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Flat knot 6.836

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,3,2,3,1,1,1,0,1,1,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.836']
Arrow polynomial of the knot is: 8*K1**3 - 12*K1**2 - 10*K1*K2 - K1 + 6*K2 + 3*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.743', '6.836']
Outer characteristic polynomial of the knot is: t^7+64t^5+60t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.836']
2-strand cable arrow polynomial of the knot is: -128*K1**6 - 512*K1**4*K2**2 + 1728*K1**4*K2 - 4448*K1**4 + 928*K1**3*K2*K3 - 1152*K1**3*K3 - 384*K1**2*K2**4 + 2848*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 11344*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 1120*K1**2*K2*K4 + 11944*K1**2*K2 - 960*K1**2*K3**2 - 32*K1**2*K3*K5 - 4888*K1**2 + 1984*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 2528*K1*K2**2*K3 - 544*K1*K2**2*K5 + 128*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 10520*K1*K2*K3 - 32*K1*K3**2*K5 + 1264*K1*K3*K4 + 40*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 3584*K2**4 - 64*K2**3*K6 - 1840*K2**2*K3**2 - 136*K2**2*K4**2 + 3088*K2**2*K4 - 3594*K2**2 + 1248*K2*K3*K5 + 88*K2*K4*K6 - 32*K3**4 + 16*K3**2*K6 - 2236*K3**2 - 588*K4**2 - 180*K5**2 - 14*K6**2 + 4674
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.836']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19939', 'vk6.19980', 'vk6.21180', 'vk6.21247', 'vk6.26896', 'vk6.26999', 'vk6.28650', 'vk6.28723', 'vk6.38315', 'vk6.38409', 'vk6.40452', 'vk6.40590', 'vk6.45191', 'vk6.45299', 'vk6.47019', 'vk6.47079', 'vk6.56724', 'vk6.56786', 'vk6.57820', 'vk6.57918', 'vk6.61143', 'vk6.61278', 'vk6.62390', 'vk6.62469', 'vk6.66417', 'vk6.66486', 'vk6.67185', 'vk6.67277', 'vk6.69068', 'vk6.69144', 'vk6.69855', 'vk6.69901']
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 O1O2O3O4U1U2O5O6U3U6U4U5
R3 orbit {'O1O2O3O4U1U2O5O6U3U6U4U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U1U6U2O6O5U3U4
Gauss code of K* O1O2O3O4U5U6U1U3O5O6U4U2
Gauss code of -K* O1O2O3O4U3U1O5O6U2U4U5U6
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 -3 -1 -1 2 2 1],[ 3 0 1 2 3 2 1],[ 1 -1 0 1 2 2 1],[ 1 -2 -1 0 2 3 1],[-2 -3 -2 -2 0 1 0],[-2 -2 -2 -3 -1 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 1 0 -2 -2 -3],[-2 -1 0 0 -2 -3 -2],[-1 0 0 0 -1 -1 -1],[ 1 2 2 1 0 1 -1],[ 1 2 3 1 -1 0 -2],[ 3 3 2 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,-1,0,2,2,3,0,2,3,2,1,1,1,-1,1,2]
Phi over symmetry [-3,-1,-1,1,2,2,0,1,3,2,3,1,1,1,0,1,1,1,1,1,-1]
Phi of -K [-3,-1,-1,1,2,2,0,1,3,2,3,1,1,1,0,1,1,1,1,1,-1]
Phi of K* [-2,-2,-1,1,1,3,-1,1,0,1,3,1,1,1,2,1,1,3,-1,0,1]
Phi of -K* [-3,-1,-1,1,2,2,1,2,1,2,3,1,1,2,2,1,3,2,0,0,-1]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 5z^2+25z+31
Enhanced Jones-Krushkal polynomial 5w^3z^2+25w^2z+31w
Inner characteristic polynomial t^6+44t^4+20t^2+1
Outer characteristic polynomial t^7+64t^5+60t^3+7t
Flat arrow polynomial 8*K1**3 - 12*K1**2 - 10*K1*K2 - K1 + 6*K2 + 3*K3 + 7
2-strand cable arrow polynomial -128*K1**6 - 512*K1**4*K2**2 + 1728*K1**4*K2 - 4448*K1**4 + 928*K1**3*K2*K3 - 1152*K1**3*K3 - 384*K1**2*K2**4 + 2848*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 11344*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 1120*K1**2*K2*K4 + 11944*K1**2*K2 - 960*K1**2*K3**2 - 32*K1**2*K3*K5 - 4888*K1**2 + 1984*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 2528*K1*K2**2*K3 - 544*K1*K2**2*K5 + 128*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 10520*K1*K2*K3 - 32*K1*K3**2*K5 + 1264*K1*K3*K4 + 40*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 3584*K2**4 - 64*K2**3*K6 - 1840*K2**2*K3**2 - 136*K2**2*K4**2 + 3088*K2**2*K4 - 3594*K2**2 + 1248*K2*K3*K5 + 88*K2*K4*K6 - 32*K3**4 + 16*K3**2*K6 - 2236*K3**2 - 588*K4**2 - 180*K5**2 - 14*K6**2 + 4674
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {2, 5}, {1, 3}], [{6}, {2, 5}, {1, 4}, {3}]]
If K is slice False
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