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Flat knot 6.49

Min(phi) over symmetries of the knot is: [-5,0,0,1,2,2,1,3,5,2,4,1,1,1,1,1,1,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.49']
Arrow polynomial of the knot is: -8*K1**3*K2 + 20*K1**3 + 4*K1**2*K3 - 8*K1**2 - 10*K1*K2 - 7*K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.49', '6.50']
Outer characteristic polynomial of the knot is: t^7+99t^5+127t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.49']
2-strand cable arrow polynomial of the knot is: -640*K1**4*K2**2 + 1152*K1**4*K2 - 1792*K1**4 + 256*K1**3*K2*K3 - 192*K1**3*K3 - 2240*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 4768*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 10912*K1**2*K2**2 - 800*K1**2*K2*K4 + 8272*K1**2*K2 - 128*K1**2*K3**2 - 4400*K1**2 + 640*K1*K2**5*K3 - 512*K1*K2**4*K3 - 128*K1*K2**4*K5 - 128*K1*K2**3*K3*K4 + 4640*K1*K2**3*K3 + 640*K1*K2**2*K3*K4 - 2304*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 544*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 7400*K1*K2*K3 + 592*K1*K3*K4 + 64*K1*K4*K5 - 128*K2**6*K4**2 + 384*K2**6*K4 - 1120*K2**6 + 128*K2**5*K4*K6 - 128*K2**5*K6 - 768*K2**4*K3**2 - 704*K2**4*K4**2 + 2560*K2**4*K4 - 32*K2**4*K6**2 - 5984*K2**4 + 384*K2**3*K3*K5 + 384*K2**3*K4*K6 - 256*K2**3*K6 - 2048*K2**2*K3**2 - 1352*K2**2*K4**2 + 3880*K2**2*K4 - 64*K2**2*K5**2 - 32*K2**2*K6**2 - 934*K2**2 + 568*K2*K3*K5 + 280*K2*K4*K6 + 8*K2*K5*K7 + 8*K3**2*K6 - 1460*K3**2 - 756*K4**2 - 76*K5**2 - 18*K6**2 + 3826
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.49']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20119', 'vk6.20123', 'vk6.21399', 'vk6.21407', 'vk6.27207', 'vk6.27215', 'vk6.28885', 'vk6.28889', 'vk6.38619', 'vk6.38627', 'vk6.40809', 'vk6.40825', 'vk6.45493', 'vk6.45509', 'vk6.47223', 'vk6.47231', 'vk6.56932', 'vk6.56940', 'vk6.58068', 'vk6.58084', 'vk6.61482', 'vk6.61498', 'vk6.62625', 'vk6.62633', 'vk6.66642', 'vk6.66646', 'vk6.67427', 'vk6.67435', 'vk6.69276', 'vk6.69284', 'vk6.70013', 'vk6.70017']
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 O1O2O3O4O5O6U1U5U6U3U4U2
R3 orbit {'O1O2O3O4O5O6U1U5U6U3U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U3U4U1U2U6
Gauss code of K* O1O2O3O4O5O6U1U6U4U5U2U3
Gauss code of -K* O1O2O3O4O5O6U4U5U2U3U1U6
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 -5 1 0 2 0 2],[ 5 0 5 3 4 1 2],[-1 -5 0 -1 1 -1 1],[ 0 -3 1 0 1 -1 1],[-2 -4 -1 -1 0 -1 1],[ 0 -1 1 1 1 0 1],[-2 -2 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 2 1 0 0 -5],[-2 0 1 -1 -1 -1 -4],[-2 -1 0 -1 -1 -1 -2],[-1 1 1 0 -1 -1 -5],[ 0 1 1 1 0 1 -1],[ 0 1 1 1 -1 0 -3],[ 5 4 2 5 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,0,0,5,-1,1,1,1,4,1,1,1,2,1,1,5,-1,1,3]
Phi over symmetry [-5,0,0,1,2,2,1,3,5,2,4,1,1,1,1,1,1,1,1,1,-1]
Phi of -K [-5,0,0,1,2,2,2,4,1,3,5,1,0,1,1,0,1,1,0,0,-1]
Phi of K* [-2,-2,-1,0,0,5,-1,0,1,1,5,0,1,1,3,0,0,1,-1,2,4]
Phi of -K* [-5,0,0,1,2,2,1,3,5,2,4,1,1,1,1,1,1,1,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^5-2t^2-t
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial -2w^4z^2+7w^3z^2-2w^3z+24w^2z+25w
Inner characteristic polynomial t^6+65t^4+37t^2+1
Outer characteristic polynomial t^7+99t^5+127t^3+9t
Flat arrow polynomial -8*K1**3*K2 + 20*K1**3 + 4*K1**2*K3 - 8*K1**2 - 10*K1*K2 - 7*K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -640*K1**4*K2**2 + 1152*K1**4*K2 - 1792*K1**4 + 256*K1**3*K2*K3 - 192*K1**3*K3 - 2240*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 4768*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 10912*K1**2*K2**2 - 800*K1**2*K2*K4 + 8272*K1**2*K2 - 128*K1**2*K3**2 - 4400*K1**2 + 640*K1*K2**5*K3 - 512*K1*K2**4*K3 - 128*K1*K2**4*K5 - 128*K1*K2**3*K3*K4 + 4640*K1*K2**3*K3 + 640*K1*K2**2*K3*K4 - 2304*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 544*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 7400*K1*K2*K3 + 592*K1*K3*K4 + 64*K1*K4*K5 - 128*K2**6*K4**2 + 384*K2**6*K4 - 1120*K2**6 + 128*K2**5*K4*K6 - 128*K2**5*K6 - 768*K2**4*K3**2 - 704*K2**4*K4**2 + 2560*K2**4*K4 - 32*K2**4*K6**2 - 5984*K2**4 + 384*K2**3*K3*K5 + 384*K2**3*K4*K6 - 256*K2**3*K6 - 2048*K2**2*K3**2 - 1352*K2**2*K4**2 + 3880*K2**2*K4 - 64*K2**2*K5**2 - 32*K2**2*K6**2 - 934*K2**2 + 568*K2*K3*K5 + 280*K2*K4*K6 + 8*K2*K5*K7 + 8*K3**2*K6 - 1460*K3**2 - 756*K4**2 - 76*K5**2 - 18*K6**2 + 3826
Genus of based matrix 2
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{1, 6}, {5}, {4}, {3}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{2, 6}, {5}, {4}, {3}, {1}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{4, 6}, {5}, {3}, {2}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {3}, {1, 2}]]
If K is slice False
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