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Flat knot 6.1788

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,1,2,0,1,1,1,0,0,1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1788']
Arrow polynomial of the knot is: -10*K1**2 - 8*K1*K2 + 4*K1 + 5*K2 + 4*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.372', '6.930', '6.1007', '6.1701', '6.1714', '6.1760', '6.1788']
Outer characteristic polynomial of the knot is: t^7+20t^5+27t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1614', '6.1788']
2-strand cable arrow polynomial of the knot is: -448*K1**6 - 192*K1**4*K2**2 + 1440*K1**4*K2 - 4912*K1**4 + 416*K1**3*K2*K3 + 64*K1**3*K3*K4 - 928*K1**3*K3 + 64*K1**2*K2**2*K4 - 3744*K1**2*K2**2 - 736*K1**2*K2*K4 + 9952*K1**2*K2 - 1488*K1**2*K3**2 - 64*K1**2*K3*K5 - 800*K1**2*K4**2 - 6412*K1**2 - 544*K1*K2**2*K3 - 96*K1*K2**2*K5 - 416*K1*K2*K3*K4 + 7648*K1*K2*K3 + 3504*K1*K3*K4 + 1040*K1*K4*K5 - 264*K2**4 - 224*K2**2*K3**2 - 128*K2**2*K4**2 + 1360*K2**2*K4 - 5584*K2**2 + 400*K2*K3*K5 + 128*K2*K4*K6 - 3188*K3**2 - 1714*K4**2 - 336*K5**2 - 32*K6**2 + 6232
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1788']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20007', 'vk6.20082', 'vk6.21277', 'vk6.21364', 'vk6.27058', 'vk6.27143', 'vk6.28761', 'vk6.28832', 'vk6.38455', 'vk6.38540', 'vk6.40642', 'vk6.40737', 'vk6.45339', 'vk6.45436', 'vk6.47106', 'vk6.47178', 'vk6.56806', 'vk6.56903', 'vk6.57938', 'vk6.58041', 'vk6.61324', 'vk6.61429', 'vk6.62498', 'vk6.62586', 'vk6.66526', 'vk6.66603', 'vk6.67313', 'vk6.67394', 'vk6.69172', 'vk6.69251', 'vk6.69921', 'vk6.69992']
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 O1O2O3U4U1O5O6U2U6O4U5U3
R3 orbit {'O1O2O3U4U1O5O6U2U6O4U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4O5U6U2O6O4U3U5
Gauss code of K* O1O2U3O4O5U6U1U5O3O6U4U2
Gauss code of -K* O1O2U3O4O5U4U2O6O3U1U5U6
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 2 -1 0 1],[ 1 0 1 1 0 0 1],[ 1 -1 0 2 0 1 1],[-2 -1 -2 0 -1 -1 0],[ 1 0 0 1 0 0 1],[ 0 0 -1 1 0 0 0],[-1 -1 -1 0 -1 0 0]]
Primitive based matrix [[ 0 2 1 0 -1 -1 -1],[-2 0 0 -1 -1 -1 -2],[-1 0 0 0 -1 -1 -1],[ 0 1 0 0 0 0 -1],[ 1 1 1 0 0 0 1],[ 1 1 1 0 0 0 0],[ 1 2 1 1 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,1,1,1,0,1,1,1,2,0,1,1,1,0,0,1,0,-1,0]
Phi over symmetry [-2,-1,0,1,1,1,0,1,1,1,2,0,1,1,1,0,0,1,0,-1,0]
Phi of -K [-1,-1,-1,0,1,2,-1,0,1,1,2,0,0,1,1,1,1,2,1,1,1]
Phi of K* [-2,-1,0,1,1,1,1,1,1,2,2,1,1,1,1,0,1,1,-1,0,0]
Phi of -K* [-1,-1,-1,0,1,2,-1,0,1,1,2,0,0,1,1,0,1,1,0,1,0]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial z^2+22z+41
Enhanced Jones-Krushkal polynomial w^3z^2+22w^2z+41w
Inner characteristic polynomial t^6+12t^4+14t^2+1
Outer characteristic polynomial t^7+20t^5+27t^3+4t
Flat arrow polynomial -10*K1**2 - 8*K1*K2 + 4*K1 + 5*K2 + 4*K3 + 6
2-strand cable arrow polynomial -448*K1**6 - 192*K1**4*K2**2 + 1440*K1**4*K2 - 4912*K1**4 + 416*K1**3*K2*K3 + 64*K1**3*K3*K4 - 928*K1**3*K3 + 64*K1**2*K2**2*K4 - 3744*K1**2*K2**2 - 736*K1**2*K2*K4 + 9952*K1**2*K2 - 1488*K1**2*K3**2 - 64*K1**2*K3*K5 - 800*K1**2*K4**2 - 6412*K1**2 - 544*K1*K2**2*K3 - 96*K1*K2**2*K5 - 416*K1*K2*K3*K4 + 7648*K1*K2*K3 + 3504*K1*K3*K4 + 1040*K1*K4*K5 - 264*K2**4 - 224*K2**2*K3**2 - 128*K2**2*K4**2 + 1360*K2**2*K4 - 5584*K2**2 + 400*K2*K3*K5 + 128*K2*K4*K6 - 3188*K3**2 - 1714*K4**2 - 336*K5**2 - 32*K6**2 + 6232
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{6}, {1, 5}, {3, 4}, {2}]]
If K is slice False
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