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Flat knot 6.1763

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,0,0,1,1,0,1,0,1,0,1,0,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1763', '6.1779', '7.45946', '7.45965']
Arrow polynomial of the knot is: 8*K1**3 - 12*K1**2 - 8*K1*K2 - 2*K1 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.218', '6.554', '6.929', '6.932', '6.1014', '6.1024', '6.1068', '6.1526', '6.1664', '6.1676', '6.1755', '6.1763', '6.2065', '6.2078']
Outer characteristic polynomial of the knot is: t^7+16t^5+23t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1763', '6.1779']
2-strand cable arrow polynomial of the knot is: -1024*K1**6 - 2560*K1**4*K2**2 + 5888*K1**4*K2 - 7968*K1**4 + 2944*K1**3*K2*K3 - 1696*K1**3*K3 - 1664*K1**2*K2**4 + 5760*K1**2*K2**3 + 512*K1**2*K2**2*K4 - 16192*K1**2*K2**2 - 1888*K1**2*K2*K4 + 13392*K1**2*K2 - 640*K1**2*K3**2 - 1920*K1**2 + 2560*K1*K2**3*K3 - 2656*K1*K2**2*K3 - 704*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 9024*K1*K2*K3 + 496*K1*K3*K4 + 32*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 3664*K2**4 - 64*K2**3*K6 - 800*K2**2*K3**2 - 128*K2**2*K4**2 + 2400*K2**2*K4 - 1732*K2**2 + 352*K2*K3*K5 + 48*K2*K4*K6 - 912*K3**2 - 164*K4**2 - 16*K5**2 - 4*K6**2 + 3178
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1763']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.400', 'vk6.433', 'vk6.434', 'vk6.838', 'vk6.845', 'vk6.879', 'vk6.884', 'vk6.1584', 'vk6.2028', 'vk6.2039', 'vk6.2057', 'vk6.2068', 'vk6.2702', 'vk6.2734', 'vk6.2741', 'vk6.3143', 'vk6.13535', 'vk6.13536', 'vk6.13725', 'vk6.13726', 'vk6.19463', 'vk6.19466', 'vk6.19758', 'vk6.19759', 'vk6.25806', 'vk6.25808', 'vk6.26633', 'vk6.37914', 'vk6.37916', 'vk6.44909', 'vk6.53667', 'vk6.66252']
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 O1O2O3U4U5O4O6U3U6O5U1U2
R3 orbit {'O1O2O3U4U5O4O6U3U6O5U1U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U3O4U5U1O5O6U4U6
Gauss code of K* O1O2U3O4O5U4U5U1O6O3U6U2
Gauss code of -K* O1O2U3O4O5U4U6O3O6U5U1U2
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 1 0 -1 0 1],[ 1 0 1 0 0 1 1],[-1 -1 0 0 -2 -1 1],[ 0 0 0 0 0 0 1],[ 1 0 2 0 0 0 1],[ 0 -1 1 0 0 0 1],[-1 -1 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 1 0 -1 -1 -2],[-1 -1 0 -1 -1 -1 -1],[ 0 0 1 0 0 0 0],[ 0 1 1 0 0 -1 0],[ 1 1 1 0 1 0 0],[ 1 2 1 0 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,1,1,-1,0,1,1,2,1,1,1,1,0,0,0,1,0,0]
Phi over symmetry [-1,-1,0,0,1,1,-1,0,0,1,1,0,1,0,1,0,1,0,1,1,0]
Phi of -K [-1,-1,0,0,1,1,0,0,1,1,1,1,1,0,1,0,0,0,1,0,-1]
Phi of K* [-1,-1,0,0,1,1,-1,0,0,1,1,0,1,0,1,0,1,0,1,1,0]
Phi of -K* [-1,-1,0,0,1,1,0,0,0,1,2,0,1,1,1,0,1,0,1,1,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+12t^4+15t^2+4
Outer characteristic polynomial t^7+16t^5+23t^3+8t
Flat arrow polynomial 8*K1**3 - 12*K1**2 - 8*K1*K2 - 2*K1 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -1024*K1**6 - 2560*K1**4*K2**2 + 5888*K1**4*K2 - 7968*K1**4 + 2944*K1**3*K2*K3 - 1696*K1**3*K3 - 1664*K1**2*K2**4 + 5760*K1**2*K2**3 + 512*K1**2*K2**2*K4 - 16192*K1**2*K2**2 - 1888*K1**2*K2*K4 + 13392*K1**2*K2 - 640*K1**2*K3**2 - 1920*K1**2 + 2560*K1*K2**3*K3 - 2656*K1*K2**2*K3 - 704*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 9024*K1*K2*K3 + 496*K1*K3*K4 + 32*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 3664*K2**4 - 64*K2**3*K6 - 800*K2**2*K3**2 - 128*K2**2*K4**2 + 2400*K2**2*K4 - 1732*K2**2 + 352*K2*K3*K5 + 48*K2*K4*K6 - 912*K3**2 - 164*K4**2 - 16*K5**2 - 4*K6**2 + 3178
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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