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Flat knot 6.1765

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,0,0,1,2,0,0,2,1,-1,1,0,1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1765']
Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 8*K1*K2 + K1 + 6*K2 + 3*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.906', '6.1223', '6.1338', '6.1351', '6.1571', '6.1670', '6.1718', '6.1743', '6.1765', '6.1793', '6.1852', '6.2070']
Outer characteristic polynomial of the knot is: t^7+14t^5+24t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1765']
2-strand cable arrow polynomial of the knot is: -448*K1**6 - 640*K1**4*K2**2 + 2016*K1**4*K2 - 5856*K1**4 + 1056*K1**3*K2*K3 + 64*K1**3*K3*K4 - 704*K1**3*K3 - 256*K1**2*K2**4 + 1600*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 8464*K1**2*K2**2 - 1056*K1**2*K2*K4 + 11504*K1**2*K2 - 1152*K1**2*K3**2 - 64*K1**2*K3*K5 - 240*K1**2*K4**2 - 4604*K1**2 + 576*K1*K2**3*K3 - 1536*K1*K2**2*K3 - 288*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 8664*K1*K2*K3 + 2032*K1*K3*K4 + 360*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 1280*K2**4 - 32*K2**3*K6 - 336*K2**2*K3**2 - 128*K2**2*K4**2 + 1672*K2**2*K4 - 4498*K2**2 + 328*K2*K3*K5 + 104*K2*K4*K6 - 2388*K3**2 - 900*K4**2 - 136*K5**2 - 22*K6**2 + 5018
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1765']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4252', 'vk6.4256', 'vk6.4333', 'vk6.4336', 'vk6.5527', 'vk6.5531', 'vk6.5648', 'vk6.5652', 'vk6.7720', 'vk6.7723', 'vk6.9118', 'vk6.9121', 'vk6.9198', 'vk6.9202', 'vk6.19823', 'vk6.19831', 'vk6.26258', 'vk6.26264', 'vk6.26703', 'vk6.26709', 'vk6.38214', 'vk6.38222', 'vk6.44987', 'vk6.44995', 'vk6.48574', 'vk6.48578', 'vk6.49285', 'vk6.49289', 'vk6.50421', 'vk6.50424', 'vk6.66360', 'vk6.66366']
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 O1O2O3U4U1O5O4U5U3O6U2U6
R3 orbit {'O1O2O3U4U1O5O4U5U3O6U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U2O4U1U5O6O5U3U6
Gauss code of K* O1O2U3O4O3U5U4U2O6O5U1U6
Gauss code of -K* O1O2U1O3O4U5U4O6O5U3U2U6
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 0 1 0 -1 1],[ 1 0 1 0 1 -1 1],[ 0 -1 0 0 1 -1 1],[-1 0 0 0 0 -1 0],[ 0 -1 -1 0 0 -1 1],[ 1 1 1 1 1 0 0],[-1 -1 -1 0 -1 0 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 0 0 0 0 -1],[-1 0 0 -1 -1 -1 0],[ 0 0 1 0 1 -1 -1],[ 0 0 1 -1 0 -1 -1],[ 1 0 1 1 1 0 -1],[ 1 1 0 1 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,1,1,0,0,0,0,1,1,1,1,0,-1,1,1,1,1,1]
Phi over symmetry [-1,-1,0,0,1,1,-1,0,0,1,2,0,0,2,1,-1,1,0,1,0,0]
Phi of -K [-1,-1,0,0,1,1,-1,0,0,1,2,0,0,2,1,-1,1,0,1,0,0]
Phi of K* [-1,-1,0,0,1,1,0,0,0,1,2,1,1,2,1,-1,0,0,0,0,-1]
Phi of -K* [-1,-1,0,0,1,1,-1,1,1,0,1,1,1,1,0,-1,0,1,0,1,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+10t^4+8t^2+1
Outer characteristic polynomial t^7+14t^5+24t^3+7t
Flat arrow polynomial 4*K1**3 - 12*K1**2 - 8*K1*K2 + K1 + 6*K2 + 3*K3 + 7
2-strand cable arrow polynomial -448*K1**6 - 640*K1**4*K2**2 + 2016*K1**4*K2 - 5856*K1**4 + 1056*K1**3*K2*K3 + 64*K1**3*K3*K4 - 704*K1**3*K3 - 256*K1**2*K2**4 + 1600*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 8464*K1**2*K2**2 - 1056*K1**2*K2*K4 + 11504*K1**2*K2 - 1152*K1**2*K3**2 - 64*K1**2*K3*K5 - 240*K1**2*K4**2 - 4604*K1**2 + 576*K1*K2**3*K3 - 1536*K1*K2**2*K3 - 288*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 8664*K1*K2*K3 + 2032*K1*K3*K4 + 360*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 1280*K2**4 - 32*K2**3*K6 - 336*K2**2*K3**2 - 128*K2**2*K4**2 + 1672*K2**2*K4 - 4498*K2**2 + 328*K2*K3*K5 + 104*K2*K4*K6 - 2388*K3**2 - 900*K4**2 - 136*K5**2 - 22*K6**2 + 5018
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {1, 5}, {3, 4}, {2}]]
If K is slice False
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