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Flat knot 6.1724

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,2,2,1,1,0,1,1,0,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1724']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 8*K1*K2 + K1 + K2 + 3*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1030', '6.1062', '6.1226', '6.1508', '6.1525', '6.1596', '6.1724', '6.1729', '6.1735', '6.1738', '6.1789', '6.1809', '6.1921']
Outer characteristic polynomial of the knot is: t^7+24t^5+41t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1724', '6.1735']
2-strand cable arrow polynomial of the knot is: 1280*K1**4*K2 - 4320*K1**4 + 768*K1**3*K2*K3 - 1024*K1**3*K3 + 1888*K1**2*K2**3 - 7728*K1**2*K2**2 - 1472*K1**2*K2*K4 + 7792*K1**2*K2 - 640*K1**2*K3**2 - 2124*K1**2 + 1088*K1*K2**3*K3 - 1504*K1*K2**2*K3 - 512*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 7008*K1*K2*K3 + 1488*K1*K3*K4 + 144*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 2040*K2**4 - 32*K2**3*K6 - 960*K2**2*K3**2 - 128*K2**2*K4**2 + 2168*K2**2*K4 - 2250*K2**2 + 864*K2*K3*K5 + 104*K2*K4*K6 - 1596*K3**2 - 770*K4**2 - 216*K5**2 - 22*K6**2 + 2904
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1724']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20177', 'vk6.20185', 'vk6.20189', 'vk6.20193', 'vk6.21459', 'vk6.21467', 'vk6.27309', 'vk6.27325', 'vk6.27331', 'vk6.27339', 'vk6.28965', 'vk6.28981', 'vk6.28987', 'vk6.28991', 'vk6.38746', 'vk6.38762', 'vk6.38768', 'vk6.38776', 'vk6.40924', 'vk6.40940', 'vk6.47304', 'vk6.47312', 'vk6.47316', 'vk6.47324', 'vk6.57014', 'vk6.57018', 'vk6.57026', 'vk6.57034', 'vk6.62695', 'vk6.62703', 'vk6.70057', 'vk6.70061']
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 O1O2O3U2U1O4O5U6U3O6U4U5
R3 orbit {'O1O2O3U2U1O4O5U6U3O6U4U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O6U1U6O4O5U3U2
Gauss code of K* O1O2U1O3O4U5U6U2O6O5U3U4
Gauss code of -K* O1O2U3O4O3U1U2O5O6U4U6U5
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 1 0 2 -1],[ 1 0 0 2 1 1 0],[ 1 0 0 1 1 1 0],[-1 -2 -1 0 0 1 -1],[ 0 -1 -1 0 0 1 0],[-2 -1 -1 -1 -1 0 -2],[ 1 0 0 1 0 2 0]]
Primitive based matrix [[ 0 2 1 0 -1 -1 -1],[-2 0 -1 -1 -1 -1 -2],[-1 1 0 0 -1 -2 -1],[ 0 1 0 0 -1 -1 0],[ 1 1 1 1 0 0 0],[ 1 1 2 1 0 0 0],[ 1 2 1 0 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,1,1,1,1,1,1,1,2,0,1,2,1,1,1,0,0,0,0]
Phi over symmetry [-2,-1,0,1,1,1,0,1,1,2,2,1,1,0,1,1,0,0,0,0,0]
Phi of -K [-1,-1,-1,0,1,2,0,0,0,0,2,0,0,1,2,1,1,1,1,1,0]
Phi of K* [-2,-1,0,1,1,1,0,1,1,2,2,1,1,0,1,1,0,0,0,0,0]
Phi of -K* [-1,-1,-1,0,1,2,0,0,0,1,2,0,1,1,1,1,2,1,0,1,1]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial 7w^3z^2+24w^2z+21w
Inner characteristic polynomial t^6+16t^4+24t^2+4
Outer characteristic polynomial t^7+24t^5+41t^3+8t
Flat arrow polynomial 4*K1**3 - 2*K1**2 - 8*K1*K2 + K1 + K2 + 3*K3 + 2
2-strand cable arrow polynomial 1280*K1**4*K2 - 4320*K1**4 + 768*K1**3*K2*K3 - 1024*K1**3*K3 + 1888*K1**2*K2**3 - 7728*K1**2*K2**2 - 1472*K1**2*K2*K4 + 7792*K1**2*K2 - 640*K1**2*K3**2 - 2124*K1**2 + 1088*K1*K2**3*K3 - 1504*K1*K2**2*K3 - 512*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 7008*K1*K2*K3 + 1488*K1*K3*K4 + 144*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 2040*K2**4 - 32*K2**3*K6 - 960*K2**2*K3**2 - 128*K2**2*K4**2 + 2168*K2**2*K4 - 2250*K2**2 + 864*K2*K3*K5 + 104*K2*K4*K6 - 1596*K3**2 - 770*K4**2 - 216*K5**2 - 22*K6**2 + 2904
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{6}, {5}, {1, 4}, {2, 3}]]
If K is slice False
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