3 - 32*K1*K3**2*K5 + 1824*K1*K3*K4 + 240*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 616*K2**4 + 32*K2**3*K3*K5 + 64*K2**2*K3**2*K4 - 400*K2**2*K3**2 + 32*K2**2*K4**3 - 280*K2**2*K4**2 + 1144*K2**2*K4 - 3812*K2**2 - 32*K2*K3*K4*K5 + 416*K2*K3*K5 + 112*K2*K4*K6 - 64*K3**4 - 48*K3**2*K4**2 + 48*K3**2*K6 - 2120*K3**2 + 16*K3*K4*K7 - 8*K4**4 - 742*K4**2 - 124*K5**2 - 28*K6**2 + 4100
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.505']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16950', 'vk6.17193', 'vk6.20534', 'vk6.21933', 'vk6.23350', 'vk6.23645', 'vk6.27992', 'vk6.29457', 'vk6.35398', 'vk6.35819', 'vk6.39396', 'vk6.41587', 'vk6.42875', 'vk6.43154', 'vk6.45976', 'vk6.47650', 'vk6.55101', 'vk6.55358', 'vk6.57410', 'vk6.58583', 'vk6.59503', 'vk6.59799', 'vk6.62081', 'vk6.63061', 'vk6.64950', 'vk6.65158', 'vk6.66954', 'vk6.67813', 'vk6.68243', 'vk6.68386', 'vk6.69569', 'vk6.70264']
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 O1O2O3O4O5U1U6U3O6U5U2U4
R3 orbit {'O1O2O3O4O5U1U6U3O6U5U2U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U4U1O6U3U6U5
Gauss code of K* O1O2O3U2O4O5O6U1U5U3U6U4
Gauss code of -K* O1O2O3U4O5O4O6U3U1U5U2U6
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 0 0 3 2 -1],[ 4 0 3 1 4 2 3],[ 0 -3 0 0 2 1 -1],[ 0 -1 0 0 1 0 0],[-3 -4 -2 -1 0 0 -3],[-2 -2 -1 0 0 0 -2],[ 1 -3 1 0 3 2 0]]
Primitive based matrix [[ 0 3 2 0 0 -1 -4],[-3 0 0 -1 -2 -3 -4],[-2 0 0 0 -1 -2 -2],[ 0 1 0 0 0 0 -1],[ 0 2 1 0 0 -1 -3],[ 1 3 2 0 1 0 -3],[ 4 4 2 1 3 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,0,0,1,4,0,1,2,3,4,0,1,2,2,0,0,1,1,3,3]
Phi over symmetry [-4,-1,0,0,2,3,0,1,3,4,3,0,1,1,1,0,1,1,2,2,1]
Phi of -K [-4,-1,0,0,2,3,0,1,3,4,3,0,1,1,1,0,1,1,2,2,1]
Phi of K* [-3,-2,0,0,1,4,1,1,2,1,3,1,2,1,4,0,0,1,1,3,0]
Phi of -K* [-4,-1,0,0,2,3,3,1,3,2,4,0,1,2,3,0,0,1,1,2,0]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial 2z^2+19z+31
Enhanced Jones-Krushkal polynomial 2w^3z^2+19w^2z+31w
Inner characteristic polynomial t^6+59t^4+24t^2
Outer characteristic polynomial t^7+89t^5+49t^3+4t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 6*K1*K2 - 2*K2**2 + 3*K2 + 2*K3 + 6
2-strand cable arrow polynomial -64*K1**6 + 448*K1**4*K2 - 2800*K1**4 + 256*K1**3*K2*K3 + 32*K1**3*K3*K4 - 992*K1**3*K3 - 128*K1**2*K2**4 + 320*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3184*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 768*K1**2*K2*K4 + 7264*K1**2*K2 - 1104*K1**2*K3**2 - 96*K1**2*K3*K5 - 256*K1**2*K4**2 - 4716*K1**2 + 256*K1*K2**3*K3 - 480*K1*K2**2*K3 - 64*K1*K2**2*K5 + 64*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 352*K1*K2*K3*K4 + 6248*K1*K2*K3 - 32*K1*K3**2*K5 + 1824*K1*K3*K4 + 240*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 616*K2**4 + 32*K2**3*K3*K5 + 64*K2**2*K3**2*K4 - 400*K2**2*K3**2 + 32*K2**2*K4**3 - 280*K2**2*K4**2 + 1144*K2**2*K4 - 3812*K2**2 - 32*K2*K3*K4*K5 + 416*K2*K3*K5 + 112*K2*K4*K6 - 64*K3**4 - 48*K3**2*K4**2 + 48*K3**2*K6 - 2120*K3**2 + 16*K3*K4*K7 - 8*K4**4 - 742*K4**2 - 124*K5**2 - 28*K6**2 + 4100
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{6}, {2, 5}, {1, 4}, {3}]]
If K is slice False
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