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Flat knot 6.676

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,2,2,2,3,1,1,0,1,1,2,2,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.676']
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+56t^5+41t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.676']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 1696*K1**4*K2 - 3184*K1**4 + 448*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1376*K1**3*K3 - 192*K1**2*K2**4 + 1056*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 6864*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 864*K1**2*K2*K4 + 10800*K1**2*K2 - 848*K1**2*K3**2 - 160*K1**2*K3*K5 - 48*K1**2*K4**2 - 7440*K1**2 + 448*K1*K2**3*K3 - 1728*K1*K2**2*K3 - 224*K1*K2**2*K5 - 288*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9368*K1*K2*K3 + 2264*K1*K3*K4 + 336*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 1120*K2**4 - 240*K2**2*K3**2 - 24*K2**2*K4**2 + 2000*K2**2*K4 - 5636*K2**2 + 416*K2*K3*K5 + 40*K2*K4*K6 + 24*K3**2*K6 - 3060*K3**2 - 1172*K4**2 - 196*K5**2 - 28*K6**2 + 5938
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.676']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11430', 'vk6.11726', 'vk6.12743', 'vk6.13087', 'vk6.20336', 'vk6.21679', 'vk6.27640', 'vk6.29186', 'vk6.31174', 'vk6.31515', 'vk6.32342', 'vk6.32759', 'vk6.39064', 'vk6.41324', 'vk6.45820', 'vk6.47493', 'vk6.52195', 'vk6.52453', 'vk6.53025', 'vk6.53343', 'vk6.57195', 'vk6.58412', 'vk6.61809', 'vk6.62936', 'vk6.63759', 'vk6.63870', 'vk6.64186', 'vk6.64373', 'vk6.66804', 'vk6.67674', 'vk6.69444', 'vk6.70168']
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 O1O2O3O4U3O5U1U4O6U2U5U6
R3 orbit {'O1O2O3O4U3O5U1U4O6U2U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U3O5U1U4O6U2
Gauss code of K* O1O2O3U4U1U5U6O5U2O4O6U3
Gauss code of -K* O1O2O3U1O4O5U2O6U4U6U3U5
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 1 2 2],[ 3 0 2 0 2 3 2],[ 1 -2 0 -1 1 2 2],[ 1 0 1 0 1 1 0],[-1 -2 -1 -1 0 1 1],[-2 -3 -2 -1 -1 0 1],[-2 -2 -2 0 -1 -1 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 1 -1 -1 -2 -3],[-2 -1 0 -1 0 -2 -2],[-1 1 1 0 -1 -1 -2],[ 1 1 0 1 0 1 0],[ 1 2 2 1 -1 0 -2],[ 3 3 2 2 0 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,-1,1,1,2,3,1,0,2,2,1,1,2,-1,0,2]
Phi over symmetry [-3,-1,-1,1,2,2,0,2,2,2,3,1,1,0,1,1,2,2,1,1,-1]
Phi of -K [-3,-1,-1,1,2,2,0,2,2,2,3,1,1,1,1,1,2,3,0,0,-1]
Phi of K* [-2,-2,-1,1,1,3,-1,0,1,3,3,0,1,2,2,1,1,2,-1,0,2]
Phi of -K* [-3,-1,-1,1,2,2,0,2,2,2,3,1,1,0,1,1,2,2,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+36t^4+19t^2+1
Outer characteristic polynomial t^7+56t^5+41t^3+5t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 6*K1*K2 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -256*K1**4*K2**2 + 1696*K1**4*K2 - 3184*K1**4 + 448*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1376*K1**3*K3 - 192*K1**2*K2**4 + 1056*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 6864*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 864*K1**2*K2*K4 + 10800*K1**2*K2 - 848*K1**2*K3**2 - 160*K1**2*K3*K5 - 48*K1**2*K4**2 - 7440*K1**2 + 448*K1*K2**3*K3 - 1728*K1*K2**2*K3 - 224*K1*K2**2*K5 - 288*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9368*K1*K2*K3 + 2264*K1*K3*K4 + 336*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 1120*K2**4 - 240*K2**2*K3**2 - 24*K2**2*K4**2 + 2000*K2**2*K4 - 5636*K2**2 + 416*K2*K3*K5 + 40*K2*K4*K6 + 24*K3**2*K6 - 3060*K3**2 - 1172*K4**2 - 196*K5**2 - 28*K6**2 + 5938
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {1, 3}]]
If K is slice False
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