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Flat knot 6.1138

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,1,0,2,3,4,-1,0,2,2,-1,2,0,2,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1138']
Arrow polynomial of the knot is: -8*K1**4 + 4*K1**3 + 8*K1**2*K2 - 2*K1**2 - 6*K1*K2 - 2*K1*K3 + 2*K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1138']
Outer characteristic polynomial of the knot is: t^7+47t^5+162t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1138']
2-strand cable arrow polynomial of the knot is: -32*K1**4 + 128*K1**3*K2*K3 - 96*K1**3*K3 + 256*K1**2*K2**5 - 2048*K1**2*K2**4 + 3264*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 6528*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 256*K1**2*K2*K4 + 4368*K1**2*K2 - 176*K1**2*K3**2 - 3128*K1**2 + 512*K1*K2**5*K3 - 640*K1*K2**4*K3 - 128*K1*K2**4*K5 + 4288*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 2208*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 416*K1*K2**2*K5 + 64*K1*K2*K3**3 - 256*K1*K2*K3*K4 + 5504*K1*K2*K3 + 472*K1*K3*K4 + 56*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1632*K2**6 - 640*K2**4*K3**2 - 192*K2**4*K4**2 + 1696*K2**4*K4 - 4056*K2**4 + 256*K2**3*K3*K5 + 64*K2**3*K4*K6 - 96*K2**3*K6 + 64*K2**2*K3**2*K4 - 1952*K2**2*K3**2 - 32*K2**2*K3*K7 - 536*K2**2*K4**2 + 2712*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 256*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 680*K2*K3*K5 + 88*K2*K4*K6 + 32*K2*K5*K7 - 16*K3**4 + 8*K3**2*K6 - 1424*K3**2 - 404*K4**2 - 104*K5**2 - 8*K6**2 + 2498
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1138']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70463', 'vk6.70480', 'vk6.70519', 'vk6.70592', 'vk6.70637', 'vk6.70662', 'vk6.70746', 'vk6.70836', 'vk6.70920', 'vk6.70951', 'vk6.71000', 'vk6.71103', 'vk6.71157', 'vk6.71174', 'vk6.71235', 'vk6.71296', 'vk6.71314', 'vk6.71331', 'vk6.73554', 'vk6.74351', 'vk6.74995', 'vk6.75313', 'vk6.76567', 'vk6.76647', 'vk6.76975', 'vk6.78295', 'vk6.79391', 'vk6.79943', 'vk6.81494', 'vk6.86864', 'vk6.88065', 'vk6.89225']
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 O1O2O3O4U1U2U4O5O6U3U5U6
R3 orbit {'O1O2O3O4U1U2U4O5O6U3U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U2O5O6U1U3U4
Gauss code of K* O1O2O3U4U5O6O4O5U1U2U6U3
Gauss code of -K* O1O2O3U1U2O4O5O6U4U3U5U6
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 -3 -1 0 2 0 2],[ 3 0 1 3 2 1 1],[ 1 -1 0 2 1 1 1],[ 0 -3 -2 0 0 1 2],[-2 -2 -1 0 0 0 0],[ 0 -1 -1 -1 0 0 1],[-2 -1 -1 -2 0 -1 0]]
Primitive based matrix [[ 0 2 2 0 0 -1 -3],[-2 0 0 0 0 -1 -2],[-2 0 0 -1 -2 -1 -1],[ 0 0 1 0 -1 -1 -1],[ 0 0 2 1 0 -2 -3],[ 1 1 1 1 2 0 -1],[ 3 2 1 1 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,0,1,3,0,0,0,1,2,1,2,1,1,1,1,1,2,3,1]
Phi over symmetry [-3,-1,0,0,2,2,1,0,2,3,4,-1,0,2,2,-1,2,0,2,1,0]
Phi of -K [-3,-1,0,0,2,2,1,0,2,3,4,-1,0,2,2,-1,2,0,2,1,0]
Phi of K* [-2,-2,0,0,1,3,0,0,1,2,4,2,2,2,3,1,-1,0,0,2,1]
Phi of -K* [-3,-1,0,0,2,2,1,1,3,1,2,1,2,1,1,-1,1,0,2,0,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 5z^2+18z+17
Enhanced Jones-Krushkal polynomial -4w^4z^2+9w^3z^2-4w^3z+22w^2z+17w
Inner characteristic polynomial t^6+29t^4+40t^2
Outer characteristic polynomial t^7+47t^5+162t^3+7t
Flat arrow polynomial -8*K1**4 + 4*K1**3 + 8*K1**2*K2 - 2*K1**2 - 6*K1*K2 - 2*K1*K3 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial -32*K1**4 + 128*K1**3*K2*K3 - 96*K1**3*K3 + 256*K1**2*K2**5 - 2048*K1**2*K2**4 + 3264*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 6528*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 256*K1**2*K2*K4 + 4368*K1**2*K2 - 176*K1**2*K3**2 - 3128*K1**2 + 512*K1*K2**5*K3 - 640*K1*K2**4*K3 - 128*K1*K2**4*K5 + 4288*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 2208*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 416*K1*K2**2*K5 + 64*K1*K2*K3**3 - 256*K1*K2*K3*K4 + 5504*K1*K2*K3 + 472*K1*K3*K4 + 56*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1632*K2**6 - 640*K2**4*K3**2 - 192*K2**4*K4**2 + 1696*K2**4*K4 - 4056*K2**4 + 256*K2**3*K3*K5 + 64*K2**3*K4*K6 - 96*K2**3*K6 + 64*K2**2*K3**2*K4 - 1952*K2**2*K3**2 - 32*K2**2*K3*K7 - 536*K2**2*K4**2 + 2712*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 256*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 680*K2*K3*K5 + 88*K2*K4*K6 + 32*K2*K5*K7 - 16*K3**4 + 8*K3**2*K6 - 1424*K3**2 - 404*K4**2 - 104*K5**2 - 8*K6**2 + 2498
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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