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Flat knot 6.703

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,1,3,2,3,0,2,1,3,0,1,2,1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.703']
Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 2*K1*K2 - 2*K1 + 3*K2 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.573', '6.628', '6.703', '6.704', '6.723', '6.796', '6.1083', '6.1099', '6.1315']
Outer characteristic polynomial of the knot is: t^7+65t^5+214t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.703']
2-strand cable arrow polynomial of the knot is: -320*K1**4*K2**2 + 608*K1**4*K2 - 1696*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 - 480*K1**3*K3 - 128*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 1888*K1**2*K2**3 - 6368*K1**2*K2**2 - 576*K1**2*K2*K4 + 7528*K1**2*K2 - 224*K1**2*K3**2 - 32*K1**2*K3*K5 - 48*K1**2*K4**2 - 4928*K1**2 + 928*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 544*K1*K2**2*K3 - 96*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 5384*K1*K2*K3 + 664*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**6 + 160*K2**4*K4 - 1640*K2**4 - 576*K2**2*K3**2 - 136*K2**2*K4**2 + 912*K2**2*K4 - 2424*K2**2 + 144*K2*K3*K5 - 1332*K3**2 - 366*K4**2 - 36*K5**2 + 3524
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.703']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73269', 'vk6.73412', 'vk6.74019', 'vk6.74563', 'vk6.75173', 'vk6.75409', 'vk6.76039', 'vk6.76767', 'vk6.78130', 'vk6.78371', 'vk6.79000', 'vk6.79563', 'vk6.79967', 'vk6.80124', 'vk6.80528', 'vk6.80990', 'vk6.81873', 'vk6.82158', 'vk6.82185', 'vk6.82591', 'vk6.83585', 'vk6.83765', 'vk6.84045', 'vk6.84611', 'vk6.84942', 'vk6.85589', 'vk6.85706', 'vk6.85936', 'vk6.86726', 'vk6.87661', 'vk6.88944', 'vk6.89973']
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 O1O2O3O4U5O6U2U3O5U1U4U6
R3 orbit {'O1O2O3O4U5O6U2U3O5U1U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U1U4O6U2U3O5U6
Gauss code of K* O1O2O3U1U4U5U2O6U3O4O5U6
Gauss code of -K* O1O2O3U4O5O6U1O4U2U5U6U3
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 -2 -2 0 2 -1 3],[ 2 0 0 2 3 0 3],[ 2 0 0 1 1 1 2],[ 0 -2 -1 0 0 0 1],[-2 -3 -1 0 0 -2 0],[ 1 0 -1 0 2 0 3],[-3 -3 -2 -1 0 -3 0]]
Primitive based matrix [[ 0 3 2 0 -1 -2 -2],[-3 0 0 -1 -3 -2 -3],[-2 0 0 0 -2 -1 -3],[ 0 1 0 0 0 -1 -2],[ 1 3 2 0 0 -1 0],[ 2 2 1 1 1 0 0],[ 2 3 3 2 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,0,1,2,2,0,1,3,2,3,0,2,1,3,0,1,2,1,0,0]
Phi over symmetry [-3,-2,0,1,2,2,0,1,3,2,3,0,2,1,3,0,1,2,1,0,0]
Phi of -K [-2,-2,-1,0,2,3,0,0,1,3,3,1,0,1,2,1,1,1,2,2,1]
Phi of K* [-3,-2,0,1,2,2,1,2,1,2,3,2,1,1,3,1,0,1,1,0,0]
Phi of -K* [-2,-2,-1,0,2,3,0,0,2,3,3,1,1,1,2,0,2,3,0,1,0]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 5z^2+24z+29
Enhanced Jones-Krushkal polynomial 5w^3z^2-2w^3z+26w^2z+29w
Inner characteristic polynomial t^6+43t^4+117t^2+4
Outer characteristic polynomial t^7+65t^5+214t^3+13t
Flat arrow polynomial 4*K1**3 - 6*K1**2 - 2*K1*K2 - 2*K1 + 3*K2 + 4
2-strand cable arrow polynomial -320*K1**4*K2**2 + 608*K1**4*K2 - 1696*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 - 480*K1**3*K3 - 128*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 1888*K1**2*K2**3 - 6368*K1**2*K2**2 - 576*K1**2*K2*K4 + 7528*K1**2*K2 - 224*K1**2*K3**2 - 32*K1**2*K3*K5 - 48*K1**2*K4**2 - 4928*K1**2 + 928*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 544*K1*K2**2*K3 - 96*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 5384*K1*K2*K3 + 664*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**6 + 160*K2**4*K4 - 1640*K2**4 - 576*K2**2*K3**2 - 136*K2**2*K4**2 + 912*K2**2*K4 - 2424*K2**2 + 144*K2*K3*K5 - 1332*K3**2 - 366*K4**2 - 36*K5**2 + 3524
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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