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Flat knot 6.1709

Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,1,1,2,0,1,0,0,0,1,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1709']
Arrow polynomial of the knot is: -14*K1**2 - 4*K1*K2 + 2*K1 + 7*K2 + 2*K3 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.665', '6.1301', '6.1514', '6.1646', '6.1669', '6.1709', '6.1710', '6.1744', '6.1776']
Outer characteristic polynomial of the knot is: t^7+21t^5+31t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1709']
2-strand cable arrow polynomial of the knot is: -512*K1**6 - 320*K1**4*K2**2 + 1824*K1**4*K2 - 6688*K1**4 + 608*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1152*K1**3*K3 - 5888*K1**2*K2**2 - 704*K1**2*K2*K4 + 12816*K1**2*K2 - 1088*K1**2*K3**2 - 240*K1**2*K4**2 - 6032*K1**2 - 320*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 8496*K1*K2*K3 + 1792*K1*K3*K4 + 192*K1*K4*K5 - 504*K2**4 - 224*K2**2*K3**2 - 48*K2**2*K4**2 + 976*K2**2*K4 - 5676*K2**2 + 192*K2*K3*K5 + 32*K2*K4*K6 - 2684*K3**2 - 762*K4**2 - 68*K5**2 - 4*K6**2 + 5984
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1709']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3657', 'vk6.3754', 'vk6.3943', 'vk6.4040', 'vk6.4492', 'vk6.4587', 'vk6.5878', 'vk6.6005', 'vk6.7150', 'vk6.7323', 'vk6.7416', 'vk6.7923', 'vk6.8042', 'vk6.9357', 'vk6.17902', 'vk6.17999', 'vk6.18747', 'vk6.24437', 'vk6.24872', 'vk6.25333', 'vk6.37494', 'vk6.43872', 'vk6.44227', 'vk6.44530', 'vk6.48297', 'vk6.48362', 'vk6.50088', 'vk6.50198', 'vk6.50584', 'vk6.50647', 'vk6.55863', 'vk6.60711']
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 O1O2O3U1U4O5O4U3U5O6U2U6
R3 orbit {'O1O2O3U1U4O5O4U3U5O6U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U2O4U5U1O6O5U6U3
Gauss code of K* O1O2U3O4O3U5U4U1O5O6U2U6
Gauss code of -K* O1O2U1O3O4U5U3O5O6U4U2U6
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 0 1 0 1],[ 2 0 2 1 2 1 1],[ 0 -2 0 -1 1 0 1],[ 0 -1 1 0 0 0 0],[-1 -2 -1 0 0 0 1],[ 0 -1 0 0 0 0 0],[-1 -1 -1 0 -1 0 0]]
Primitive based matrix [[ 0 1 1 0 0 0 -2],[-1 0 1 0 0 -1 -2],[-1 -1 0 0 0 -1 -1],[ 0 0 0 0 0 1 -1],[ 0 0 0 0 0 0 -1],[ 0 1 1 -1 0 0 -2],[ 2 2 1 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,0,2,-1,0,0,1,2,0,0,1,1,0,-1,1,0,1,2]
Phi over symmetry [-2,0,0,0,1,1,0,1,1,1,2,0,1,0,0,0,1,1,1,1,-1]
Phi of -K [-2,0,0,0,1,1,0,1,1,1,2,0,1,0,0,0,1,1,1,1,-1]
Phi of K* [-1,-1,0,0,0,2,-1,0,1,1,2,0,1,1,1,-1,0,0,0,1,1]
Phi of -K* [-2,0,0,0,1,1,1,1,2,1,2,0,0,0,0,1,0,0,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 21z+43
Enhanced Jones-Krushkal polynomial 21w^2z+43w
Inner characteristic polynomial t^6+15t^4+20t^2+1
Outer characteristic polynomial t^7+21t^5+31t^3+4t
Flat arrow polynomial -14*K1**2 - 4*K1*K2 + 2*K1 + 7*K2 + 2*K3 + 8
2-strand cable arrow polynomial -512*K1**6 - 320*K1**4*K2**2 + 1824*K1**4*K2 - 6688*K1**4 + 608*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1152*K1**3*K3 - 5888*K1**2*K2**2 - 704*K1**2*K2*K4 + 12816*K1**2*K2 - 1088*K1**2*K3**2 - 240*K1**2*K4**2 - 6032*K1**2 - 320*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 8496*K1*K2*K3 + 1792*K1*K3*K4 + 192*K1*K4*K5 - 504*K2**4 - 224*K2**2*K3**2 - 48*K2**2*K4**2 + 976*K2**2*K4 - 5676*K2**2 + 192*K2*K3*K5 + 32*K2*K4*K6 - 2684*K3**2 - 762*K4**2 - 68*K5**2 - 4*K6**2 + 5984
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{6}, {4, 5}, {3}, {1, 2}]]
If K is slice False
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