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Flat knot 6.611

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,-1,1,1,3,3,1,0,1,1,0,0,1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.611']
Arrow polynomial of the knot is: -6*K1**2 - 6*K1*K2 + 3*K1 + 3*K2 + 3*K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.458', '6.601', '6.611', '6.995', '6.1026', '6.1037']
Outer characteristic polynomial of the knot is: t^7+42t^5+40t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.611', '6.1017', '6.1020']
2-strand cable arrow polynomial of the knot is: 96*K1**4*K2 - 2224*K1**4 + 480*K1**3*K2*K3 + 32*K1**3*K3*K4 - 384*K1**3*K3 + 128*K1**2*K2**2*K4 - 2880*K1**2*K2**2 - 512*K1**2*K2*K4 + 5544*K1**2*K2 - 704*K1**2*K3**2 - 96*K1**2*K4**2 - 3108*K1**2 - 992*K1*K2**2*K3 - 96*K1*K2**2*K5 + 32*K1*K2*K3**3 - 96*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5024*K1*K2*K3 + 1496*K1*K3*K4 + 104*K1*K4*K5 + 8*K1*K5*K6 - 152*K2**4 - 304*K2**2*K3**2 - 56*K2**2*K4**2 + 888*K2**2*K4 - 3062*K2**2 + 320*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 16*K3**2*K6 - 1752*K3**2 - 650*K4**2 - 84*K5**2 - 10*K6**2 + 3000
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.611']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4346', 'vk6.4377', 'vk6.5664', 'vk6.5695', 'vk6.7737', 'vk6.7768', 'vk6.9215', 'vk6.9246', 'vk6.10486', 'vk6.10564', 'vk6.10661', 'vk6.10705', 'vk6.10736', 'vk6.10850', 'vk6.14625', 'vk6.15325', 'vk6.15450', 'vk6.16248', 'vk6.17968', 'vk6.24412', 'vk6.30165', 'vk6.30243', 'vk6.30340', 'vk6.30469', 'vk6.33963', 'vk6.34364', 'vk6.34418', 'vk6.43839', 'vk6.50439', 'vk6.50470', 'vk6.54211', 'vk6.63425']
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 O1O2O3O4U3O5U2O6U5U6U1U4
R3 orbit {'O1O2O3O4U3O5U2O6U5U6U1U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U4U5U6O5U3O6U2
Gauss code of K* O1O2O3O4U3U5U6U4O6U1O5U2
Gauss code of -K* O1O2O3O4U3O5U4O6U1U6U5U2
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 0 1],[ 1 0 -1 0 3 0 1],[ 2 1 0 0 3 1 1],[ 1 0 0 0 1 0 0],[-3 -3 -3 -1 0 -1 1],[ 0 0 -1 0 1 0 1],[-1 -1 -1 0 -1 -1 0]]
Primitive based matrix [[ 0 3 1 0 -1 -1 -2],[-3 0 1 -1 -1 -3 -3],[-1 -1 0 -1 0 -1 -1],[ 0 1 1 0 0 0 -1],[ 1 1 0 0 0 0 0],[ 1 3 1 0 0 0 -1],[ 2 3 1 1 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,1,1,2,-1,1,1,3,3,1,0,1,1,0,0,1,0,0,1]
Phi over symmetry [-3,-1,0,1,1,2,-1,1,1,3,3,1,0,1,1,0,0,1,0,0,1]
Phi of -K [-2,-1,-1,0,1,3,0,1,1,2,2,0,1,1,1,1,2,3,0,2,3]
Phi of K* [-3,-1,0,1,1,2,3,2,1,3,2,0,1,2,2,1,1,1,0,0,1]
Phi of -K* [-2,-1,-1,0,1,3,0,1,1,1,3,0,0,0,1,0,1,3,1,1,-1]
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+26t^4+15t^2
Outer characteristic polynomial t^7+42t^5+40t^3+4t
Flat arrow polynomial -6*K1**2 - 6*K1*K2 + 3*K1 + 3*K2 + 3*K3 + 4
2-strand cable arrow polynomial 96*K1**4*K2 - 2224*K1**4 + 480*K1**3*K2*K3 + 32*K1**3*K3*K4 - 384*K1**3*K3 + 128*K1**2*K2**2*K4 - 2880*K1**2*K2**2 - 512*K1**2*K2*K4 + 5544*K1**2*K2 - 704*K1**2*K3**2 - 96*K1**2*K4**2 - 3108*K1**2 - 992*K1*K2**2*K3 - 96*K1*K2**2*K5 + 32*K1*K2*K3**3 - 96*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5024*K1*K2*K3 + 1496*K1*K3*K4 + 104*K1*K4*K5 + 8*K1*K5*K6 - 152*K2**4 - 304*K2**2*K3**2 - 56*K2**2*K4**2 + 888*K2**2*K4 - 3062*K2**2 + 320*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 16*K3**2*K6 - 1752*K3**2 - 650*K4**2 - 84*K5**2 - 10*K6**2 + 3000
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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