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Flat knot 6.750

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,0,2,3,3,2,1,2,3,2,0,0,1,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.750']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 6*K1*K2 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.211', '6.557', '6.676', '6.685', '6.750', '6.751', '6.856', '6.919', '6.1093', '6.1371']
Outer characteristic polynomial of the knot is: t^7+70t^5+75t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.750']
2-strand cable arrow polynomial of the knot is: -64*K1**6 - 64*K1**4*K2**2 + 1248*K1**4*K2 - 4544*K1**4 + 480*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1216*K1**3*K3 - 192*K1**2*K2**4 + 480*K1**2*K2**3 + 224*K1**2*K2**2*K4 - 5744*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 768*K1**2*K2*K4 + 10608*K1**2*K2 - 1344*K1**2*K3**2 - 64*K1**2*K3*K5 - 80*K1**2*K4**2 - 5388*K1**2 + 320*K1*K2**3*K3 - 576*K1*K2**2*K3 - 32*K1*K2**2*K5 + 64*K1*K2*K3**3 - 352*K1*K2*K3*K4 + 7688*K1*K2*K3 - 32*K1*K3**2*K5 + 1392*K1*K3*K4 + 104*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1264*K2**4 - 688*K2**2*K3**2 - 56*K2**2*K4**2 + 1304*K2**2*K4 - 4244*K2**2 + 536*K2*K3*K5 + 24*K2*K4*K6 - 64*K3**4 + 48*K3**2*K6 - 1992*K3**2 - 472*K4**2 - 84*K5**2 - 12*K6**2 + 4686
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.750']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13886', 'vk6.13981', 'vk6.14121', 'vk6.14342', 'vk6.14957', 'vk6.15078', 'vk6.15571', 'vk6.16041', 'vk6.16301', 'vk6.16326', 'vk6.17425', 'vk6.22612', 'vk6.22645', 'vk6.23933', 'vk6.33697', 'vk6.33772', 'vk6.34134', 'vk6.34255', 'vk6.34600', 'vk6.36202', 'vk6.36229', 'vk6.42297', 'vk6.53872', 'vk6.53913', 'vk6.54096', 'vk6.54417', 'vk6.54584', 'vk6.55569', 'vk6.59027', 'vk6.59056', 'vk6.60059', 'vk6.64556']
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 O1O2O3O4U2O5U3U6U4O6U1U5
R3 orbit {'O1O2O3O4U2O5U3U6U4O6U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U4O6U1U6U2O5U3
Gauss code of K* O1O2O3U2O4O5U4U6U1U3O6U5
Gauss code of -K* O1O2O3U4O5U1U3U5U6O4O6U2
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 2 3 -1],[ 1 0 -2 0 3 3 0],[ 2 2 0 1 2 2 1],[ 1 0 -1 0 1 2 0],[-2 -3 -2 -1 0 0 -2],[-3 -3 -2 -2 0 0 -3],[ 1 0 -1 0 2 3 0]]
Primitive based matrix [[ 0 3 2 -1 -1 -1 -2],[-3 0 0 -2 -3 -3 -2],[-2 0 0 -1 -2 -3 -2],[ 1 2 1 0 0 0 -1],[ 1 3 2 0 0 0 -1],[ 1 3 3 0 0 0 -2],[ 2 2 2 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,1,1,1,2,0,2,3,3,2,1,2,3,2,0,0,1,0,1,2]
Phi over symmetry [-3,-2,1,1,1,2,0,2,3,3,2,1,2,3,2,0,0,1,0,1,2]
Phi of -K [-2,-1,-1,-1,2,3,-1,0,0,2,3,0,0,0,1,0,1,1,2,2,1]
Phi of K* [-3,-2,1,1,1,2,1,1,1,2,3,0,1,2,2,0,0,-1,0,0,0]
Phi of -K* [-2,-1,-1,-1,2,3,1,1,2,2,2,0,0,1,2,0,2,3,3,3,0]
Symmetry type of based matrix c
u-polynomial -t^3+3t
Normalized Jones-Krushkal polynomial 3z^2+23z+35
Enhanced Jones-Krushkal polynomial 3w^3z^2+23w^2z+35w
Inner characteristic polynomial t^6+50t^4+41t^2+4
Outer characteristic polynomial t^7+70t^5+75t^3+10t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 6*K1*K2 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -64*K1**6 - 64*K1**4*K2**2 + 1248*K1**4*K2 - 4544*K1**4 + 480*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1216*K1**3*K3 - 192*K1**2*K2**4 + 480*K1**2*K2**3 + 224*K1**2*K2**2*K4 - 5744*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 768*K1**2*K2*K4 + 10608*K1**2*K2 - 1344*K1**2*K3**2 - 64*K1**2*K3*K5 - 80*K1**2*K4**2 - 5388*K1**2 + 320*K1*K2**3*K3 - 576*K1*K2**2*K3 - 32*K1*K2**2*K5 + 64*K1*K2*K3**3 - 352*K1*K2*K3*K4 + 7688*K1*K2*K3 - 32*K1*K3**2*K5 + 1392*K1*K3*K4 + 104*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1264*K2**4 - 688*K2**2*K3**2 - 56*K2**2*K4**2 + 1304*K2**2*K4 - 4244*K2**2 + 536*K2*K3*K5 + 24*K2*K4*K6 - 64*K3**4 + 48*K3**2*K6 - 1992*K3**2 - 472*K4**2 - 84*K5**2 - 12*K6**2 + 4686
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {3, 5}, {2, 4}, {1}]]
If K is slice False
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