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Flat knot 6.115

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,-1,1,1,2,3,1,1,1,1,1,1,2,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.115', '7.10416']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.115', '6.407', '6.413', '6.448', '6.844', '6.879', '6.888', '6.926', '6.934', '6.1140', '6.1143', '6.1161', '6.1177']
Outer characteristic polynomial of the knot is: t^7+76t^5+106t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.115']
2-strand cable arrow polynomial of the knot is: -576*K1**4 + 1536*K1**3*K2*K3 - 512*K1**3*K3 - 512*K1**2*K2**4 + 1344*K1**2*K2**3 - 1024*K1**2*K2**2*K3**2 - 5648*K1**2*K2**2 + 384*K1**2*K2*K3**2 + 128*K1**2*K2*K3*K5 - 736*K1**2*K2*K4 + 4320*K1**2*K2 - 1536*K1**2*K3**2 - 64*K1**2*K3*K5 - 2476*K1**2 + 256*K1*K2**5*K3 - 512*K1*K2**4*K3 + 4448*K1*K2**3*K3 + 864*K1*K2**2*K3*K4 - 2432*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 736*K1*K2**2*K5 + 128*K1*K2*K3**3 - 384*K1*K2*K3*K4 - 160*K1*K2*K3*K6 + 6304*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K5*K6 + 1376*K1*K3*K4 + 64*K1*K4*K5 + 16*K1*K5*K6 + 8*K1*K6*K7 - 128*K2**6 - 1280*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 2368*K2**4 + 704*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 + 128*K2**2*K3**2*K4 - 3072*K2**2*K3**2 - 64*K2**2*K3*K7 - 200*K2**2*K4**2 + 1696*K2**2*K4 - 80*K2**2*K5**2 - 48*K2**2*K6**2 - 918*K2**2 + 1328*K2*K3*K5 + 120*K2*K4*K6 + 16*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1408*K3**2 - 288*K4**2 - 112*K5**2 - 26*K6**2 - 4*K7**2 - 2*K8**2 + 1984
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.115']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.472', 'vk6.534', 'vk6.570', 'vk6.931', 'vk6.982', 'vk6.1028', 'vk6.1070', 'vk6.1715', 'vk6.1790', 'vk6.2111', 'vk6.2215', 'vk6.2250', 'vk6.2541', 'vk6.2822', 'vk6.2855', 'vk6.3160', 'vk6.20315', 'vk6.20644', 'vk6.21656', 'vk6.22075', 'vk6.27615', 'vk6.28131', 'vk6.29161', 'vk6.39038', 'vk6.39564', 'vk6.41297', 'vk6.41794', 'vk6.46179', 'vk6.57177', 'vk6.57548', 'vk6.58382', 'vk6.66789']
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 O1O2O3O4O5O6U3U4U6U5U1U2
R3 orbit {'O1O2O3O4O5O6U3U4U6U5U1U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U6U2U1U3U4
Gauss code of K* O1O2O3O4O5O6U5U6U1U2U4U3
Gauss code of -K* O1O2O3O4O5O6U4U3U5U6U1U2
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 -3 -1 2 2],[ 1 0 1 -3 -1 2 2],[-1 -1 0 -3 -1 2 2],[ 3 3 3 0 1 3 2],[ 1 1 1 -1 0 2 1],[-2 -2 -2 -3 -2 0 0],[-2 -2 -2 -2 -1 0 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 0 -2 -1 -2 -2],[-2 0 0 -2 -2 -2 -3],[-1 2 2 0 -1 -1 -3],[ 1 1 2 1 0 1 -1],[ 1 2 2 1 -1 0 -3],[ 3 2 3 3 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,0,2,1,2,2,2,2,2,3,1,1,3,-1,1,3]
Phi over symmetry [-3,-1,-1,1,2,2,-1,1,1,2,3,1,1,1,1,1,1,2,-1,-1,0]
Phi of -K [-3,-1,-1,1,2,2,-1,1,1,2,3,1,1,1,1,1,1,2,-1,-1,0]
Phi of K* [-2,-2,-1,1,1,3,0,-1,1,1,2,-1,1,2,3,1,1,1,-1,-1,1]
Phi of -K* [-3,-1,-1,1,2,2,1,3,3,2,3,1,1,1,2,1,2,2,2,2,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial -4w^4z^2+11w^3z^2+24w^2z+21w
Inner characteristic polynomial t^6+56t^4+40t^2+1
Outer characteristic polynomial t^7+76t^5+106t^3+13t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial -576*K1**4 + 1536*K1**3*K2*K3 - 512*K1**3*K3 - 512*K1**2*K2**4 + 1344*K1**2*K2**3 - 1024*K1**2*K2**2*K3**2 - 5648*K1**2*K2**2 + 384*K1**2*K2*K3**2 + 128*K1**2*K2*K3*K5 - 736*K1**2*K2*K4 + 4320*K1**2*K2 - 1536*K1**2*K3**2 - 64*K1**2*K3*K5 - 2476*K1**2 + 256*K1*K2**5*K3 - 512*K1*K2**4*K3 + 4448*K1*K2**3*K3 + 864*K1*K2**2*K3*K4 - 2432*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 736*K1*K2**2*K5 + 128*K1*K2*K3**3 - 384*K1*K2*K3*K4 - 160*K1*K2*K3*K6 + 6304*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K5*K6 + 1376*K1*K3*K4 + 64*K1*K4*K5 + 16*K1*K5*K6 + 8*K1*K6*K7 - 128*K2**6 - 1280*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 2368*K2**4 + 704*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 + 128*K2**2*K3**2*K4 - 3072*K2**2*K3**2 - 64*K2**2*K3*K7 - 200*K2**2*K4**2 + 1696*K2**2*K4 - 80*K2**2*K5**2 - 48*K2**2*K6**2 - 918*K2**2 + 1328*K2*K3*K5 + 120*K2*K4*K6 + 16*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1408*K3**2 - 288*K4**2 - 112*K5**2 - 26*K6**2 - 4*K7**2 - 2*K8**2 + 1984
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {3}, {1, 2}]]
If K is slice False
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