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Flat knot 6.73

Min(phi) over symmetries of the knot is: [-4,-2,-1,1,3,3,1,0,3,3,4,0,2,2,3,1,1,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.73']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 - 4*K1 + 3*K2 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.73', '6.171', '6.183']
Outer characteristic polynomial of the knot is: t^7+116t^5+40t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.73']
2-strand cable arrow polynomial of the knot is: -720*K1**4 - 64*K1**3*K3 - 128*K1**2*K2**4 + 512*K1**2*K2**3 - 3440*K1**2*K2**2 - 416*K1**2*K2*K4 + 3968*K1**2*K2 - 16*K1**2*K3**2 - 64*K1**2*K4**2 - 2468*K1**2 + 960*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 640*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 352*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 3592*K1*K2*K3 - 64*K1*K2*K4*K5 + 440*K1*K3*K4 + 80*K1*K4*K5 - 64*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1488*K2**4 + 160*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 736*K2**2*K3**2 - 272*K2**2*K4**2 + 1360*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 1380*K2**2 + 360*K2*K3*K5 + 104*K2*K4*K6 - 868*K3**2 - 314*K4**2 - 32*K5**2 - 4*K6**2 + 1936
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.73']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11413', 'vk6.11696', 'vk6.12711', 'vk6.13068', 'vk6.20271', 'vk6.21592', 'vk6.27535', 'vk6.29107', 'vk6.31146', 'vk6.31471', 'vk6.32292', 'vk6.32737', 'vk6.38938', 'vk6.41165', 'vk6.45700', 'vk6.47412', 'vk6.52160', 'vk6.52385', 'vk6.52969', 'vk6.53305', 'vk6.57100', 'vk6.58270', 'vk6.61675', 'vk6.62836', 'vk6.63738', 'vk6.63834', 'vk6.64152', 'vk6.64352', 'vk6.66735', 'vk6.67607', 'vk6.69384', 'vk6.70117', 'vk6.81986', 'vk6.82714', 'vk6.84385', 'vk6.85978', 'vk6.85995', 'vk6.88175', 'vk6.88751', 'vk6.89118']
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 O1O2O3O4O5O6U2U3U5U6U1U4
R3 orbit {'O1O2O3O4O5O6U2U3U5U6U1U4', 'O1O2O3O4O5U1O6U3U5U2U6U4'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5O6U3U6U1U2U4U5
Gauss code of K* O1O2O3O4O5O6U5U1U2U6U3U4
Gauss code of -K* O1O2O3O4O5O6U3U4U1U5U6U2
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 3 1 3],[ 1 0 -3 -1 3 1 3],[ 4 3 0 1 4 2 3],[ 2 1 -1 0 3 1 2],[-3 -3 -4 -3 0 -1 1],[-1 -1 -2 -1 1 0 1],[-3 -3 -3 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 3 1 -1 -2 -4],[-3 0 1 -1 -3 -3 -4],[-3 -1 0 -1 -3 -2 -3],[-1 1 1 0 -1 -1 -2],[ 1 3 3 1 0 -1 -3],[ 2 3 2 1 1 0 -1],[ 4 4 3 2 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,-1,1,2,4,-1,1,3,3,4,1,3,2,3,1,1,2,1,3,1]
Phi over symmetry [-4,-2,-1,1,3,3,1,0,3,3,4,0,2,2,3,1,1,1,1,1,-1]
Phi of -K [-4,-2,-1,1,3,3,1,0,3,3,4,0,2,2,3,1,1,1,1,1,-1]
Phi of K* [-3,-3,-1,1,2,4,-1,1,1,3,4,1,1,2,3,1,2,3,0,0,1]
Phi of -K* [-4,-2,-1,1,3,3,1,3,2,3,4,1,1,2,3,1,3,3,1,1,-1]
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+76t^4+17t^2
Outer characteristic polynomial t^7+116t^5+40t^3+3t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 - 4*K1 + 3*K2 + 4
2-strand cable arrow polynomial -720*K1**4 - 64*K1**3*K3 - 128*K1**2*K2**4 + 512*K1**2*K2**3 - 3440*K1**2*K2**2 - 416*K1**2*K2*K4 + 3968*K1**2*K2 - 16*K1**2*K3**2 - 64*K1**2*K4**2 - 2468*K1**2 + 960*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 640*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 352*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 3592*K1*K2*K3 - 64*K1*K2*K4*K5 + 440*K1*K3*K4 + 80*K1*K4*K5 - 64*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1488*K2**4 + 160*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 736*K2**2*K3**2 - 272*K2**2*K4**2 + 1360*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 1380*K2**2 + 360*K2*K3*K5 + 104*K2*K4*K6 - 868*K3**2 - 314*K4**2 - 32*K5**2 - 4*K6**2 + 1936
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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