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Flat knot 6.688

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,1,0,2,3,1,0,1,1,1,2,2,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.688']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.315', '6.337', '6.389', '6.418', '6.599', '6.675', '6.686', '6.688', '6.746', '6.747', '6.809', '6.1034', '6.1128', '6.1133', '6.1334', '6.1363', '6.1489', '6.1539', '6.1564', '6.1821', '6.1863']
Outer characteristic polynomial of the knot is: t^7+40t^5+46t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.688']
2-strand cable arrow polynomial of the knot is: -384*K1**6 - 320*K1**4*K2**2 + 3104*K1**4*K2 - 6608*K1**4 + 640*K1**3*K2*K3 - 1504*K1**3*K3 - 192*K1**2*K2**4 + 1248*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 8288*K1**2*K2**2 - 1344*K1**2*K2*K4 + 12976*K1**2*K2 - 304*K1**2*K3**2 - 48*K1**2*K4**2 - 5680*K1**2 + 320*K1*K2**3*K3 - 1280*K1*K2**2*K3 - 256*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 8680*K1*K2*K3 + 1240*K1*K3*K4 + 104*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1072*K2**4 - 32*K2**3*K6 - 160*K2**2*K3**2 - 16*K2**2*K4**2 + 1752*K2**2*K4 - 5414*K2**2 + 216*K2*K3*K5 + 16*K2*K4*K6 - 2208*K3**2 - 720*K4**2 - 80*K5**2 - 2*K6**2 + 5454
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.688']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4470', 'vk6.4565', 'vk6.5852', 'vk6.5979', 'vk6.7902', 'vk6.8020', 'vk6.9331', 'vk6.9450', 'vk6.13404', 'vk6.13501', 'vk6.13688', 'vk6.14058', 'vk6.15035', 'vk6.15155', 'vk6.17794', 'vk6.17827', 'vk6.18848', 'vk6.19427', 'vk6.19720', 'vk6.24337', 'vk6.25445', 'vk6.25478', 'vk6.26599', 'vk6.33246', 'vk6.33307', 'vk6.37567', 'vk6.44876', 'vk6.48657', 'vk6.50549', 'vk6.53654', 'vk6.55827', 'vk6.65491']
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 O1O2O3O4U3O5U2U5O6U4U1U6
R3 orbit {'O1O2O3O4U3O5U2U5O6U4U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U4U1O5U6U3O6U2
Gauss code of K* O1O2O3U2U4U5U1O5U6O4O6U3
Gauss code of -K* O1O2O3U1O4O5U4O6U3U6U5U2
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 -2 -1 1 1 2],[ 1 0 -2 -1 2 1 2],[ 2 2 0 0 3 1 1],[ 1 1 0 0 1 0 1],[-1 -2 -3 -1 0 0 1],[-1 -1 -1 0 0 0 0],[-2 -2 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -1 -1 -2 -1],[-1 0 0 0 0 -1 -1],[-1 1 0 0 -1 -2 -3],[ 1 1 0 1 0 1 0],[ 1 2 1 2 -1 0 -2],[ 2 1 1 3 0 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,0,1,1,2,1,0,0,1,1,1,2,3,-1,0,2]
Phi over symmetry [-2,-1,-1,1,1,2,-1,1,0,2,3,1,0,1,1,1,2,2,0,0,1]
Phi of -K [-2,-1,-1,1,1,2,-1,1,0,2,3,1,0,1,1,1,2,2,0,0,1]
Phi of K* [-2,-1,-1,1,1,2,0,1,1,2,3,0,0,1,0,1,2,2,-1,-1,1]
Phi of -K* [-2,-1,-1,1,1,2,0,2,1,3,1,1,0,1,1,1,2,2,0,0,1]
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+28t^4+20t^2+1
Outer characteristic polynomial t^7+40t^5+46t^3+8t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -384*K1**6 - 320*K1**4*K2**2 + 3104*K1**4*K2 - 6608*K1**4 + 640*K1**3*K2*K3 - 1504*K1**3*K3 - 192*K1**2*K2**4 + 1248*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 8288*K1**2*K2**2 - 1344*K1**2*K2*K4 + 12976*K1**2*K2 - 304*K1**2*K3**2 - 48*K1**2*K4**2 - 5680*K1**2 + 320*K1*K2**3*K3 - 1280*K1*K2**2*K3 - 256*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 8680*K1*K2*K3 + 1240*K1*K3*K4 + 104*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1072*K2**4 - 32*K2**3*K6 - 160*K2**2*K3**2 - 16*K2**2*K4**2 + 1752*K2**2*K4 - 5414*K2**2 + 216*K2*K3*K5 + 16*K2*K4*K6 - 2208*K3**2 - 720*K4**2 - 80*K5**2 - 2*K6**2 + 5454
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {5}, {2, 4}, {1}], [{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
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