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Flat knot 6.1048

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,0,3,2,1,0,2,1,1,-1,0,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1048']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 6*K1*K2 + K2 + 2*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.362', '6.624', '6.789', '6.859', '6.882', '6.975', '6.989', '6.1048', '6.1057', '6.1158']
Outer characteristic polynomial of the knot is: t^7+66t^5+332t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1048']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 96*K1**4*K2 - 208*K1**4 + 128*K1**3*K2**3*K3 + 512*K1**3*K2*K3 - 320*K1**2*K2**4 + 192*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 1952*K1**2*K2**2 + 1096*K1**2*K2 - 448*K1**2*K3**2 - 848*K1**2 + 576*K1*K2**3*K3 + 96*K1*K2*K3**3 + 2312*K1*K2*K3 + 280*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 536*K2**4 - 416*K2**2*K3**2 - 56*K2**2*K4**2 + 336*K2**2*K4 - 620*K2**2 + 112*K2*K3*K5 + 24*K2*K4*K6 - 16*K3**4 - 740*K3**2 - 170*K4**2 - 4*K5**2 - 4*K6**2 + 992
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1048']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4711', 'vk6.5024', 'vk6.6232', 'vk6.6688', 'vk6.8212', 'vk6.8648', 'vk6.9590', 'vk6.9921', 'vk6.20300', 'vk6.21635', 'vk6.27596', 'vk6.29150', 'vk6.39018', 'vk6.41268', 'vk6.45786', 'vk6.47465', 'vk6.48751', 'vk6.48950', 'vk6.49548', 'vk6.49766', 'vk6.50765', 'vk6.50969', 'vk6.51242', 'vk6.51451', 'vk6.57163', 'vk6.58353', 'vk6.61789', 'vk6.62910', 'vk6.66780', 'vk6.67658', 'vk6.69428', 'vk6.70152']
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 O1O2O3O4U5U6O5U2O6U1U3U4
R3 orbit {'O1O2O3O4U5U6O5U2O6U1U3U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U2U4O5U3O6U5U6
Gauss code of K* O1O2O3U1U4U2U3O5O6U5O4U6
Gauss code of -K* O1O2O3U4O5U6O4O6U1U2U5U3
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 -1 1 3 -1 0],[ 2 0 1 2 3 1 2],[ 1 -1 0 0 1 1 2],[-1 -2 0 0 1 -2 0],[-3 -3 -1 -1 0 -4 -2],[ 1 -1 -1 2 4 0 0],[ 0 -2 -2 0 2 0 0]]
Primitive based matrix [[ 0 3 1 0 -1 -1 -2],[-3 0 -1 -2 -1 -4 -3],[-1 1 0 0 0 -2 -2],[ 0 2 0 0 -2 0 -2],[ 1 1 0 2 0 1 -1],[ 1 4 2 0 -1 0 -1],[ 2 3 2 2 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,1,1,2,1,2,1,4,3,0,0,2,2,2,0,2,-1,1,1]
Phi over symmetry [-3,-1,0,1,1,2,1,1,0,3,2,1,0,2,1,1,-1,0,-1,0,0]
Phi of -K [-2,-1,-1,0,1,3,0,0,0,1,2,-1,-1,2,3,1,0,0,1,1,1]
Phi of K* [-3,-1,0,1,1,2,1,1,0,3,2,1,0,2,1,1,-1,0,-1,0,0]
Phi of -K* [-2,-1,-1,0,1,3,1,1,2,2,3,-1,0,2,4,2,0,1,0,2,1]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 5z+11
Enhanced Jones-Krushkal polynomial -8w^3z+13w^2z+11w
Inner characteristic polynomial t^6+50t^4+243t^2
Outer characteristic polynomial t^7+66t^5+332t^3
Flat arrow polynomial 4*K1**3 - 2*K1**2 - 6*K1*K2 + K2 + 2*K3 + 2
2-strand cable arrow polynomial -256*K1**4*K2**2 + 96*K1**4*K2 - 208*K1**4 + 128*K1**3*K2**3*K3 + 512*K1**3*K2*K3 - 320*K1**2*K2**4 + 192*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 1952*K1**2*K2**2 + 1096*K1**2*K2 - 448*K1**2*K3**2 - 848*K1**2 + 576*K1*K2**3*K3 + 96*K1*K2*K3**3 + 2312*K1*K2*K3 + 280*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 536*K2**4 - 416*K2**2*K3**2 - 56*K2**2*K4**2 + 336*K2**2*K4 - 620*K2**2 + 112*K2*K3*K5 + 24*K2*K4*K6 - 16*K3**4 - 740*K3**2 - 170*K4**2 - 4*K5**2 - 4*K6**2 + 992
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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