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Flat knot 6.1098

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,1,0,2,2,3,-1,0,1,1,0,0,0,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1098']
Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 6*K1*K2 - 3*K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.238', '6.431', '6.945', '6.977', '6.981', '6.997', '6.1050', '6.1070', '6.1098', '6.1376']
Outer characteristic polynomial of the knot is: t^7+65t^5+55t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1098']
2-strand cable arrow polynomial of the knot is: 128*K1**4*K2**3 - 192*K1**4*K2**2 + 96*K1**4*K2 - 1440*K1**4 - 128*K1**3*K2**2*K3 + 256*K1**3*K2*K3 - 480*K1**3*K3 + 512*K1**2*K2**5 - 1792*K1**2*K2**4 + 3648*K1**2*K2**3 - 7584*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 480*K1**2*K2*K4 + 7808*K1**2*K2 - 256*K1**2*K3**2 - 4948*K1**2 - 512*K1*K2**4*K3 + 1824*K1*K2**3*K3 - 800*K1*K2**2*K3 - 96*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 5816*K1*K2*K3 + 600*K1*K3*K4 + 48*K1*K4*K5 - 704*K2**6 + 640*K2**4*K4 - 2464*K2**4 - 32*K2**3*K6 - 688*K2**2*K3**2 - 88*K2**2*K4**2 + 1272*K2**2*K4 - 1830*K2**2 + 280*K2*K3*K5 + 24*K2*K4*K6 - 1424*K3**2 - 336*K4**2 - 36*K5**2 - 2*K6**2 + 3518
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1098']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73702', 'vk6.73821', 'vk6.74193', 'vk6.74804', 'vk6.75623', 'vk6.75813', 'vk6.76354', 'vk6.76870', 'vk6.78602', 'vk6.78802', 'vk6.79226', 'vk6.79698', 'vk6.80244', 'vk6.80386', 'vk6.80703', 'vk6.81068', 'vk6.81613', 'vk6.81795', 'vk6.81922', 'vk6.82170', 'vk6.82302', 'vk6.82643', 'vk6.83189', 'vk6.84063', 'vk6.84221', 'vk6.84686', 'vk6.85003', 'vk6.86024', 'vk6.87744', 'vk6.88217', 'vk6.89399', 'vk6.89606']
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 O1O2O3O4U1U5O6U3U4O5U2U6
R3 orbit {'O1O2O3O4U1U5O6U3U4O5U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U3O6U1U2O5U6U4
Gauss code of K* O1O2U3O4O5U2O6O3U1U6U4U5
Gauss code of -K* O1O2U3O4O5U1O3O6U4U5U2U6
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 0 0 2 -1 2],[ 3 0 3 1 2 1 3],[ 0 -3 0 0 2 -2 2],[ 0 -1 0 0 1 -1 1],[-2 -2 -2 -1 0 -2 0],[ 1 -1 2 1 2 0 2],[-2 -3 -2 -1 0 -2 0]]
Primitive based matrix [[ 0 2 2 0 0 -1 -3],[-2 0 0 -1 -2 -2 -2],[-2 0 0 -1 -2 -2 -3],[ 0 1 1 0 0 -1 -1],[ 0 2 2 0 0 -2 -3],[ 1 2 2 1 2 0 -1],[ 3 2 3 1 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,0,1,3,0,1,2,2,2,1,2,2,3,0,1,1,2,3,1]
Phi over symmetry [-3,-1,0,0,2,2,1,0,2,2,3,-1,0,1,1,0,0,0,1,1,0]
Phi of -K [-3,-1,0,0,2,2,1,0,2,2,3,-1,0,1,1,0,0,0,1,1,0]
Phi of K* [-2,-2,0,0,1,3,0,0,1,1,2,0,1,1,3,0,-1,0,0,2,1]
Phi of -K* [-3,-1,0,0,2,2,1,1,3,2,3,1,2,2,2,0,1,1,2,2,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 2z^2+15z+23
Enhanced Jones-Krushkal polynomial -2w^4z^2+4w^3z^2-6w^3z+21w^2z+23w
Inner characteristic polynomial t^6+47t^4+21t^2
Outer characteristic polynomial t^7+65t^5+55t^3+7t
Flat arrow polynomial 8*K1**3 - 8*K1**2 - 6*K1*K2 - 3*K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial 128*K1**4*K2**3 - 192*K1**4*K2**2 + 96*K1**4*K2 - 1440*K1**4 - 128*K1**3*K2**2*K3 + 256*K1**3*K2*K3 - 480*K1**3*K3 + 512*K1**2*K2**5 - 1792*K1**2*K2**4 + 3648*K1**2*K2**3 - 7584*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 480*K1**2*K2*K4 + 7808*K1**2*K2 - 256*K1**2*K3**2 - 4948*K1**2 - 512*K1*K2**4*K3 + 1824*K1*K2**3*K3 - 800*K1*K2**2*K3 - 96*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 5816*K1*K2*K3 + 600*K1*K3*K4 + 48*K1*K4*K5 - 704*K2**6 + 640*K2**4*K4 - 2464*K2**4 - 32*K2**3*K6 - 688*K2**2*K3**2 - 88*K2**2*K4**2 + 1272*K2**2*K4 - 1830*K2**2 + 280*K2*K3*K5 + 24*K2*K4*K6 - 1424*K3**2 - 336*K4**2 - 36*K5**2 - 2*K6**2 + 3518
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{6}, {1, 5}, {2, 4}, {3}]]
If K is slice False
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