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Flat knot 6.1701

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,2,2,3,-1,1,2,1,1,1,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1701']
Arrow polynomial of the knot is: -10*K1**2 - 8*K1*K2 + 4*K1 + 5*K2 + 4*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.372', '6.930', '6.1007', '6.1701', '6.1714', '6.1760', '6.1788']
Outer characteristic polynomial of the knot is: t^7+22t^5+49t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1701']
2-strand cable arrow polynomial of the knot is: -384*K1**6 - 192*K1**4*K2**2 + 1120*K1**4*K2 - 3184*K1**4 + 704*K1**3*K2*K3 + 96*K1**3*K3*K4 - 1088*K1**3*K3 - 2848*K1**2*K2**2 - 480*K1**2*K2*K4 + 7264*K1**2*K2 - 880*K1**2*K3**2 - 224*K1**2*K4**2 - 4392*K1**2 - 480*K1*K2**2*K3 - 32*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 5352*K1*K2*K3 + 1568*K1*K3*K4 + 368*K1*K4*K5 - 168*K2**4 - 224*K2**2*K3**2 - 128*K2**2*K4**2 + 864*K2**2*K4 - 3848*K2**2 + 344*K2*K3*K5 + 128*K2*K4*K6 - 1968*K3**2 - 826*K4**2 - 184*K5**2 - 32*K6**2 + 4008
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1701']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4447', 'vk6.4544', 'vk6.5833', 'vk6.5962', 'vk6.7891', 'vk6.8007', 'vk6.9320', 'vk6.9441', 'vk6.13398', 'vk6.13495', 'vk6.13686', 'vk6.14075', 'vk6.15050', 'vk6.15172', 'vk6.17788', 'vk6.17821', 'vk6.18828', 'vk6.19430', 'vk6.19725', 'vk6.24335', 'vk6.25425', 'vk6.25458', 'vk6.26604', 'vk6.33252', 'vk6.33313', 'vk6.37555', 'vk6.44887', 'vk6.48636', 'vk6.50538', 'vk6.53652', 'vk6.55821', 'vk6.65493']
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 O1O2O3U1U3O4O5U2U5O6U4U6
R3 orbit {'O1O2O3U1U3O4O5U2U5O6U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O4U6U2O6O5U1U3
Gauss code of K* O1O2U3O4O3U5U1U6O5O6U4U2
Gauss code of -K* O1O2U1O3O4U3U2O5O6U5U4U6
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 -1 1 0 1 1],[ 2 0 2 1 1 1 0],[ 1 -2 0 0 2 1 1],[-1 -1 0 0 0 0 0],[ 0 -1 -2 0 0 0 1],[-1 -1 -1 0 0 0 0],[-1 0 -1 0 -1 0 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 -1 -1],[-1 0 0 0 -1 -1 0],[ 0 0 0 1 0 -2 -1],[ 1 0 1 1 2 0 -2],[ 2 1 1 0 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,0,0,0,0,1,0,0,1,1,1,1,0,2,1,2]
Phi over symmetry [-2,-1,0,1,1,1,-1,1,2,2,3,-1,1,2,1,1,1,0,0,0,0]
Phi of -K [-2,-1,0,1,1,1,-1,1,2,2,3,-1,1,2,1,1,1,0,0,0,0]
Phi of K* [-1,-1,-1,0,1,2,0,0,0,1,3,0,1,1,2,1,2,2,-1,1,-1]
Phi of -K* [-2,-1,0,1,1,1,2,1,0,1,1,2,1,0,1,1,0,0,0,0,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 17z+35
Enhanced Jones-Krushkal polynomial 17w^2z+35w
Inner characteristic polynomial t^6+14t^4+12t^2+1
Outer characteristic polynomial t^7+22t^5+49t^3+5t
Flat arrow polynomial -10*K1**2 - 8*K1*K2 + 4*K1 + 5*K2 + 4*K3 + 6
2-strand cable arrow polynomial -384*K1**6 - 192*K1**4*K2**2 + 1120*K1**4*K2 - 3184*K1**4 + 704*K1**3*K2*K3 + 96*K1**3*K3*K4 - 1088*K1**3*K3 - 2848*K1**2*K2**2 - 480*K1**2*K2*K4 + 7264*K1**2*K2 - 880*K1**2*K3**2 - 224*K1**2*K4**2 - 4392*K1**2 - 480*K1*K2**2*K3 - 32*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 5352*K1*K2*K3 + 1568*K1*K3*K4 + 368*K1*K4*K5 - 168*K2**4 - 224*K2**2*K3**2 - 128*K2**2*K4**2 + 864*K2**2*K4 - 3848*K2**2 + 344*K2*K3*K5 + 128*K2*K4*K6 - 1968*K3**2 - 826*K4**2 - 184*K5**2 - 32*K6**2 + 4008
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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