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Flat knot 6.175

Min(phi) over symmetries of the knot is: [-4,-1,0,1,2,2,1,3,1,3,4,1,0,1,1,0,1,1,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.175']
Arrow polynomial of the knot is: -2*K1**2 - 2*K1*K3 + 2*K2 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.175', '6.485']
Outer characteristic polynomial of the knot is: t^7+81t^5+44t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.175']
2-strand cable arrow polynomial of the knot is: -256*K1**4 - 160*K1**2*K2**2 + 472*K1**2*K2 - 176*K1**2*K3**2 - 528*K1**2 + 32*K1*K2*K3**3 + 808*K1*K2*K3 + 240*K1*K3*K4 + 16*K1*K4*K5 + 16*K1*K5*K6 - 24*K2**4 - 128*K2**2*K3**2 + 48*K2**2*K4 - 8*K2**2*K6**2 - 480*K2**2 + 88*K2*K3*K5 + 24*K2*K4*K6 + 8*K2*K6*K8 - 48*K3**4 + 40*K3**2*K6 - 384*K3**2 - 116*K4**2 - 32*K5**2 - 32*K6**2 - 2*K8**2 + 580
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.175']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16522', 'vk6.16613', 'vk6.18091', 'vk6.18427', 'vk6.22949', 'vk6.23044', 'vk6.23507', 'vk6.23844', 'vk6.24538', 'vk6.24955', 'vk6.35029', 'vk6.35652', 'vk6.36673', 'vk6.37095', 'vk6.39450', 'vk6.41649', 'vk6.42487', 'vk6.42598', 'vk6.43949', 'vk6.44264', 'vk6.46034', 'vk6.47700', 'vk6.54749', 'vk6.54844', 'vk6.56191', 'vk6.57448', 'vk6.59209', 'vk6.59272', 'vk6.59657', 'vk6.60003', 'vk6.60786', 'vk6.62119', 'vk6.64818', 'vk6.65053', 'vk6.65541', 'vk6.65851', 'vk6.68053', 'vk6.68116', 'vk6.68619', 'vk6.68832', 'vk6.73728', 'vk6.73847', 'vk6.78303', 'vk6.78487', 'vk6.78652', 'vk6.78847', 'vk6.85140', 'vk6.89434']
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 O1O2O3O4O5U1O6U5U3U6U4U2
R3 orbit {'O1O2O3O4O5U1O6U5U3U6U4U2', 'O1O2O3O4O5U1U4O6U3U5U6U2', 'O1O2O3O4O5U1U4U2O6U5U3U6'}
R3 orbit length 3
Gauss code of -K O1O2O3O4O5U4U2U6U3U1O6U5
Gauss code of K* O1O2O3O4O5U6U5U2U4U1O6U3
Gauss code of -K* O1O2O3O4O5U3O6U5U2U4U1U6
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -4 1 -1 2 0 2],[ 4 0 4 2 3 1 2],[-1 -4 0 -2 1 -1 2],[ 1 -2 2 0 2 0 2],[-2 -3 -1 -2 0 -1 1],[ 0 -1 1 0 1 0 1],[-2 -2 -2 -2 -1 -1 0]]
Primitive based matrix [[ 0 2 2 1 0 -1 -4],[-2 0 1 -1 -1 -2 -3],[-2 -1 0 -2 -1 -2 -2],[-1 1 2 0 -1 -2 -4],[ 0 1 1 1 0 0 -1],[ 1 2 2 2 0 0 -2],[ 4 3 2 4 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,0,1,4,-1,1,1,2,3,2,1,2,2,1,2,4,0,1,2]
Phi over symmetry [-4,-1,0,1,2,2,1,3,1,3,4,1,0,1,1,0,1,1,0,-1,-1]
Phi of -K [-4,-1,0,1,2,2,1,3,1,3,4,1,0,1,1,0,1,1,0,-1,-1]
Phi of K* [-2,-2,-1,0,1,4,-1,-1,1,1,4,0,1,1,3,0,0,1,1,3,1]
Phi of -K* [-4,-1,0,1,2,2,2,1,4,2,3,0,2,2,2,1,1,1,2,1,-1]
Symmetry type of based matrix c
u-polynomial t^4-2t^2
Normalized Jones-Krushkal polynomial 8z+17
Enhanced Jones-Krushkal polynomial 8w^2z+17w
Inner characteristic polynomial t^6+55t^4
Outer characteristic polynomial t^7+81t^5+44t^3
Flat arrow polynomial -2*K1**2 - 2*K1*K3 + 2*K2 + K4 + 2
2-strand cable arrow polynomial -256*K1**4 - 160*K1**2*K2**2 + 472*K1**2*K2 - 176*K1**2*K3**2 - 528*K1**2 + 32*K1*K2*K3**3 + 808*K1*K2*K3 + 240*K1*K3*K4 + 16*K1*K4*K5 + 16*K1*K5*K6 - 24*K2**4 - 128*K2**2*K3**2 + 48*K2**2*K4 - 8*K2**2*K6**2 - 480*K2**2 + 88*K2*K3*K5 + 24*K2*K4*K6 + 8*K2*K6*K8 - 48*K3**4 + 40*K3**2*K6 - 384*K3**2 - 116*K4**2 - 32*K5**2 - 32*K6**2 - 2*K8**2 + 580
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {4}, {2, 3}, {1}]]
If K is slice False
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