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Flat knot 6.223

Min(phi) over symmetries of the knot is: [-4,-2,1,1,2,2,0,2,4,2,4,1,2,1,2,0,1,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.223']
Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 - 2*K2**2 + 5*K2 + 3*K3 + K4 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.223', '6.281', '6.497']
Outer characteristic polynomial of the knot is: t^7+85t^5+52t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.223']
2-strand cable arrow polynomial of the knot is: -64*K1**6 - 64*K1**4*K2**2 + 1376*K1**4*K2 - 4400*K1**4 + 512*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1216*K1**3*K3 + 416*K1**2*K2**3 - 5424*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 992*K1**2*K2*K4 + 11240*K1**2*K2 - 1216*K1**2*K3**2 - 192*K1**2*K3*K5 - 416*K1**2*K4**2 - 7812*K1**2 + 352*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1120*K1*K2**2*K3 - 64*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 9032*K1*K2*K3 - 32*K1*K2*K4*K5 + 3448*K1*K3*K4 + 600*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 888*K2**4 - 32*K2**3*K6 - 848*K2**2*K3**2 - 384*K2**2*K4**2 + 1744*K2**2*K4 - 5902*K2**2 - 160*K2*K3**2*K4 + 672*K2*K3*K5 + 280*K2*K4*K6 - 80*K3**4 - 128*K3**2*K4**2 + 80*K3**2*K6 - 3480*K3**2 + 96*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1630*K4**2 - 308*K5**2 - 50*K6**2 - 16*K7**2 - 2*K8**2 + 6798
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.223']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16566', 'vk6.16657', 'vk6.18155', 'vk6.18489', 'vk6.22965', 'vk6.23084', 'vk6.24610', 'vk6.25021', 'vk6.34958', 'vk6.35077', 'vk6.36745', 'vk6.37162', 'vk6.42527', 'vk6.42636', 'vk6.44013', 'vk6.44323', 'vk6.54797', 'vk6.54882', 'vk6.55953', 'vk6.56251', 'vk6.59225', 'vk6.59305', 'vk6.60487', 'vk6.60851', 'vk6.64771', 'vk6.64834', 'vk6.65610', 'vk6.65915', 'vk6.68069', 'vk6.68132', 'vk6.68681', 'vk6.68890']
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 O1O2O3O4O5U3O6U1U5U6U4U2
R3 orbit {'O1O2O3O4O5U3O6U1U5U6U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U2U6U1U5O6U3
Gauss code of K* O1O2O3O4O5U1U5U6U4U2O6U3
Gauss code of -K* O1O2O3O4O5U3O6U4U2U6U1U5
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 1 -2 2 1 2],[ 4 0 4 0 4 2 2],[-1 -4 0 -2 1 0 1],[ 2 0 2 0 2 1 1],[-2 -4 -1 -2 0 -1 1],[-1 -2 0 -1 1 0 1],[-2 -2 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 2 1 1 -2 -4],[-2 0 1 -1 -1 -2 -4],[-2 -1 0 -1 -1 -1 -2],[-1 1 1 0 0 -1 -2],[-1 1 1 0 0 -2 -4],[ 2 2 1 1 2 0 0],[ 4 4 2 2 4 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,-1,2,4,-1,1,1,2,4,1,1,1,2,0,1,2,2,4,0]
Phi over symmetry [-4,-2,1,1,2,2,0,2,4,2,4,1,2,1,2,0,1,1,1,1,-1]
Phi of -K [-4,-2,1,1,2,2,2,1,3,2,4,1,2,2,3,0,0,0,0,0,-1]
Phi of K* [-2,-2,-1,-1,2,4,-1,0,0,3,4,0,0,2,2,0,1,1,2,3,2]
Phi of -K* [-4,-2,1,1,2,2,0,2,4,2,4,1,2,1,2,0,1,1,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+55t^4+15t^2
Outer characteristic polynomial t^7+85t^5+52t^3+5t
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 - 2*K2**2 + 5*K2 + 3*K3 + K4 + 7
2-strand cable arrow polynomial -64*K1**6 - 64*K1**4*K2**2 + 1376*K1**4*K2 - 4400*K1**4 + 512*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1216*K1**3*K3 + 416*K1**2*K2**3 - 5424*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 992*K1**2*K2*K4 + 11240*K1**2*K2 - 1216*K1**2*K3**2 - 192*K1**2*K3*K5 - 416*K1**2*K4**2 - 7812*K1**2 + 352*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1120*K1*K2**2*K3 - 64*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 9032*K1*K2*K3 - 32*K1*K2*K4*K5 + 3448*K1*K3*K4 + 600*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 888*K2**4 - 32*K2**3*K6 - 848*K2**2*K3**2 - 384*K2**2*K4**2 + 1744*K2**2*K4 - 5902*K2**2 - 160*K2*K3**2*K4 + 672*K2*K3*K5 + 280*K2*K4*K6 - 80*K3**4 - 128*K3**2*K4**2 + 80*K3**2*K6 - 3480*K3**2 + 96*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1630*K4**2 - 308*K5**2 - 50*K6**2 - 16*K7**2 - 2*K8**2 + 6798
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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