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Flat knot 6.1708

Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,0,1,1,3,-1,0,1,0,0,1,0,1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1708']
Arrow polynomial of the knot is: 4*K1**3 + 2*K1**2 - 4*K1*K2 - K1 - K2 + K3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.140', '6.569', '6.943', '6.970', '6.1234', '6.1298', '6.1311', '6.1326', '6.1500', '6.1506', '6.1708', '6.1712', '6.1720', '6.1859']
Outer characteristic polynomial of the knot is: t^7+23t^5+43t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1708']
2-strand cable arrow polynomial of the knot is: 1248*K1**4*K2 - 2336*K1**4 + 672*K1**3*K2*K3 - 576*K1**3*K3 - 128*K1**2*K2**4 + 992*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5328*K1**2*K2**2 - 512*K1**2*K2*K4 + 6728*K1**2*K2 - 864*K1**2*K3**2 - 32*K1**2*K4**2 - 3680*K1**2 + 448*K1*K2**3*K3 - 1888*K1*K2**2*K3 - 128*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 5976*K1*K2*K3 + 1288*K1*K3*K4 + 128*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1128*K2**4 - 720*K2**2*K3**2 - 48*K2**2*K4**2 + 1680*K2**2*K4 - 3238*K2**2 + 576*K2*K3*K5 + 16*K2*K4*K6 - 1628*K3**2 - 630*K4**2 - 140*K5**2 - 2*K6**2 + 3316
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1708']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16802', 'vk6.16811', 'vk6.16859', 'vk6.16866', 'vk6.18176', 'vk6.18180', 'vk6.18511', 'vk6.18515', 'vk6.23238', 'vk6.23247', 'vk6.24635', 'vk6.25056', 'vk6.25060', 'vk6.35234', 'vk6.35261', 'vk6.36771', 'vk6.37208', 'vk6.37216', 'vk6.42749', 'vk6.42762', 'vk6.44352', 'vk6.44356', 'vk6.54995', 'vk6.55030', 'vk6.55973', 'vk6.55985', 'vk6.59393', 'vk6.59409', 'vk6.60509', 'vk6.65638', 'vk6.68183', 'vk6.68192']
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 O1O2O3U1U4O5O4U3U2O6U5U6
R3 orbit {'O1O2O3U1U4O5O4U3U2O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O4U2U1O6O5U6U3
Gauss code of K* O1O2U3O4O3U5U2U1O5O6U4U6
Gauss code of -K* O1O2U1O3O4U5U2O5O6U4U3U6
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 2 0],[ 0 -2 0 0 0 1 1],[ 0 -1 0 0 0 0 1],[-1 -2 0 0 0 0 0],[ 0 -2 -1 0 0 0 1],[-1 0 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 1 1 0 0 0 -2],[-1 0 0 0 0 0 -2],[-1 0 0 -1 -1 -1 0],[ 0 0 1 0 1 0 -2],[ 0 0 1 -1 0 0 -2],[ 0 0 1 0 0 0 -1],[ 2 2 0 2 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,0,2,0,0,0,0,2,1,1,1,0,-1,0,2,0,2,1]
Phi over symmetry [-2,0,0,0,1,1,0,0,1,1,3,-1,0,1,0,0,1,0,1,0,0]
Phi of -K [-2,0,0,0,1,1,0,0,1,1,3,-1,0,1,0,0,1,0,1,0,0]
Phi of K* [-1,-1,0,0,0,2,0,0,0,0,3,1,1,1,1,-1,0,0,0,0,1]
Phi of -K* [-2,0,0,0,1,1,1,2,2,0,2,0,0,1,0,-1,1,0,1,0,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 8z^2+27z+23
Enhanced Jones-Krushkal polynomial 8w^3z^2-2w^3z+29w^2z+23w
Inner characteristic polynomial t^6+17t^4+20t^2+4
Outer characteristic polynomial t^7+23t^5+43t^3+13t
Flat arrow polynomial 4*K1**3 + 2*K1**2 - 4*K1*K2 - K1 - K2 + K3
2-strand cable arrow polynomial 1248*K1**4*K2 - 2336*K1**4 + 672*K1**3*K2*K3 - 576*K1**3*K3 - 128*K1**2*K2**4 + 992*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5328*K1**2*K2**2 - 512*K1**2*K2*K4 + 6728*K1**2*K2 - 864*K1**2*K3**2 - 32*K1**2*K4**2 - 3680*K1**2 + 448*K1*K2**3*K3 - 1888*K1*K2**2*K3 - 128*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 5976*K1*K2*K3 + 1288*K1*K3*K4 + 128*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1128*K2**4 - 720*K2**2*K3**2 - 48*K2**2*K4**2 + 1680*K2**2*K4 - 3238*K2**2 + 576*K2*K3*K5 + 16*K2*K4*K6 - 1628*K3**2 - 630*K4**2 - 140*K5**2 - 2*K6**2 + 3316
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}]]
If K is slice False
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