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Flat knot 6.1039

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,1,2,1,0,1,1,1,1,0,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1039']
Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.241', '6.341', '6.542', '6.567', '6.699', '6.713', '6.771', '6.791', '6.1025', '6.1039', '6.1041', '6.1072', '6.1077', '6.1121', '6.1123', '6.1499', '6.1502', '6.1531', '6.1645', '6.1648', '6.1726', '6.1727', '6.1761', '6.1784', '6.1807', '6.1823', '6.1832', '6.1869', '6.1873', '6.1874']
Outer characteristic polynomial of the knot is: t^7+28t^5+31t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1039']
2-strand cable arrow polynomial of the knot is: -384*K1**6 - 320*K1**4*K2**2 + 3712*K1**4*K2 - 8576*K1**4 + 928*K1**3*K2*K3 - 1632*K1**3*K3 - 192*K1**2*K2**4 + 1504*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 10720*K1**2*K2**2 - 1120*K1**2*K2*K4 + 14936*K1**2*K2 - 512*K1**2*K3**2 - 48*K1**2*K4**2 - 4768*K1**2 + 384*K1*K2**3*K3 - 1344*K1*K2**2*K3 - 224*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 9216*K1*K2*K3 + 1008*K1*K3*K4 + 56*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1144*K2**4 - 32*K2**3*K6 - 160*K2**2*K3**2 - 16*K2**2*K4**2 + 1440*K2**2*K4 - 5078*K2**2 + 152*K2*K3*K5 + 16*K2*K4*K6 - 1960*K3**2 - 470*K4**2 - 32*K5**2 - 2*K6**2 + 5252
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1039']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4456', 'vk6.4551', 'vk6.5842', 'vk6.5969', 'vk6.7898', 'vk6.8014', 'vk6.9329', 'vk6.9448', 'vk6.13411', 'vk6.13506', 'vk6.13699', 'vk6.14078', 'vk6.15055', 'vk6.15175', 'vk6.17801', 'vk6.17832', 'vk6.18823', 'vk6.19416', 'vk6.19711', 'vk6.24348', 'vk6.25422', 'vk6.25453', 'vk6.26594', 'vk6.33257', 'vk6.33316', 'vk6.37542', 'vk6.44869', 'vk6.48659', 'vk6.50557', 'vk6.53649', 'vk6.55816', 'vk6.65480']
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 O1O2O3O4U5U3O5U1O6U4U6U2
R3 orbit {'O1O2O3O4U5U3O5U1O6U4U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5U1O5U4O6U2U6
Gauss code of K* O1O2O3U4U3U5U1O6O5U6O4U2
Gauss code of -K* O1O2O3U2O4U5O6O5U3U6U1U4
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 0 1 -1 1],[ 2 0 2 1 2 1 1],[-1 -2 0 0 0 -2 1],[ 0 -1 0 0 0 0 1],[-1 -2 0 0 0 -1 1],[ 1 -1 2 0 1 0 1],[-1 -1 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 0 -1 -2],[-1 -1 0 -1 -1 -1 -1],[-1 0 1 0 0 -2 -2],[ 0 0 1 0 0 0 -1],[ 1 1 1 2 0 0 -1],[ 2 2 1 2 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,-1,0,0,1,2,1,1,1,1,0,2,2,0,1,1]
Phi over symmetry [-2,-1,0,1,1,1,0,1,1,1,2,1,0,1,1,1,1,0,0,-1,-1]
Phi of -K [-2,-1,0,1,1,1,0,1,1,1,2,1,0,1,1,1,1,0,0,-1,-1]
Phi of K* [-1,-1,-1,0,1,2,-1,-1,0,1,2,0,1,0,1,1,1,1,1,1,0]
Phi of -K* [-2,-1,0,1,1,1,1,1,1,2,2,0,1,1,2,1,0,0,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+20t^4+18t^2+1
Outer characteristic polynomial t^7+28t^5+31t^3+5t
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial -384*K1**6 - 320*K1**4*K2**2 + 3712*K1**4*K2 - 8576*K1**4 + 928*K1**3*K2*K3 - 1632*K1**3*K3 - 192*K1**2*K2**4 + 1504*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 10720*K1**2*K2**2 - 1120*K1**2*K2*K4 + 14936*K1**2*K2 - 512*K1**2*K3**2 - 48*K1**2*K4**2 - 4768*K1**2 + 384*K1*K2**3*K3 - 1344*K1*K2**2*K3 - 224*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 9216*K1*K2*K3 + 1008*K1*K3*K4 + 56*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1144*K2**4 - 32*K2**3*K6 - 160*K2**2*K3**2 - 16*K2**2*K4**2 + 1440*K2**2*K4 - 5078*K2**2 + 152*K2*K3*K5 + 16*K2*K4*K6 - 1960*K3**2 - 470*K4**2 - 32*K5**2 - 2*K6**2 + 5252
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{6}, {2, 5}, {1, 4}, {3}]]
If K is slice False
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