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Flat knot 6.928

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,1,3,2,1,0,1,1,1,1,2,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.928']
Arrow polynomial of the knot is: -2*K1**2 - 6*K1*K2 + 3*K1 - 4*K2**2 + K2 + 3*K3 + 2*K4 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.325', '6.928']
Outer characteristic polynomial of the knot is: t^7+50t^5+35t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.928']
2-strand cable arrow polynomial of the knot is: -2640*K1**4 + 320*K1**3*K2*K3 + 160*K1**3*K3*K4 - 320*K1**3*K3 + 128*K1**2*K2**2*K4 - 1872*K1**2*K2**2 + 32*K1**2*K2*K3*K5 - 704*K1**2*K2*K4 + 5040*K1**2*K2 - 1184*K1**2*K3**2 - 352*K1**2*K3*K5 - 304*K1**2*K4**2 - 64*K1**2*K5**2 - 3208*K1**2 - 576*K1*K2**2*K3 - 160*K1*K2**2*K5 - 480*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 4368*K1*K2*K3 + 2928*K1*K3*K4 + 984*K1*K4*K5 + 64*K1*K5*K6 - 72*K2**4 - 96*K2**2*K3**2 - 88*K2**2*K4**2 + 1008*K2**2*K4 - 2926*K2**2 - 32*K2*K3*K4*K5 + 568*K2*K3*K5 + 96*K2*K4*K6 + 8*K2*K5*K7 - 16*K3**4 - 48*K3**2*K4**2 + 32*K3**2*K6 - 2120*K3**2 + 88*K3*K4*K7 - 16*K4**4 + 32*K4**2*K8 - 1478*K4**2 - 452*K5**2 - 42*K6**2 - 28*K7**2 - 12*K8**2 + 3496
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.928']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4360', 'vk6.4391', 'vk6.5682', 'vk6.5713', 'vk6.7751', 'vk6.7782', 'vk6.9233', 'vk6.9264', 'vk6.10500', 'vk6.10549', 'vk6.10644', 'vk6.10723', 'vk6.10754', 'vk6.10835', 'vk6.14605', 'vk6.15305', 'vk6.15430', 'vk6.16228', 'vk6.17992', 'vk6.24436', 'vk6.30179', 'vk6.30228', 'vk6.30323', 'vk6.30454', 'vk6.33951', 'vk6.34352', 'vk6.34406', 'vk6.43863', 'vk6.50453', 'vk6.50484', 'vk6.54191', 'vk6.63435']
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 O1O2O3O4U3U5O6O5U2U6U1U4
R3 orbit {'O1O2O3O4U3U5O6O5U2U6U1U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U4U5U3O6O5U6U2
Gauss code of K* O1O2O3O4U3U1U5U4O5O6U2U6
Gauss code of -K* O1O2O3O4U5U3O5O6U1U6U4U2
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 3 1 0],[ 1 0 -1 0 3 2 0],[ 2 1 0 0 3 2 0],[ 1 0 0 0 1 1 0],[-3 -3 -3 -1 0 -2 -1],[-1 -2 -2 -1 2 0 0],[ 0 0 0 0 1 0 0]]
Primitive based matrix [[ 0 3 1 0 -1 -1 -2],[-3 0 -2 -1 -1 -3 -3],[-1 2 0 0 -1 -2 -2],[ 0 1 0 0 0 0 0],[ 1 1 1 0 0 0 0],[ 1 3 2 0 0 0 -1],[ 2 3 2 0 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,1,1,2,2,1,1,3,3,0,1,2,2,0,0,0,0,0,1]
Phi over symmetry [-3,-1,0,1,1,2,0,2,1,3,2,1,0,1,1,1,1,2,0,0,1]
Phi of -K [-2,-1,-1,0,1,3,0,1,2,1,2,0,1,0,1,1,1,3,1,2,0]
Phi of K* [-3,-1,0,1,1,2,0,2,1,3,2,1,0,1,1,1,1,2,0,0,1]
Phi of -K* [-2,-1,-1,0,1,3,0,1,0,2,3,0,0,1,1,0,2,3,0,1,2]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial 4w^3z^2+21w^2z+27w
Inner characteristic polynomial t^6+34t^4+18t^2+1
Outer characteristic polynomial t^7+50t^5+35t^3+5t
Flat arrow polynomial -2*K1**2 - 6*K1*K2 + 3*K1 - 4*K2**2 + K2 + 3*K3 + 2*K4 + 4
2-strand cable arrow polynomial -2640*K1**4 + 320*K1**3*K2*K3 + 160*K1**3*K3*K4 - 320*K1**3*K3 + 128*K1**2*K2**2*K4 - 1872*K1**2*K2**2 + 32*K1**2*K2*K3*K5 - 704*K1**2*K2*K4 + 5040*K1**2*K2 - 1184*K1**2*K3**2 - 352*K1**2*K3*K5 - 304*K1**2*K4**2 - 64*K1**2*K5**2 - 3208*K1**2 - 576*K1*K2**2*K3 - 160*K1*K2**2*K5 - 480*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 4368*K1*K2*K3 + 2928*K1*K3*K4 + 984*K1*K4*K5 + 64*K1*K5*K6 - 72*K2**4 - 96*K2**2*K3**2 - 88*K2**2*K4**2 + 1008*K2**2*K4 - 2926*K2**2 - 32*K2*K3*K4*K5 + 568*K2*K3*K5 + 96*K2*K4*K6 + 8*K2*K5*K7 - 16*K3**4 - 48*K3**2*K4**2 + 32*K3**2*K6 - 2120*K3**2 + 88*K3*K4*K7 - 16*K4**4 + 32*K4**2*K8 - 1478*K4**2 - 452*K5**2 - 42*K6**2 - 28*K7**2 - 12*K8**2 + 3496
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {5}, {1, 4}, {2}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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