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Flat knot 6.478

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,1,1,4,4,3,0,2,3,2,0,2,1,1,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.478']
Arrow polynomial of the knot is: 8*K1**3 - 2*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.245', '6.255', '6.478']
Outer characteristic polynomial of the knot is: t^7+85t^5+142t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.478']
2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 352*K1**4*K2 - 528*K1**4 + 608*K1**3*K2*K3 - 672*K1**3*K3 - 512*K1**2*K2**4 + 2688*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7200*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 704*K1**2*K2*K4 + 6816*K1**2*K2 - 624*K1**2*K3**2 - 4968*K1**2 - 512*K1*K2**4*K3 - 128*K1*K2**3*K3*K4 + 2496*K1*K2**3*K3 + 608*K1*K2**2*K3*K4 - 2176*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 320*K1*K2**2*K5 + 192*K1*K2*K3**3 - 704*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 7824*K1*K2*K3 - 64*K1*K3**2*K5 + 1080*K1*K3*K4 + 120*K1*K4*K5 + 8*K1*K5*K6 - 192*K2**6 + 672*K2**4*K4 - 3384*K2**4 + 160*K2**3*K3*K5 - 32*K2**3*K6 - 2368*K2**2*K3**2 - 32*K2**2*K3*K7 - 504*K2**2*K4**2 + 2648*K2**2*K4 - 144*K2**2*K5**2 - 8*K2**2*K6**2 - 2474*K2**2 + 1432*K2*K3*K5 + 64*K2*K4*K6 + 40*K2*K5*K7 + 8*K2*K6*K8 - 96*K3**4 + 72*K3**2*K6 - 2180*K3**2 + 8*K3*K4*K7 - 656*K4**2 - 248*K5**2 - 22*K6**2 - 4*K7**2 - 2*K8**2 + 3880
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.478']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16339', 'vk6.16382', 'vk6.18061', 'vk6.18399', 'vk6.22678', 'vk6.22757', 'vk6.24504', 'vk6.24927', 'vk6.34622', 'vk6.34705', 'vk6.36641', 'vk6.37065', 'vk6.42313', 'vk6.42344', 'vk6.43927', 'vk6.44246', 'vk6.54602', 'vk6.54641', 'vk6.55889', 'vk6.56177', 'vk6.59088', 'vk6.59127', 'vk6.60413', 'vk6.60772', 'vk6.64635', 'vk6.64681', 'vk6.65523', 'vk6.65839', 'vk6.67992', 'vk6.68018', 'vk6.68609', 'vk6.68826']
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 O1O2O3O4O5U1U2U5O6U3U6U4
R3 orbit {'O1O2O3O4O5U1U2U5O6U3U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U6U3O6U1U4U5
Gauss code of K* O1O2O3U4O5O4O6U1U2U5U6U3
Gauss code of -K* O1O2O3U2O4O5O6U4U1U3U5U6
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 0 3 2 1],[ 4 0 1 3 4 2 1],[ 2 -1 0 2 3 1 1],[ 0 -3 -2 0 2 0 1],[-3 -4 -3 -2 0 0 0],[-2 -2 -1 0 0 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 3 2 1 0 -2 -4],[-3 0 0 0 -2 -3 -4],[-2 0 0 0 0 -1 -2],[-1 0 0 0 -1 -1 -1],[ 0 2 0 1 0 -2 -3],[ 2 3 1 1 2 0 -1],[ 4 4 2 1 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,0,2,4,0,0,2,3,4,0,0,1,2,1,1,1,2,3,1]
Phi over symmetry [-4,-2,0,1,2,3,1,1,4,4,3,0,2,3,2,0,2,1,1,2,1]
Phi of -K [-4,-2,0,1,2,3,1,1,4,4,3,0,2,3,2,0,2,1,1,2,1]
Phi of K* [-3,-2,-1,0,2,4,1,2,1,2,3,1,2,3,4,0,2,4,0,1,1]
Phi of -K* [-4,-2,0,1,2,3,1,3,1,2,4,2,1,1,3,1,0,2,0,0,0]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t
Normalized Jones-Krushkal polynomial 7z^2+26z+25
Enhanced Jones-Krushkal polynomial 7w^3z^2-2w^3z+28w^2z+25w
Inner characteristic polynomial t^6+51t^4+38t^2
Outer characteristic polynomial t^7+85t^5+142t^3+9t
Flat arrow polynomial 8*K1**3 - 2*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial -384*K1**4*K2**2 + 352*K1**4*K2 - 528*K1**4 + 608*K1**3*K2*K3 - 672*K1**3*K3 - 512*K1**2*K2**4 + 2688*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7200*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 704*K1**2*K2*K4 + 6816*K1**2*K2 - 624*K1**2*K3**2 - 4968*K1**2 - 512*K1*K2**4*K3 - 128*K1*K2**3*K3*K4 + 2496*K1*K2**3*K3 + 608*K1*K2**2*K3*K4 - 2176*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 320*K1*K2**2*K5 + 192*K1*K2*K3**3 - 704*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 7824*K1*K2*K3 - 64*K1*K3**2*K5 + 1080*K1*K3*K4 + 120*K1*K4*K5 + 8*K1*K5*K6 - 192*K2**6 + 672*K2**4*K4 - 3384*K2**4 + 160*K2**3*K3*K5 - 32*K2**3*K6 - 2368*K2**2*K3**2 - 32*K2**2*K3*K7 - 504*K2**2*K4**2 + 2648*K2**2*K4 - 144*K2**2*K5**2 - 8*K2**2*K6**2 - 2474*K2**2 + 1432*K2*K3*K5 + 64*K2*K4*K6 + 40*K2*K5*K7 + 8*K2*K6*K8 - 96*K3**4 + 72*K3**2*K6 - 2180*K3**2 + 8*K3*K4*K7 - 656*K4**2 - 248*K5**2 - 22*K6**2 - 4*K7**2 - 2*K8**2 + 3880
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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