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Flat knot 6.1751

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,0,1,1,0,1,0,0,0,0,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1751', '7.33031']
Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.374', '6.446', '6.527', '6.1218', '6.1237', '6.1276', '6.1498', '6.1523', '6.1595', '6.1703', '6.1751', '6.1766', '6.1849', '6.1926']
Outer characteristic polynomial of the knot is: t^7+38t^5+104t^3+23t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1751', '7.33031']
2-strand cable arrow polynomial of the knot is: -768*K1**6 - 1472*K1**4*K2**2 + 3488*K1**4*K2 - 6064*K1**4 + 2400*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1632*K1**3*K3 - 832*K1**2*K2**4 + 2592*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 10560*K1**2*K2**2 - 1824*K1**2*K2*K4 + 10768*K1**2*K2 - 1424*K1**2*K3**2 - 384*K1**2*K4**2 - 2780*K1**2 + 1952*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1536*K1*K2**2*K3 - 640*K1*K2**2*K5 - 480*K1*K2*K3*K4 + 9000*K1*K2*K3 - 96*K1*K2*K4*K5 + 1768*K1*K3*K4 + 336*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 2264*K2**4 - 32*K2**3*K6 - 1456*K2**2*K3**2 - 320*K2**2*K4**2 + 2016*K2**2*K4 - 2674*K2**2 - 64*K2*K3**2*K4 + 888*K2*K3*K5 + 232*K2*K4*K6 - 1756*K3**2 - 618*K4**2 - 128*K5**2 - 22*K6**2 + 3616
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1751']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.56', 'vk6.113', 'vk6.210', 'vk6.259', 'vk6.289', 'vk6.675', 'vk6.1216', 'vk6.1265', 'vk6.1356', 'vk6.1405', 'vk6.1441', 'vk6.1927', 'vk6.2382', 'vk6.2446', 'vk6.2936', 'vk6.2996', 'vk6.5737', 'vk6.5770', 'vk6.7806', 'vk6.7839', 'vk6.13271', 'vk6.13304', 'vk6.14773', 'vk6.14809', 'vk6.15931', 'vk6.15967', 'vk6.18045', 'vk6.24487', 'vk6.33028', 'vk6.33383', 'vk6.43917', 'vk6.50490']
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 O1O2O3U4U5O4O5U6U2O6U1U3
R3 orbit {'O1O2O3U4U5O4O5U6U2O6U1U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U3O4U2U4O5O6U5U6
Gauss code of K* O1O2U1O3O4U3U2U4O5O6U5U6
Gauss code of -K* O1O2U3O4O3U5U6O5O6U1U4U2
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 0 2 -1 1 -1],[ 1 0 1 2 0 2 0],[ 0 -1 0 0 -1 1 0],[-2 -2 0 0 -3 -1 -2],[ 1 0 1 3 0 1 0],[-1 -2 -1 1 -1 0 -2],[ 1 0 0 2 0 2 0]]
Primitive based matrix [[ 0 2 1 0 -1 -1 -1],[-2 0 -1 0 -2 -2 -3],[-1 1 0 -1 -2 -2 -1],[ 0 0 1 0 0 -1 -1],[ 1 2 2 0 0 0 0],[ 1 2 2 1 0 0 0],[ 1 3 1 1 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,1,1,1,1,0,2,2,3,1,2,2,1,0,1,1,0,0,0]
Phi over symmetry [-2,-1,0,1,1,1,0,2,0,1,1,0,1,0,0,0,0,1,0,0,0]
Phi of -K [-1,-1,-1,0,1,2,0,0,0,0,1,0,0,1,0,1,0,1,0,2,0]
Phi of K* [-2,-1,0,1,1,1,0,2,0,1,1,0,1,0,0,0,0,1,0,0,0]
Phi of -K* [-1,-1,-1,0,1,2,0,0,0,2,2,0,1,1,3,1,2,2,1,0,1]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+30t^4+79t^2+16
Outer characteristic polynomial t^7+38t^5+104t^3+23t
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6
2-strand cable arrow polynomial -768*K1**6 - 1472*K1**4*K2**2 + 3488*K1**4*K2 - 6064*K1**4 + 2400*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1632*K1**3*K3 - 832*K1**2*K2**4 + 2592*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 10560*K1**2*K2**2 - 1824*K1**2*K2*K4 + 10768*K1**2*K2 - 1424*K1**2*K3**2 - 384*K1**2*K4**2 - 2780*K1**2 + 1952*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1536*K1*K2**2*K3 - 640*K1*K2**2*K5 - 480*K1*K2*K3*K4 + 9000*K1*K2*K3 - 96*K1*K2*K4*K5 + 1768*K1*K3*K4 + 336*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 2264*K2**4 - 32*K2**3*K6 - 1456*K2**2*K3**2 - 320*K2**2*K4**2 + 2016*K2**2*K4 - 2674*K2**2 - 64*K2*K3**2*K4 + 888*K2*K3*K5 + 232*K2*K4*K6 - 1756*K3**2 - 618*K4**2 - 128*K5**2 - 22*K6**2 + 3616
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{6}, {4, 5}, {3}, {1, 2}]]
If K is slice False
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