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Flat knot 6.476

Min(phi) over symmetries of the knot is: [-4,-2,1,1,2,2,1,1,3,3,4,0,2,2,2,1,0,-1,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.476']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 3*K2 + K3 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.156', '6.476', '6.483']
Outer characteristic polynomial of the knot is: t^7+91t^5+135t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.476']
2-strand cable arrow polynomial of the knot is: -528*K1**4 + 224*K1**3*K2*K3 - 192*K1**3*K3 + 1152*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 6064*K1**2*K2**2 + 32*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 320*K1**2*K2*K4 + 6528*K1**2*K2 - 448*K1**2*K3**2 - 5288*K1**2 + 1344*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 960*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 448*K1*K2**2*K5 + 128*K1*K2*K3**3 - 160*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6872*K1*K2*K3 - 32*K1*K2*K4*K5 + 832*K1*K3*K4 + 176*K1*K4*K5 + 8*K1*K5*K6 - 288*K2**6 + 448*K2**4*K4 - 2592*K2**4 + 64*K2**3*K3*K5 - 32*K2**3*K6 - 1520*K2**2*K3**2 - 32*K2**2*K3*K7 - 240*K2**2*K4**2 + 1856*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 2666*K2**2 - 96*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 984*K2*K3*K5 + 64*K2*K4*K6 + 56*K2*K5*K7 + 8*K2*K6*K8 - 48*K3**4 + 32*K3**2*K6 - 2012*K3**2 + 16*K3*K4*K7 - 562*K4**2 - 228*K5**2 - 14*K6**2 - 8*K7**2 - 2*K8**2 + 4026
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.476']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73708', 'vk6.73827', 'vk6.74188', 'vk6.74796', 'vk6.75637', 'vk6.75823', 'vk6.76347', 'vk6.76867', 'vk6.78620', 'vk6.78815', 'vk6.79221', 'vk6.79692', 'vk6.80254', 'vk6.80392', 'vk6.80696', 'vk6.81066', 'vk6.81609', 'vk6.81791', 'vk6.81925', 'vk6.82166', 'vk6.82299', 'vk6.82647', 'vk6.83196', 'vk6.84051', 'vk6.84209', 'vk6.84689', 'vk6.85007', 'vk6.86004', 'vk6.87757', 'vk6.88197', 'vk6.89404', 'vk6.89608']
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 O1O2O3O4O5U1U2U4O6U5U3U6
R3 orbit {'O1O2O3O4O5U1U2U4O6U5U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U3U1O6U2U4U5
Gauss code of K* O1O2O3U4O5O6O4U1U2U6U3U5
Gauss code of -K* O1O2O3U1O4O5O6U3U4U2U5U6
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 -4 -2 1 1 2 2],[ 4 0 1 4 2 3 2],[ 2 -1 0 3 1 2 2],[-1 -4 -3 0 -1 1 2],[-1 -2 -1 1 0 1 1],[-2 -3 -2 -1 -1 0 1],[-2 -2 -2 -2 -1 -1 0]]
Primitive based matrix [[ 0 2 2 1 1 -2 -4],[-2 0 1 -1 -1 -2 -3],[-2 -1 0 -1 -2 -2 -2],[-1 1 1 0 1 -1 -2],[-1 1 2 -1 0 -3 -4],[ 2 2 2 1 3 0 -1],[ 4 3 2 2 4 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,-1,2,4,-1,1,1,2,3,1,2,2,2,-1,1,2,3,4,1]
Phi over symmetry [-4,-2,1,1,2,2,1,1,3,3,4,0,2,2,2,1,0,-1,0,0,-1]
Phi of -K [-4,-2,1,1,2,2,1,1,3,3,4,0,2,2,2,1,0,-1,0,0,-1]
Phi of K* [-2,-2,-1,-1,2,4,-1,-1,0,2,4,0,0,2,3,-1,0,1,2,3,1]
Phi of -K* [-4,-2,1,1,2,2,1,2,4,2,3,1,3,2,2,1,1,1,2,1,-1]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial -2w^4z^2+6w^3z^2-4w^3z+25w^2z+27w
Inner characteristic polynomial t^6+61t^4+16t^2+1
Outer characteristic polynomial t^7+91t^5+135t^3+13t
Flat arrow polynomial 4*K1**3 - 4*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 3*K2 + K3 + K4 + 3
2-strand cable arrow polynomial -528*K1**4 + 224*K1**3*K2*K3 - 192*K1**3*K3 + 1152*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 6064*K1**2*K2**2 + 32*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 320*K1**2*K2*K4 + 6528*K1**2*K2 - 448*K1**2*K3**2 - 5288*K1**2 + 1344*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 960*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 448*K1*K2**2*K5 + 128*K1*K2*K3**3 - 160*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6872*K1*K2*K3 - 32*K1*K2*K4*K5 + 832*K1*K3*K4 + 176*K1*K4*K5 + 8*K1*K5*K6 - 288*K2**6 + 448*K2**4*K4 - 2592*K2**4 + 64*K2**3*K3*K5 - 32*K2**3*K6 - 1520*K2**2*K3**2 - 32*K2**2*K3*K7 - 240*K2**2*K4**2 + 1856*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 2666*K2**2 - 96*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 984*K2*K3*K5 + 64*K2*K4*K6 + 56*K2*K5*K7 + 8*K2*K6*K8 - 48*K3**4 + 32*K3**2*K6 - 2012*K3**2 + 16*K3*K4*K7 - 562*K4**2 - 228*K5**2 - 14*K6**2 - 8*K7**2 - 2*K8**2 + 4026
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}]]
If K is slice False
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