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Flat knot 6.1332

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,3,2,0,-1,1,1,0,1,2,-1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1332']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1*K2 - K1 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.395', '6.430', '6.440', '6.548', '6.551', '6.774', '6.832', '6.887', '6.908', '6.911', '6.1205', '6.1332', '6.1339', '6.1341', '6.1346', '6.1382', '6.1488', '6.1651', '6.1655', '6.1686']
Outer characteristic polynomial of the knot is: t^7+46t^5+87t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1332']
2-strand cable arrow polynomial of the knot is: -576*K1**2*K2**4 + 1024*K1**2*K2**3 - 4496*K1**2*K2**2 - 288*K1**2*K2*K4 + 3568*K1**2*K2 - 2392*K1**2 + 672*K1*K2**3*K3 - 864*K1*K2**2*K3 - 32*K1*K2**2*K5 + 4232*K1*K2*K3 + 392*K1*K3*K4 - 288*K2**6 + 448*K2**4*K4 - 1888*K2**4 - 32*K2**3*K6 - 240*K2**2*K3**2 - 208*K2**2*K4**2 + 1776*K2**2*K4 - 1374*K2**2 + 40*K2*K3*K5 + 48*K2*K4*K6 - 1040*K3**2 - 468*K4**2 - 2*K6**2 + 1986
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1332']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17534', 'vk6.17542', 'vk6.17591', 'vk6.17599', 'vk6.24040', 'vk6.24050', 'vk6.24135', 'vk6.24142', 'vk6.36322', 'vk6.36334', 'vk6.36394', 'vk6.36401', 'vk6.43447', 'vk6.43455', 'vk6.43495', 'vk6.43501', 'vk6.55632', 'vk6.55640', 'vk6.55658', 'vk6.55665', 'vk6.60150', 'vk6.60155', 'vk6.60206', 'vk6.60215', 'vk6.65341', 'vk6.65347', 'vk6.65370', 'vk6.65379', 'vk6.68507', 'vk6.68511', 'vk6.68528', 'vk6.68535']
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 O1O2O3U1O4O5U3U2U5O6U4U6
R3 orbit {'O1O2O3U1O4O5U3U2U5O6U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O4U6U2U1O6O5U3
Gauss code of K* O1O2O3U4O5O4U6U2U1O6U5U3
Gauss code of -K* O1O2O3U1U4O5U3U2U5O6O4U6
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 -2 -1 -1 1 2 1],[ 2 0 2 1 2 2 0],[ 1 -2 0 0 3 2 1],[ 1 -1 0 0 2 1 1],[-1 -2 -3 -2 0 0 1],[-2 -2 -2 -1 0 0 0],[-1 0 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 0 0 -1 -2 -2],[-1 0 0 1 -2 -3 -2],[-1 0 -1 0 -1 -1 0],[ 1 1 2 1 0 0 -1],[ 1 2 3 1 0 0 -2],[ 2 2 2 0 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,0,0,1,2,2,-1,2,3,2,1,1,0,0,1,2]
Phi over symmetry [-2,-1,-1,1,1,2,-1,0,1,3,2,0,-1,1,1,0,1,2,-1,1,1]
Phi of -K [-2,-1,-1,1,1,2,-1,0,1,3,2,0,-1,1,1,0,1,2,-1,1,1]
Phi of K* [-2,-1,-1,1,1,2,1,1,1,2,2,-1,1,1,3,-1,0,1,0,-1,0]
Phi of -K* [-2,-1,-1,1,1,2,1,2,0,2,2,0,1,2,1,1,3,2,-1,0,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 2z^2+7z+7
Enhanced Jones-Krushkal polynomial -4w^4z^2+6w^3z^2-12w^3z+19w^2z+7w
Inner characteristic polynomial t^6+34t^4+31t^2
Outer characteristic polynomial t^7+46t^5+87t^3+10t
Flat arrow polynomial 4*K1**3 - 4*K1*K2 - K1 + K3 + 1
2-strand cable arrow polynomial -576*K1**2*K2**4 + 1024*K1**2*K2**3 - 4496*K1**2*K2**2 - 288*K1**2*K2*K4 + 3568*K1**2*K2 - 2392*K1**2 + 672*K1*K2**3*K3 - 864*K1*K2**2*K3 - 32*K1*K2**2*K5 + 4232*K1*K2*K3 + 392*K1*K3*K4 - 288*K2**6 + 448*K2**4*K4 - 1888*K2**4 - 32*K2**3*K6 - 240*K2**2*K3**2 - 208*K2**2*K4**2 + 1776*K2**2*K4 - 1374*K2**2 + 40*K2*K3*K5 + 48*K2*K4*K6 - 1040*K3**2 - 468*K4**2 - 2*K6**2 + 1986
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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