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Flat knot 6.1191

Min(phi) over symmetries of the knot is: [-2,-2,-1,1,2,2,0,-1,2,3,3,1,3,2,3,2,0,1,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1191']
Arrow polynomial of the knot is: 4*K1**2*K2 - 2*K1**2 - 2*K1*K3 - 2*K2**2 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1191', '6.1918']
Outer characteristic polynomial of the knot is: t^7+43t^5+129t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1191']
2-strand cable arrow polynomial of the knot is: -64*K1**3*K3 + 224*K1**2*K2**2*K4 - 1952*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 928*K1**2*K2*K4 + 3984*K1**2*K2 - 128*K1**2*K3**2 - 288*K1**2*K4**2 - 4104*K1**2 + 288*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1376*K1*K2**2*K3 - 192*K1*K2**2*K5 - 576*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 4832*K1*K2*K3 - 224*K1*K2*K4*K5 - 96*K1*K2*K4*K7 + 1920*K1*K3*K4 + 496*K1*K4*K5 + 32*K1*K5*K6 + 24*K1*K6*K7 - 32*K2**4*K4**2 + 128*K2**4*K4 - 760*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 768*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 384*K2**2*K4**2 - 32*K2**2*K4*K8 + 2264*K2**2*K4 - 8*K2**2*K6**2 - 3636*K2**2 - 64*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 856*K2*K3*K5 + 392*K2*K4*K6 + 112*K2*K5*K7 + 8*K2*K6*K8 + 48*K3**2*K6 - 1944*K3**2 + 88*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1320*K4**2 - 264*K5**2 - 68*K6**2 - 64*K7**2 - 2*K8**2 + 3416
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1191']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4227', 'vk6.4308', 'vk6.5494', 'vk6.5610', 'vk6.7599', 'vk6.7692', 'vk6.9096', 'vk6.9177', 'vk6.18377', 'vk6.18715', 'vk6.24830', 'vk6.25287', 'vk6.37022', 'vk6.37470', 'vk6.44191', 'vk6.44510', 'vk6.48539', 'vk6.48596', 'vk6.49238', 'vk6.49352', 'vk6.50326', 'vk6.50387', 'vk6.51069', 'vk6.51102', 'vk6.56158', 'vk6.56385', 'vk6.60687', 'vk6.61038', 'vk6.65830', 'vk6.66082', 'vk6.68819', 'vk6.69027']
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 O1O2O3O4U2U1U4O5O6U5U3U6
R3 orbit {'O1O2O3O4U2U1U4O5O6U5U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U2U6O5O6U1U4U3
Gauss code of K* O1O2O3U4U5O4O6O5U2U1U6U3
Gauss code of -K* O1O2O3U1U3O4O5O6U4U2U6U5
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 -2 -2 1 2 -1 2],[ 2 0 0 3 2 0 1],[ 2 0 0 2 1 0 1],[-1 -3 -2 0 0 0 2],[-2 -2 -1 0 0 0 0],[ 1 0 0 0 0 0 1],[-2 -1 -1 -2 0 -1 0]]
Primitive based matrix [[ 0 2 2 1 -1 -2 -2],[-2 0 0 0 0 -1 -2],[-2 0 0 -2 -1 -1 -1],[-1 0 2 0 0 -2 -3],[ 1 0 1 0 0 0 0],[ 2 1 1 2 0 0 0],[ 2 2 1 3 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,2,2,0,0,0,1,2,2,1,1,1,0,2,3,0,0,0]
Phi over symmetry [-2,-2,-1,1,2,2,0,-1,2,3,3,1,3,2,3,2,0,1,1,1,0]
Phi of -K [-2,-2,-1,1,2,2,0,1,0,2,3,1,1,3,3,2,3,2,1,-1,0]
Phi of K* [-2,-2,-1,1,2,2,0,-1,2,3,3,1,3,2,3,2,0,1,1,1,0]
Phi of -K* [-2,-2,-1,1,2,2,0,0,2,1,1,0,3,1,2,0,1,0,2,0,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 8z^2+27z+23
Enhanced Jones-Krushkal polynomial 8w^3z^2+27w^2z+23w
Inner characteristic polynomial t^6+25t^4+41t^2+1
Outer characteristic polynomial t^7+43t^5+129t^3+6t
Flat arrow polynomial 4*K1**2*K2 - 2*K1**2 - 2*K1*K3 - 2*K2**2 + K4 + 2
2-strand cable arrow polynomial -64*K1**3*K3 + 224*K1**2*K2**2*K4 - 1952*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 928*K1**2*K2*K4 + 3984*K1**2*K2 - 128*K1**2*K3**2 - 288*K1**2*K4**2 - 4104*K1**2 + 288*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1376*K1*K2**2*K3 - 192*K1*K2**2*K5 - 576*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 4832*K1*K2*K3 - 224*K1*K2*K4*K5 - 96*K1*K2*K4*K7 + 1920*K1*K3*K4 + 496*K1*K4*K5 + 32*K1*K5*K6 + 24*K1*K6*K7 - 32*K2**4*K4**2 + 128*K2**4*K4 - 760*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 768*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 384*K2**2*K4**2 - 32*K2**2*K4*K8 + 2264*K2**2*K4 - 8*K2**2*K6**2 - 3636*K2**2 - 64*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 856*K2*K3*K5 + 392*K2*K4*K6 + 112*K2*K5*K7 + 8*K2*K6*K8 + 48*K3**2*K6 - 1944*K3**2 + 88*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1320*K4**2 - 264*K5**2 - 68*K6**2 - 64*K7**2 - 2*K8**2 + 3416
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}]]
If K is slice False
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