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Flat knot 6.1705

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,0,2,2,3,0,1,1,1,-1,1,0,1,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.1705']
Arrow polynomial of the knot is: -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.546', '6.591', '6.598', '6.666', '6.680', '6.742', '6.778', '6.805', '6.822', '6.824', '6.1129', '6.1512', '6.1647', '6.1678', '6.1705', '6.1847', '6.1857']
Outer characteristic polynomial of the knot is: t^7+31t^5+44t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1705']
2-strand cable arrow polynomial of the knot is: -384*K1**6 - 192*K1**4*K2**2 + 1152*K1**4*K2 - 5472*K1**4 + 1120*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1056*K1**3*K3 - 5120*K1**2*K2**2 - 704*K1**2*K2*K4 + 11176*K1**2*K2 - 2304*K1**2*K3**2 - 64*K1**2*K3*K5 - 384*K1**2*K4**2 - 6516*K1**2 - 576*K1*K2**2*K3 - 160*K1*K2*K3*K4 + 9384*K1*K2*K3 + 3024*K1*K3*K4 + 464*K1*K4*K5 - 144*K2**4 - 80*K2**2*K3**2 - 16*K2**2*K4**2 + 760*K2**2*K4 - 5836*K2**2 + 160*K2*K3*K5 + 32*K2*K4*K6 - 3440*K3**2 - 1112*K4**2 - 156*K5**2 - 12*K6**2 + 6326
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1705']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3649', 'vk6.3746', 'vk6.3935', 'vk6.4032', 'vk6.4493', 'vk6.4590', 'vk6.5875', 'vk6.6004', 'vk6.7142', 'vk6.7315', 'vk6.7408', 'vk6.7924', 'vk6.8045', 'vk6.9354', 'vk6.17919', 'vk6.18014', 'vk6.18741', 'vk6.24454', 'vk6.24862', 'vk6.25323', 'vk6.37488', 'vk6.43889', 'vk6.44217', 'vk6.44520', 'vk6.48273', 'vk6.48338', 'vk6.50064', 'vk6.50174', 'vk6.50585', 'vk6.50650', 'vk6.55878', 'vk6.60717']
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 O1O2O3U1U3O4O5U6U4O6U2U5
R3 orbit {'O1O2O3U1U3O4O5U6U4O6U2U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U2O5U6U5O4O6U1U3
Gauss code of K* O1O2U1O3O4U5U3U6O5O6U2U4
Gauss code of -K* O1O2U3O4O3U1U4O5O6U5U2U6
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 0 1 0 2 -1],[ 2 0 2 1 0 1 2],[ 0 -2 0 0 1 2 -1],[-1 -1 0 0 0 0 -1],[ 0 0 -1 0 0 0 0],[-2 -1 -2 0 0 0 -2],[ 1 -2 1 1 0 2 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 0 0 -2 -2 -1],[-1 0 0 0 0 -1 -1],[ 0 0 0 0 -1 0 0],[ 0 2 0 1 0 -1 -2],[ 1 2 1 0 1 0 -2],[ 2 1 1 0 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,0,0,2,2,1,0,0,1,1,1,0,0,1,2,2]
Phi over symmetry [-2,-1,0,0,1,2,-1,0,2,2,3,0,1,1,1,-1,1,0,1,2,1]
Phi of -K [-2,-1,0,0,1,2,-1,0,2,2,3,0,1,1,1,-1,1,0,1,2,1]
Phi of K* [-2,-1,0,0,1,2,1,0,2,1,3,1,1,1,2,1,0,0,1,2,-1]
Phi of -K* [-2,-1,0,0,1,2,2,0,2,1,1,0,1,1,2,-1,0,0,0,2,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 21z+43
Enhanced Jones-Krushkal polynomial 21w^2z+43w
Inner characteristic polynomial t^6+21t^4+22t^2+1
Outer characteristic polynomial t^7+31t^5+44t^3+5t
Flat arrow polynomial -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -384*K1**6 - 192*K1**4*K2**2 + 1152*K1**4*K2 - 5472*K1**4 + 1120*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1056*K1**3*K3 - 5120*K1**2*K2**2 - 704*K1**2*K2*K4 + 11176*K1**2*K2 - 2304*K1**2*K3**2 - 64*K1**2*K3*K5 - 384*K1**2*K4**2 - 6516*K1**2 - 576*K1*K2**2*K3 - 160*K1*K2*K3*K4 + 9384*K1*K2*K3 + 3024*K1*K3*K4 + 464*K1*K4*K5 - 144*K2**4 - 80*K2**2*K3**2 - 16*K2**2*K4**2 + 760*K2**2*K4 - 5836*K2**2 + 160*K2*K3*K5 + 32*K2*K4*K6 - 3440*K3**2 - 1112*K4**2 - 156*K5**2 - 12*K6**2 + 6326
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}]]
If K is slice False
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