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Flat knot 6.104

Min(phi) over symmetries of the knot is: [-4,0,2,2,2,2,3,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.104']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 8*K1*K2 - 2*K1*K3 + K1 + 4*K2 + 3*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.104']
Outer characteristic polynomial of the knot is: t^5+56t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.104']
2-strand cable arrow polynomial of the knot is: -512*K1**6 + 256*K1**4*K2**3 - 1728*K1**4*K2**2 + 2624*K1**4*K2 - 4000*K1**4 + 864*K1**3*K2*K3 - 128*K1**3*K3 - 384*K1**2*K2**4 + 1440*K1**2*K2**3 - 7472*K1**2*K2**2 - 576*K1**2*K2*K4 + 8568*K1**2*K2 - 1056*K1**2*K3**2 - 96*K1**2*K3*K5 - 352*K1**2*K4**2 - 32*K1**2*K4*K6 - 4084*K1**2 + 896*K1*K2**3*K3 + 896*K1*K2**2*K3*K4 - 1376*K1*K2**2*K3 + 288*K1*K2**2*K4*K5 - 192*K1*K2**2*K5 - 1088*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 7720*K1*K2*K3 - 288*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2512*K1*K3*K4 + 880*K1*K4*K5 + 56*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 224*K2**4*K4 - 1784*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 1840*K2**2*K3**2 - 1120*K2**2*K4**2 + 2528*K2**2*K4 - 240*K2**2*K5**2 - 8*K2**2*K6**2 - 3646*K2**2 - 224*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1632*K2*K3*K5 + 592*K2*K4*K6 + 72*K2*K5*K7 - 2424*K3**2 - 1448*K4**2 - 484*K5**2 - 50*K6**2 + 4742
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.104']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4694', 'vk6.4997', 'vk6.6180', 'vk6.6651', 'vk6.8173', 'vk6.8591', 'vk6.9559', 'vk6.9898', 'vk6.17388', 'vk6.20924', 'vk6.20988', 'vk6.22334', 'vk6.22410', 'vk6.23555', 'vk6.23892', 'vk6.28400', 'vk6.36148', 'vk6.40062', 'vk6.40189', 'vk6.42113', 'vk6.43059', 'vk6.43363', 'vk6.46590', 'vk6.46694', 'vk6.48726', 'vk6.49514', 'vk6.49717', 'vk6.51422', 'vk6.55538', 'vk6.58922', 'vk6.65284', 'vk6.69768']
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 O1O2O3O4O5O6U2U6U4U5U1U3
R3 orbit {'O1O2O3O4O5O6U2U6U4U5U1U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U4U6U2U3U1U5
Gauss code of K* O1O2O3O4O5O6U5U1U6U3U4U2
Gauss code of -K* O1O2O3O4O5O6U5U3U4U1U6U2
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 -4 2 0 2 1],[ 1 0 -3 2 0 2 1],[ 4 3 0 4 2 3 1],[-2 -2 -4 0 -1 1 0],[ 0 0 -2 1 0 1 0],[-2 -2 -3 -1 -1 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 2 2 0 -4],[-2 0 1 -1 -4],[-2 -1 0 -1 -3],[ 0 1 1 0 -2],[ 4 4 3 2 0]]
If based matrix primitive False
Phi of primitive based matrix [-2,-2,0,4,-1,1,4,1,3,2]
Phi over symmetry [-4,0,2,2,2,2,3,1,1,-1]
Phi of -K [-4,0,2,2,2,2,3,1,1,-1]
Phi of K* [-2,-2,0,4,-1,1,3,1,2,2]
Phi of -K* [-4,0,2,2,2,3,4,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^4-2t^2
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^4+32t^2+1
Outer characteristic polynomial t^5+56t^3+5t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 8*K1*K2 - 2*K1*K3 + K1 + 4*K2 + 3*K3 + 5
2-strand cable arrow polynomial -512*K1**6 + 256*K1**4*K2**3 - 1728*K1**4*K2**2 + 2624*K1**4*K2 - 4000*K1**4 + 864*K1**3*K2*K3 - 128*K1**3*K3 - 384*K1**2*K2**4 + 1440*K1**2*K2**3 - 7472*K1**2*K2**2 - 576*K1**2*K2*K4 + 8568*K1**2*K2 - 1056*K1**2*K3**2 - 96*K1**2*K3*K5 - 352*K1**2*K4**2 - 32*K1**2*K4*K6 - 4084*K1**2 + 896*K1*K2**3*K3 + 896*K1*K2**2*K3*K4 - 1376*K1*K2**2*K3 + 288*K1*K2**2*K4*K5 - 192*K1*K2**2*K5 - 1088*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 7720*K1*K2*K3 - 288*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2512*K1*K3*K4 + 880*K1*K4*K5 + 56*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 224*K2**4*K4 - 1784*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 1840*K2**2*K3**2 - 1120*K2**2*K4**2 + 2528*K2**2*K4 - 240*K2**2*K5**2 - 8*K2**2*K6**2 - 3646*K2**2 - 224*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1632*K2*K3*K5 + 592*K2*K4*K6 + 72*K2*K5*K7 - 2424*K3**2 - 1448*K4**2 - 484*K5**2 - 50*K6**2 + 4742
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {5}, {4}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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