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Flat knot 6.308

Min(phi) over symmetries of the knot is: [-3,-3,-1,2,2,3,-1,1,2,3,5,1,1,2,3,1,1,2,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.308']
Arrow polynomial of the knot is: 12*K1**3 - 4*K1**2 - 6*K1*K2 - 6*K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.308']
Outer characteristic polynomial of the knot is: t^7+100t^5+56t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.308']
2-strand cable arrow polynomial of the knot is: -112*K1**4 + 512*K1**2*K2**5 - 1792*K1**2*K2**4 + 1472*K1**2*K2**3 - 2592*K1**2*K2**2 + 2184*K1**2*K2 - 16*K1**2*K3**2 - 80*K1**2*K4**2 - 1704*K1**2 + 896*K1*K2**3*K3 + 1440*K1*K2*K3 + 256*K1*K3*K4 + 120*K1*K4*K5 + 8*K1*K5*K6 - 736*K2**6 + 288*K2**4*K4 - 896*K2**4 - 160*K2**2*K3**2 - 88*K2**2*K4**2 + 656*K2**2*K4 - 224*K2**2 + 16*K2*K3*K5 + 56*K2*K4*K6 - 404*K3**2 - 268*K4**2 - 44*K5**2 - 16*K6**2 + 1226
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.308']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71625', 'vk6.71790', 'vk6.72212', 'vk6.72354', 'vk6.73369', 'vk6.73531', 'vk6.75271', 'vk6.75539', 'vk6.77245', 'vk6.77330', 'vk6.77580', 'vk6.77685', 'vk6.78257', 'vk6.78508', 'vk6.80075', 'vk6.80224', 'vk6.81112', 'vk6.81175', 'vk6.81196', 'vk6.81237', 'vk6.81341', 'vk6.81529', 'vk6.82010', 'vk6.82429', 'vk6.82741', 'vk6.85453', 'vk6.86349', 'vk6.86924', 'vk6.87142', 'vk6.88096', 'vk6.88663', 'vk6.88768']
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 O1O2O3O4O5U2U3O6U1U5U6U4
R3 orbit {'O1O2O3O4O5U2U3O6U1U5U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U6U1U5O6U3U4
Gauss code of K* O1O2O3O4U1U5U6U4U2O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U3U1U5U6U4
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 -3 -1 3 2 2],[ 3 0 -1 1 5 3 2],[ 3 1 0 1 3 2 1],[ 1 -1 -1 0 2 1 1],[-3 -5 -3 -2 0 -1 1],[-2 -3 -2 -1 1 0 1],[-2 -2 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 3 2 2 -1 -3 -3],[-3 0 1 -1 -2 -3 -5],[-2 -1 0 -1 -1 -1 -2],[-2 1 1 0 -1 -2 -3],[ 1 2 1 1 0 -1 -1],[ 3 3 1 2 1 0 1],[ 3 5 2 3 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-2,1,3,3,-1,1,2,3,5,1,1,1,2,1,2,3,1,1,-1]
Phi over symmetry [-3,-3,-1,2,2,3,-1,1,2,3,5,1,1,2,3,1,1,2,-1,-1,1]
Phi of -K [-3,-3,-1,2,2,3,-1,1,3,4,3,1,2,3,1,2,2,2,-1,0,2]
Phi of K* [-3,-2,-2,1,3,3,0,2,2,1,3,1,2,2,3,2,3,4,1,1,-1]
Phi of -K* [-3,-3,-1,2,2,3,-1,1,2,3,5,1,1,2,3,1,1,2,-1,-1,1]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 4z+9
Enhanced Jones-Krushkal polynomial 4w^4z-12w^3z+12w^2z+9w
Inner characteristic polynomial t^6+64t^4
Outer characteristic polynomial t^7+100t^5+56t^3
Flat arrow polynomial 12*K1**3 - 4*K1**2 - 6*K1*K2 - 6*K1 + 2*K2 + 3
2-strand cable arrow polynomial -112*K1**4 + 512*K1**2*K2**5 - 1792*K1**2*K2**4 + 1472*K1**2*K2**3 - 2592*K1**2*K2**2 + 2184*K1**2*K2 - 16*K1**2*K3**2 - 80*K1**2*K4**2 - 1704*K1**2 + 896*K1*K2**3*K3 + 1440*K1*K2*K3 + 256*K1*K3*K4 + 120*K1*K4*K5 + 8*K1*K5*K6 - 736*K2**6 + 288*K2**4*K4 - 896*K2**4 - 160*K2**2*K3**2 - 88*K2**2*K4**2 + 656*K2**2*K4 - 224*K2**2 + 16*K2*K3*K5 + 56*K2*K4*K6 - 404*K3**2 - 268*K4**2 - 44*K5**2 - 16*K6**2 + 1226
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {5}, {1, 4}, {3}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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