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Flat knot 6.140

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,0,1,1,2,1,1,1,1,0,1,0,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.140', '7.10425']
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+32t^5+43t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.140']
2-strand cable arrow polynomial of the knot is: 768*K1**4*K2 - 1824*K1**4 + 768*K1**3*K2*K3 - 576*K1**3*K3 - 128*K1**2*K2**4 + 288*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 2448*K1**2*K2**2 - 352*K1**2*K2*K4 + 2440*K1**2*K2 - 1056*K1**2*K3**2 - 32*K1**2*K4**2 - 208*K1**2 + 320*K1*K2**3*K3 - 288*K1*K2**2*K3 - 64*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 2304*K1*K2*K3 + 632*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 488*K2**4 - 336*K2**2*K3**2 - 48*K2**2*K4**2 + 464*K2**2*K4 - 558*K2**2 + 264*K2*K3*K5 + 16*K2*K4*K6 - 396*K3**2 - 126*K4**2 - 36*K5**2 - 2*K6**2 + 708
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.140']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.517', 'vk6.611', 'vk6.640', 'vk6.1019', 'vk6.1112', 'vk6.1155', 'vk6.1871', 'vk6.2292', 'vk6.2518', 'vk6.2566', 'vk6.2595', 'vk6.2802', 'vk6.2897', 'vk6.2920', 'vk6.3079', 'vk6.3201', 'vk6.4613', 'vk6.5900', 'vk6.6027', 'vk6.6540', 'vk6.8068', 'vk6.9379', 'vk6.17850', 'vk6.17867', 'vk6.19067', 'vk6.19885', 'vk6.22554', 'vk6.24371', 'vk6.25679', 'vk6.26327', 'vk6.26770', 'vk6.28577', 'vk6.29807', 'vk6.39904', 'vk6.43788', 'vk6.45064', 'vk6.46846', 'vk6.48009', 'vk6.48083', 'vk6.50657']
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 O1O2O3O4O5O6U4U6U5U3U1U2
R3 orbit {'O1O2O3O4O5O6U4U6U5U3U1U2', 'O1O2O3O4O5U3U5U4U6U1O6U2'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5O6U5U6U4U2U1U3
Gauss code of K* O1O2O3O4O5O6U5U6U4U1U3U2
Gauss code of -K* O1O2O3O4O5O6U5U4U6U3U1U2
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 1 0 -2 1 1],[ 1 0 1 0 -2 1 1],[-1 -1 0 0 -2 1 1],[ 0 0 0 0 -2 1 1],[ 2 2 2 2 0 2 1],[-1 -1 -1 -1 -2 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 1 1 0 -1 -2],[-1 -1 0 0 -1 -1 -1],[-1 -1 0 0 -1 -1 -2],[ 0 0 1 1 0 0 -2],[ 1 1 1 1 0 0 -2],[ 2 2 1 2 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,-1,-1,0,1,2,0,1,1,1,1,1,2,0,2,2]
Phi over symmetry [-2,-1,0,1,1,1,-1,0,1,1,2,1,1,1,1,0,1,0,1,0,-1]
Phi of -K [-2,-1,0,1,1,1,-1,0,1,1,2,1,1,1,1,0,1,0,1,0,-1]
Phi of K* [-1,-1,-1,0,1,2,-1,0,0,1,1,1,1,1,1,0,1,2,1,0,-1]
Phi of -K* [-2,-1,0,1,1,1,2,2,1,2,2,0,1,1,1,1,0,1,-1,0,1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 6z^2+19z+15
Enhanced Jones-Krushkal polynomial 6w^3z^2+19w^2z+15w
Inner characteristic polynomial t^6+24t^4+14t^2+1
Outer characteristic polynomial t^7+32t^5+43t^3+4t
Flat arrow polynomial 4*K1**3 + 2*K1**2 - 4*K1*K2 - K1 - K2 + K3
2-strand cable arrow polynomial 768*K1**4*K2 - 1824*K1**4 + 768*K1**3*K2*K3 - 576*K1**3*K3 - 128*K1**2*K2**4 + 288*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 2448*K1**2*K2**2 - 352*K1**2*K2*K4 + 2440*K1**2*K2 - 1056*K1**2*K3**2 - 32*K1**2*K4**2 - 208*K1**2 + 320*K1*K2**3*K3 - 288*K1*K2**2*K3 - 64*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 2304*K1*K2*K3 + 632*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 488*K2**4 - 336*K2**2*K3**2 - 48*K2**2*K4**2 + 464*K2**2*K4 - 558*K2**2 + 264*K2*K3*K5 + 16*K2*K4*K6 - 396*K3**2 - 126*K4**2 - 36*K5**2 - 2*K6**2 + 708
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {4, 5}, {3}, {1, 2}]]
If K is slice False
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