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Flat knot 6.421

Min(phi) over symmetries of the knot is: [-4,-2,1,1,1,3,1,1,2,3,4,1,1,2,3,-1,-1,-1,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.421']
Arrow polynomial of the knot is: -2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.58', '6.76', '6.78', '6.90', '6.98', '6.154', '6.161', '6.162', '6.198', '6.280', '6.284', '6.345', '6.417', '6.421', '6.435', '6.511']
Outer characteristic polynomial of the knot is: t^7+96t^5+89t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.421']
2-strand cable arrow polynomial of the knot is: 768*K1**4*K2 - 1488*K1**4 - 768*K1**3*K2**2*K3 + 1664*K1**3*K2*K3 - 1088*K1**3*K3 + 1152*K1**2*K2**3 - 768*K1**2*K2**2*K3**2 - 4768*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 928*K1**2*K2*K4 + 5400*K1**2*K2 - 1264*K1**2*K3**2 - 64*K1**2*K4**2 - 3832*K1**2 + 1888*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 1440*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 32*K1*K2**2*K5 + 384*K1*K2*K3**3 - 384*K1*K2*K3*K4 - 160*K1*K2*K3*K6 + 6080*K1*K2*K3 - 32*K1*K2*K4*K5 - 64*K1*K2*K4*K7 + 1608*K1*K3*K4 + 160*K1*K4*K5 + 8*K1*K5*K6 - 736*K2**4 - 1344*K2**2*K3**2 - 136*K2**2*K4**2 + 888*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 2642*K2**2 - 96*K2*K3**2*K4 + 664*K2*K3*K5 + 104*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**4 + 56*K3**2*K6 - 1948*K3**2 + 8*K3*K4*K7 - 562*K4**2 - 152*K5**2 - 22*K6**2 - 4*K7**2 - 2*K8**2 + 3106
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.421']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19913', 'vk6.19959', 'vk6.21136', 'vk6.21216', 'vk6.26832', 'vk6.26958', 'vk6.28610', 'vk6.28700', 'vk6.38266', 'vk6.38368', 'vk6.40394', 'vk6.40527', 'vk6.45139', 'vk6.45235', 'vk6.46991', 'vk6.47048', 'vk6.56696', 'vk6.56757', 'vk6.57782', 'vk6.57872', 'vk6.61100', 'vk6.61226', 'vk6.62354', 'vk6.62444', 'vk6.66384', 'vk6.66467', 'vk6.67146', 'vk6.67252', 'vk6.69045', 'vk6.69116', 'vk6.69835', 'vk6.69888']
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 O1O2O3O4O5U6U1O6U3U2U4U5
R3 orbit {'O1O2O3O4O5U6U1O6U3U2U4U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U2U4U3O6U5U6
Gauss code of K* O1O2O3O4U5U2U1U3U4O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U1U2U4U3U6
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 -1 -1 2 4 -1],[ 3 0 1 0 2 3 3],[ 1 -1 0 0 2 3 1],[ 1 0 0 0 1 2 1],[-2 -2 -2 -1 0 1 -2],[-4 -3 -3 -2 -1 0 -4],[ 1 -3 -1 -1 2 4 0]]
Primitive based matrix [[ 0 4 2 -1 -1 -1 -3],[-4 0 -1 -2 -3 -4 -3],[-2 1 0 -1 -2 -2 -2],[ 1 2 1 0 0 1 0],[ 1 3 2 0 0 1 -1],[ 1 4 2 -1 -1 0 -3],[ 3 3 2 0 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,1,1,1,3,1,2,3,4,3,1,2,2,2,0,-1,0,-1,1,3]
Phi over symmetry [-4,-2,1,1,1,3,1,1,2,3,4,1,1,2,3,-1,-1,-1,0,1,2]
Phi of -K [-3,-1,-1,-1,2,4,-1,1,2,3,4,1,1,1,1,0,1,2,2,3,1]
Phi of K* [-4,-2,1,1,1,3,1,1,2,3,4,1,1,2,3,-1,-1,-1,0,1,2]
Phi of -K* [-3,-1,-1,-1,2,4,0,1,3,2,3,0,1,1,2,1,2,3,2,4,1]
Symmetry type of based matrix c
u-polynomial -t^4+t^3-t^2+3t
Normalized Jones-Krushkal polynomial 7z^2+26z+25
Enhanced Jones-Krushkal polynomial -2w^4z^2+9w^3z^2+26w^2z+25w
Inner characteristic polynomial t^6+64t^4+32t^2+1
Outer characteristic polynomial t^7+96t^5+89t^3+10t
Flat arrow polynomial -2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + K4 + 1
2-strand cable arrow polynomial 768*K1**4*K2 - 1488*K1**4 - 768*K1**3*K2**2*K3 + 1664*K1**3*K2*K3 - 1088*K1**3*K3 + 1152*K1**2*K2**3 - 768*K1**2*K2**2*K3**2 - 4768*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 928*K1**2*K2*K4 + 5400*K1**2*K2 - 1264*K1**2*K3**2 - 64*K1**2*K4**2 - 3832*K1**2 + 1888*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 1440*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 32*K1*K2**2*K5 + 384*K1*K2*K3**3 - 384*K1*K2*K3*K4 - 160*K1*K2*K3*K6 + 6080*K1*K2*K3 - 32*K1*K2*K4*K5 - 64*K1*K2*K4*K7 + 1608*K1*K3*K4 + 160*K1*K4*K5 + 8*K1*K5*K6 - 736*K2**4 - 1344*K2**2*K3**2 - 136*K2**2*K4**2 + 888*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 2642*K2**2 - 96*K2*K3**2*K4 + 664*K2*K3*K5 + 104*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**4 + 56*K3**2*K6 - 1948*K3**2 + 8*K3*K4*K7 - 562*K4**2 - 152*K5**2 - 22*K6**2 - 4*K7**2 - 2*K8**2 + 3106
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}]]
If K is slice False
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