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Flat knot 6.1205

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,1,1,2,4,0,1,1,1,2,2,2,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1205']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1*K2 - K1 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.395', '6.430', '6.440', '6.548', '6.551', '6.774', '6.832', '6.887', '6.908', '6.911', '6.1205', '6.1332', '6.1339', '6.1341', '6.1346', '6.1382', '6.1488', '6.1651', '6.1655', '6.1686']
Outer characteristic polynomial of the knot is: t^7+28t^5+76t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1205', '6.1229']
2-strand cable arrow polynomial of the knot is: 1536*K1**4*K2 - 3360*K1**4 - 384*K1**3*K2**2*K3 + 1792*K1**3*K2*K3 - 576*K1**3*K3 - 128*K1**2*K2**4 + 1728*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 8016*K1**2*K2**2 + 768*K1**2*K2*K3**2 - 896*K1**2*K2*K4 + 7056*K1**2*K2 - 2272*K1**2*K3**2 - 192*K1**2*K3*K5 - 32*K1**2*K4**2 - 2464*K1**2 + 1472*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 2624*K1*K2**2*K3 - 416*K1*K2**2*K5 + 384*K1*K2*K3**3 - 544*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 8048*K1*K2*K3 + 2016*K1*K3*K4 + 96*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1792*K2**4 - 32*K2**3*K6 - 2128*K2**2*K3**2 - 48*K2**2*K4**2 + 1856*K2**2*K4 - 2542*K2**2 - 192*K2*K3**2*K4 + 1264*K2*K3*K5 + 80*K2*K4*K6 - 256*K3**4 + 208*K3**2*K6 - 1700*K3**2 - 468*K4**2 - 124*K5**2 - 34*K6**2 + 2946
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1205']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3249', 'vk6.3273', 'vk6.3301', 'vk6.3375', 'vk6.3410', 'vk6.3436', 'vk6.3476', 'vk6.3514', 'vk6.4612', 'vk6.5901', 'vk6.6030', 'vk6.7944', 'vk6.8067', 'vk6.9380', 'vk6.17843', 'vk6.17858', 'vk6.19060', 'vk6.19879', 'vk6.24360', 'vk6.25678', 'vk6.25692', 'vk6.26325', 'vk6.26768', 'vk6.37783', 'vk6.43781', 'vk6.43796', 'vk6.45070', 'vk6.48105', 'vk6.48124', 'vk6.48147', 'vk6.48196', 'vk6.50656']
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 O1O2O3O4U2U4U3O5O6U5U1U6
R3 orbit {'O1O2O3O4U2U4U3O5O6U5U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U4U6O5O6U2U1U3
Gauss code of K* O1O2O3U4U5O4O6O5U6U1U3U2
Gauss code of -K* O1O2O3U1U3O4O5O6U5U4U6U2
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 -2 1 1 -1 2],[ 1 0 -2 1 1 0 2],[ 2 2 0 2 1 0 0],[-1 -1 -2 0 0 0 0],[-1 -1 -1 0 0 0 0],[ 1 0 0 0 0 0 1],[-2 -2 0 0 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 0 0 -1 -2 0],[-1 0 0 0 0 -1 -1],[-1 0 0 0 0 -1 -2],[ 1 1 0 0 0 0 0],[ 1 2 1 1 0 0 -2],[ 2 0 1 2 0 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,0,0,1,2,0,0,0,1,1,0,1,2,0,0,2]
Phi over symmetry [-2,-1,-1,1,1,2,-1,1,1,2,4,0,1,1,1,2,2,2,0,1,1]
Phi of -K [-2,-1,-1,1,1,2,-1,1,1,2,4,0,1,1,1,2,2,2,0,1,1]
Phi of K* [-2,-1,-1,1,1,2,1,1,1,2,4,0,1,2,1,1,2,2,0,-1,1]
Phi of -K* [-2,-1,-1,1,1,2,0,2,1,2,0,0,0,0,1,1,1,2,0,0,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 9z^2+30z+25
Enhanced Jones-Krushkal polynomial 9w^3z^2+30w^2z+25w
Inner characteristic polynomial t^6+16t^4+32t^2+1
Outer characteristic polynomial t^7+28t^5+76t^3+8t
Flat arrow polynomial 4*K1**3 - 4*K1*K2 - K1 + K3 + 1
2-strand cable arrow polynomial 1536*K1**4*K2 - 3360*K1**4 - 384*K1**3*K2**2*K3 + 1792*K1**3*K2*K3 - 576*K1**3*K3 - 128*K1**2*K2**4 + 1728*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 8016*K1**2*K2**2 + 768*K1**2*K2*K3**2 - 896*K1**2*K2*K4 + 7056*K1**2*K2 - 2272*K1**2*K3**2 - 192*K1**2*K3*K5 - 32*K1**2*K4**2 - 2464*K1**2 + 1472*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 2624*K1*K2**2*K3 - 416*K1*K2**2*K5 + 384*K1*K2*K3**3 - 544*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 8048*K1*K2*K3 + 2016*K1*K3*K4 + 96*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1792*K2**4 - 32*K2**3*K6 - 2128*K2**2*K3**2 - 48*K2**2*K4**2 + 1856*K2**2*K4 - 2542*K2**2 - 192*K2*K3**2*K4 + 1264*K2*K3*K5 + 80*K2*K4*K6 - 256*K3**4 + 208*K3**2*K6 - 1700*K3**2 - 468*K4**2 - 124*K5**2 - 34*K6**2 + 2946
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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