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Flat knot 6.473

Min(phi) over symmetries of the knot is: [-4,0,1,1,1,1,0,2,3,4,4,1,1,0,1,0,0,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.473']
Arrow polynomial of the knot is: -2*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 + 2*K2 + 2*K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.180', '6.263', '6.295', '6.317', '6.350', '6.473', '6.504']
Outer characteristic polynomial of the knot is: t^7+68t^5+116t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.473']
2-strand cable arrow polynomial of the knot is: -432*K1**4 + 192*K1**3*K2*K3 - 192*K1**3*K3 - 720*K1**2*K2**2 - 128*K1**2*K2*K4 + 1776*K1**2*K2 - 784*K1**2*K3**2 - 128*K1**2*K3*K5 - 2244*K1**2 - 96*K1*K2**2*K3 - 32*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 2560*K1*K2*K3 + 1456*K1*K3*K4 + 528*K1*K4*K5 + 128*K1*K5*K6 - 72*K2**4 - 32*K2**3*K6 - 64*K2**2*K3**2 - 16*K2**2*K4**2 + 760*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 2096*K2**2 + 672*K2*K3*K5 + 184*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 + 32*K3**2*K6 - 1480*K3**2 - 968*K4**2 - 472*K5**2 - 128*K6**2 - 12*K7**2 - 2*K8**2 + 2368
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.473']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4707', 'vk6.5016', 'vk6.6217', 'vk6.6680', 'vk6.8204', 'vk6.8633', 'vk6.9580', 'vk6.9915', 'vk6.17403', 'vk6.20935', 'vk6.21083', 'vk6.22345', 'vk6.22513', 'vk6.23570', 'vk6.23909', 'vk6.28418', 'vk6.36178', 'vk6.40092', 'vk6.40321', 'vk6.42137', 'vk6.43392', 'vk6.46617', 'vk6.46791', 'vk6.48048', 'vk6.48745', 'vk6.49758', 'vk6.50757', 'vk6.51443', 'vk6.57739', 'vk6.58941', 'vk6.65300', 'vk6.69780']
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 O1O2O3O4O5U6U4O6U3U2U1U5
R3 orbit {'O1O2O3O4O5U6U4O6U3U2U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U5U4U3O6U2U6
Gauss code of K* O1O2O3O4U3U2U1U5U4O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U1U6U4U3U2
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 -1 -1 0 4 -1],[ 1 0 0 0 1 4 0],[ 1 0 0 0 1 3 0],[ 1 0 0 0 1 2 0],[ 0 -1 -1 -1 0 0 0],[-4 -4 -3 -2 0 0 -4],[ 1 0 0 0 0 4 0]]
Primitive based matrix [[ 0 4 0 -1 -1 -1 -1],[-4 0 0 -2 -3 -4 -4],[ 0 0 0 -1 -1 0 -1],[ 1 2 1 0 0 0 0],[ 1 3 1 0 0 0 0],[ 1 4 0 0 0 0 0],[ 1 4 1 0 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,0,1,1,1,1,0,2,3,4,4,1,1,0,1,0,0,0,0,0,0]
Phi over symmetry [-4,0,1,1,1,1,0,2,3,4,4,1,1,0,1,0,0,0,0,0,0]
Phi of -K [-1,-1,-1,-1,0,4,0,0,0,0,1,0,0,0,2,0,0,3,1,1,4]
Phi of K* [-4,0,1,1,1,1,4,1,1,2,3,0,1,0,0,0,0,0,0,0,0]
Phi of -K* [-1,-1,-1,-1,0,4,0,0,0,0,4,0,0,1,2,0,1,3,1,4,0]
Symmetry type of based matrix c
u-polynomial -t^4+4t
Normalized Jones-Krushkal polynomial 5z+11
Enhanced Jones-Krushkal polynomial -4w^4z^2+4w^3z^2-12w^3z+17w^2z+11w
Inner characteristic polynomial t^6+48t^4+54t^2
Outer characteristic polynomial t^7+68t^5+116t^3+6t
Flat arrow polynomial -2*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 + 2*K2 + 2*K3 + K4 + 2
2-strand cable arrow polynomial -432*K1**4 + 192*K1**3*K2*K3 - 192*K1**3*K3 - 720*K1**2*K2**2 - 128*K1**2*K2*K4 + 1776*K1**2*K2 - 784*K1**2*K3**2 - 128*K1**2*K3*K5 - 2244*K1**2 - 96*K1*K2**2*K3 - 32*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 2560*K1*K2*K3 + 1456*K1*K3*K4 + 528*K1*K4*K5 + 128*K1*K5*K6 - 72*K2**4 - 32*K2**3*K6 - 64*K2**2*K3**2 - 16*K2**2*K4**2 + 760*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 2096*K2**2 + 672*K2*K3*K5 + 184*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 + 32*K3**2*K6 - 1480*K3**2 - 968*K4**2 - 472*K5**2 - 128*K6**2 - 12*K7**2 - 2*K8**2 + 2368
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{6}, {4, 5}, {1, 3}, {2}]]
If K is slice False
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