Table of flat knot invariants
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Flat knot 6.1

Min(phi) over symmetries of the knot is: [-5,-3,-1,1,4,4,1,2,3,4,5,1,2,3,4,1,2,3,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.1']
Arrow polynomial of the knot is: -8*K1**3*K2 + 4*K1**3 + 4*K1**2*K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1', '6.37']
Outer characteristic polynomial of the knot is: t^7+172t^5+124t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1']
2-strand cable arrow polynomial of the knot is: -2048*K1**2*K2**6 + 768*K1**2*K2**5 - 256*K1**2*K2**4 - 640*K1**2*K2**2 + 336*K1**2*K2 - 464*K1**2 + 1024*K1*K2**7*K3 + 1536*K1*K2**5*K3 + 256*K1*K2**3*K3 + 960*K1*K2*K3 + 64*K1*K3*K4 + 64*K1*K4*K5 - 512*K2**8 - 1024*K2**6*K3**2 - 128*K2**6*K4**2 + 128*K2**6*K4 + 160*K2**6 + 256*K2**5*K3*K5 + 128*K2**5*K4*K6 - 256*K2**4*K3**2 - 64*K2**4*K4**2 - 32*K2**4*K6**2 + 192*K2**4 - 256*K2**2*K3**2 - 32*K2**2*K4**2 + 32*K2**2*K4 - 280*K2**2 + 112*K2*K3*K5 + 32*K2*K4*K6 - 376*K3**2 - 80*K4**2 - 56*K5**2 - 8*K6**2 + 462
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.82805', 'vk6.82927', 'vk6.82928', 'vk6.82957', 'vk6.82959', 'vk6.82961', 'vk6.83235', 'vk6.83237', 'vk6.83871', 'vk6.83875', 'vk6.83882', 'vk6.86030', 'vk6.86036', 'vk6.86774', 'vk6.86775', 'vk6.86779', 'vk6.86791', 'vk6.86795', 'vk6.86808', 'vk6.86811', 'vk6.89783', 'vk6.89848', 'vk6.89991', 'vk6.90095']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3O4O5O6U1U2U3U4U6U5
R3 orbit {'O1O2O3O4O5O6U1U2U3U4U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U2U1U3U4U5U6
Gauss code of K* Same
Gauss code of -K* O1O2O3O4O5O6U2U1U3U4U5U6
Diagrammatic symmetry type +
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -5 -3 -1 1 4 4],[ 5 0 1 2 3 5 4],[ 3 -1 0 1 2 4 3],[ 1 -2 -1 0 1 3 2],[-1 -3 -2 -1 0 2 1],[-4 -5 -4 -3 -2 0 0],[-4 -4 -3 -2 -1 0 0]]
Primitive based matrix [[ 0 4 4 1 -1 -3 -5],[-4 0 0 -1 -2 -3 -4],[-4 0 0 -2 -3 -4 -5],[-1 1 2 0 -1 -2 -3],[ 1 2 3 1 0 -1 -2],[ 3 3 4 2 1 0 -1],[ 5 4 5 3 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-4,-1,1,3,5,0,1,2,3,4,2,3,4,5,1,2,3,1,2,1]
Phi over symmetry [-5,-3,-1,1,4,4,1,2,3,4,5,1,2,3,4,1,2,3,1,2,0]
Phi of -K [-5,-3,-1,1,4,4,1,2,3,4,5,1,2,3,4,1,2,3,1,2,0]
Phi of K* [-4,-4,-1,1,3,5,0,1,2,3,4,2,3,4,5,1,2,3,1,2,1]
Phi of -K* [-5,-3,-1,1,4,4,1,2,3,4,5,1,2,3,4,1,2,3,1,2,0]
Symmetry type of based matrix +
u-polynomial t^5-2t^4+t^3
Normalized Jones-Krushkal polynomial 1
Enhanced Jones-Krushkal polynomial -8w^5z+12w^4z-6w^3z+2w^2z+w
Inner characteristic polynomial t^6+104t^4+20t^2
Outer characteristic polynomial t^7+172t^5+124t^3
Flat arrow polynomial -8*K1**3*K2 + 4*K1**3 + 4*K1**2*K3 + 1
2-strand cable arrow polynomial -2048*K1**2*K2**6 + 768*K1**2*K2**5 - 256*K1**2*K2**4 - 640*K1**2*K2**2 + 336*K1**2*K2 - 464*K1**2 + 1024*K1*K2**7*K3 + 1536*K1*K2**5*K3 + 256*K1*K2**3*K3 + 960*K1*K2*K3 + 64*K1*K3*K4 + 64*K1*K4*K5 - 512*K2**8 - 1024*K2**6*K3**2 - 128*K2**6*K4**2 + 128*K2**6*K4 + 160*K2**6 + 256*K2**5*K3*K5 + 128*K2**5*K4*K6 - 256*K2**4*K3**2 - 64*K2**4*K4**2 - 32*K2**4*K6**2 + 192*K2**4 - 256*K2**2*K3**2 - 32*K2**2*K4**2 + 32*K2**2*K4 - 280*K2**2 + 112*K2*K3*K5 + 32*K2*K4*K6 - 376*K3**2 - 80*K4**2 - 56*K5**2 - 8*K6**2 + 462
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{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}], [{5, 6}, {4}, {3}, {2}, {1}]]
If K is slice False
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