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Flat knot 6.204

Min(phi) over symmetries of the knot is: [-4,-2,-1,1,3,3,0,2,2,3,5,1,1,1,2,1,2,3,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.204']
Arrow polynomial of the knot is: 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 + 3*K2 + 2*K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.204']
Outer characteristic polynomial of the knot is: t^7+106t^5+41t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.204']
2-strand cable arrow polynomial of the knot is: -16*K1**4 + 64*K1**3*K2*K3 - 128*K1**3*K3 + 64*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 1888*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 224*K1**2*K2*K4 + 2616*K1**2*K2 - 208*K1**2*K3**2 - 16*K1**2*K4**2 - 2520*K1**2 + 800*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 768*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 128*K1*K2**2*K5 + 32*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3344*K1*K2*K3 + 32*K1*K3**3*K4 + 568*K1*K3*K4 + 72*K1*K4*K5 - 32*K2**4*K4**2 + 128*K2**4*K4 - 1056*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 800*K2**2*K3**2 - 192*K2**2*K4**2 + 1104*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 1576*K2**2 - 64*K2*K3**2*K4 + 408*K2*K3*K5 + 80*K2*K4*K6 + 8*K2*K5*K7 - 32*K3**4 - 32*K3**2*K4**2 + 24*K3**2*K6 - 1084*K3**2 - 346*K4**2 - 60*K5**2 - 8*K6**2 + 1904
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.204']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71573', 'vk6.71687', 'vk6.72106', 'vk6.72312', 'vk6.73750', 'vk6.73890', 'vk6.74591', 'vk6.75692', 'vk6.75709', 'vk6.75901', 'vk6.76071', 'vk6.76783', 'vk6.77189', 'vk6.77491', 'vk6.78685', 'vk6.78693', 'vk6.78891', 'vk6.79017', 'vk6.79593', 'vk6.80307', 'vk6.80315', 'vk6.80439', 'vk6.80556', 'vk6.81006', 'vk6.81097', 'vk6.81144', 'vk6.81208', 'vk6.81320', 'vk6.81703', 'vk6.82205', 'vk6.82462', 'vk6.83977', 'vk6.84435', 'vk6.86317', 'vk6.87110', 'vk6.87778', 'vk6.88026', 'vk6.88114', 'vk6.88335', 'vk6.88392']
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 O1O2O3O4O5U2O6U4U1U6U3U5
R3 orbit {'O1O2O3O4O5U2U3O6U1U4U6U5', 'O1O2O3O4O5U2O6U4U1U6U3U5'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5U1U3U6U5U2O6U4
Gauss code of K* O1O2O3O4O5U2U6U4U1U5O6U3
Gauss code of -K* O1O2O3O4O5U3O6U1U5U2U6U4
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 -3 1 -1 4 2],[ 3 0 -1 3 1 5 2],[ 3 1 0 2 1 3 1],[-1 -3 -2 0 -1 2 1],[ 1 -1 -1 1 0 2 1],[-4 -5 -3 -2 -2 0 0],[-2 -2 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 4 2 1 -1 -3 -3],[-4 0 0 -2 -2 -3 -5],[-2 0 0 -1 -1 -1 -2],[-1 2 1 0 -1 -2 -3],[ 1 2 1 1 0 -1 -1],[ 3 3 1 2 1 0 1],[ 3 5 2 3 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,-1,1,3,3,0,2,2,3,5,1,1,1,2,1,2,3,1,1,-1]
Phi over symmetry [-4,-2,-1,1,3,3,0,2,2,3,5,1,1,1,2,1,2,3,1,1,-1]
Phi of -K [-3,-3,-1,1,2,4,-1,1,2,4,4,1,1,3,2,1,2,3,0,1,2]
Phi of K* [-4,-2,-1,1,3,3,2,1,3,2,4,0,2,3,4,1,1,2,1,1,-1]
Phi of -K* [-3,-3,-1,1,2,4,-1,1,3,2,5,1,2,1,3,1,1,2,1,2,0]
Symmetry type of based matrix c
u-polynomial -t^4+2t^3-t^2
Normalized Jones-Krushkal polynomial 4z^2+17z+19
Enhanced Jones-Krushkal polynomial 4w^3z^2+17w^2z+19w
Inner characteristic polynomial t^6+66t^4+8t^2
Outer characteristic polynomial t^7+106t^5+41t^3+3t
Flat arrow polynomial 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 + 3*K2 + 2*K3 + 4
2-strand cable arrow polynomial -16*K1**4 + 64*K1**3*K2*K3 - 128*K1**3*K3 + 64*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 1888*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 224*K1**2*K2*K4 + 2616*K1**2*K2 - 208*K1**2*K3**2 - 16*K1**2*K4**2 - 2520*K1**2 + 800*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 768*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 128*K1*K2**2*K5 + 32*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3344*K1*K2*K3 + 32*K1*K3**3*K4 + 568*K1*K3*K4 + 72*K1*K4*K5 - 32*K2**4*K4**2 + 128*K2**4*K4 - 1056*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 800*K2**2*K3**2 - 192*K2**2*K4**2 + 1104*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 1576*K2**2 - 64*K2*K3**2*K4 + 408*K2*K3*K5 + 80*K2*K4*K6 + 8*K2*K5*K7 - 32*K3**4 - 32*K3**2*K4**2 + 24*K3**2*K6 - 1084*K3**2 - 346*K4**2 - 60*K5**2 - 8*K6**2 + 1904
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {3}, {1, 2}]]
If K is slice False
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