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Flat knot 6.324

Min(phi) over symmetries of the knot is: [-3,-1,0,0,1,3,-1,0,2,2,4,0,1,0,1,1,0,1,0,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.324']
Arrow polynomial of the knot is: 12*K1**3 - 8*K1**2 - 8*K1*K2 - 5*K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.324', '6.672', '6.953', '6.1196', '6.1215', '6.1216', '6.1699']
Outer characteristic polynomial of the knot is: t^7+70t^5+69t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.324']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 768*K1**4*K2 - 2592*K1**4 + 480*K1**3*K2*K3 - 768*K1**3*K3 - 192*K1**2*K2**4 + 864*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 6560*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 10960*K1**2*K2 - 800*K1**2*K3**2 - 48*K1**2*K4**2 - 7788*K1**2 + 416*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1856*K1*K2**2*K3 - 192*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9520*K1*K2*K3 - 96*K1*K2*K4*K5 + 1704*K1*K3*K4 + 96*K1*K4*K5 + 48*K1*K5*K6 - 96*K2**6 + 128*K2**4*K4 - 1648*K2**4 - 32*K2**3*K6 - 736*K2**2*K3**2 - 112*K2**2*K4**2 + 2248*K2**2*K4 - 5834*K2**2 - 32*K2*K3**2*K4 + 664*K2*K3*K5 + 136*K2*K4*K6 + 8*K3**2*K6 - 2972*K3**2 - 888*K4**2 - 160*K5**2 - 38*K6**2 + 6134
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.324']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11541', 'vk6.11872', 'vk6.12887', 'vk6.13194', 'vk6.20351', 'vk6.21693', 'vk6.27652', 'vk6.29197', 'vk6.31316', 'vk6.31711', 'vk6.32470', 'vk6.32885', 'vk6.39083', 'vk6.41337', 'vk6.45835', 'vk6.47502', 'vk6.52316', 'vk6.52576', 'vk6.53156', 'vk6.53456', 'vk6.57210', 'vk6.58429', 'vk6.61821', 'vk6.62949', 'vk6.63817', 'vk6.63949', 'vk6.64259', 'vk6.64455', 'vk6.66817', 'vk6.67686', 'vk6.69454', 'vk6.70177']
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 O1O2O3O4O5U2U4O6U5U3U1U6
R3 orbit {'O1O2O3O4O5U2U4O6U5U3U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U5U3U1O6U2U4
Gauss code of K* O1O2O3O4U3U5U2U6U1O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U4U5U3U6U2
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 -3 0 0 1 3],[ 1 0 -3 1 0 2 3],[ 3 3 0 3 1 2 2],[ 0 -1 -3 0 -1 1 2],[ 0 0 -1 1 0 1 1],[-1 -2 -2 -1 -1 0 1],[-3 -3 -2 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 1 0 0 -1 -3],[-3 0 -1 -1 -2 -3 -2],[-1 1 0 -1 -1 -2 -2],[ 0 1 1 0 1 0 -1],[ 0 2 1 -1 0 -1 -3],[ 1 3 2 0 1 0 -3],[ 3 2 2 1 3 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,0,1,3,1,1,2,3,2,1,1,2,2,-1,0,1,1,3,3]
Phi over symmetry [-3,-1,0,0,1,3,-1,0,2,2,4,0,1,0,1,1,0,1,0,2,1]
Phi of -K [-3,-1,0,0,1,3,-1,0,2,2,4,0,1,0,1,1,0,1,0,2,1]
Phi of K* [-3,-1,0,0,1,3,1,1,2,1,4,0,0,0,2,-1,0,0,1,2,-1]
Phi of -K* [-3,-1,0,0,1,3,3,1,3,2,2,0,1,2,3,1,1,1,1,2,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 4z^2+24z+33
Enhanced Jones-Krushkal polynomial 4w^3z^2+24w^2z+33w
Inner characteristic polynomial t^6+50t^4+17t^2+1
Outer characteristic polynomial t^7+70t^5+69t^3+7t
Flat arrow polynomial 12*K1**3 - 8*K1**2 - 8*K1*K2 - 5*K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -64*K1**6 + 768*K1**4*K2 - 2592*K1**4 + 480*K1**3*K2*K3 - 768*K1**3*K3 - 192*K1**2*K2**4 + 864*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 6560*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 10960*K1**2*K2 - 800*K1**2*K3**2 - 48*K1**2*K4**2 - 7788*K1**2 + 416*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1856*K1*K2**2*K3 - 192*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9520*K1*K2*K3 - 96*K1*K2*K4*K5 + 1704*K1*K3*K4 + 96*K1*K4*K5 + 48*K1*K5*K6 - 96*K2**6 + 128*K2**4*K4 - 1648*K2**4 - 32*K2**3*K6 - 736*K2**2*K3**2 - 112*K2**2*K4**2 + 2248*K2**2*K4 - 5834*K2**2 - 32*K2*K3**2*K4 + 664*K2*K3*K5 + 136*K2*K4*K6 + 8*K3**2*K6 - 2972*K3**2 - 888*K4**2 - 160*K5**2 - 38*K6**2 + 6134
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {2, 5}, {3, 4}, {1}]]
If K is slice False
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