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Flat knot 6.287

Min(phi) over symmetries of the knot is: [-4,-1,1,1,1,2,1,1,2,4,4,0,1,2,1,0,0,-1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.287']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.97', '6.99', '6.287']
Outer characteristic polynomial of the knot is: t^7+72t^5+79t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.287']
2-strand cable arrow polynomial of the knot is: -192*K1**2*K2**4 + 640*K1**2*K2**3 - 2736*K1**2*K2**2 - 320*K1**2*K2*K4 + 2368*K1**2*K2 - 1728*K1**2 + 512*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 672*K1*K2**2*K3 - 384*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 2944*K1*K2*K3 - 32*K1*K2*K4*K5 + 296*K1*K3*K4 + 56*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1200*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 560*K2**2*K3**2 - 192*K2**2*K4**2 + 1216*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 1118*K2**2 + 576*K2*K3*K5 + 80*K2*K4*K6 + 24*K2*K5*K7 - 760*K3**2 - 266*K4**2 - 136*K5**2 - 10*K6**2 + 1400
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.287']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16790', 'vk6.16799', 'vk6.16823', 'vk6.16830', 'vk6.18162', 'vk6.18166', 'vk6.18497', 'vk6.18501', 'vk6.23202', 'vk6.23211', 'vk6.23406', 'vk6.23715', 'vk6.24620', 'vk6.25033', 'vk6.25040', 'vk6.35219', 'vk6.35920', 'vk6.36758', 'vk6.37177', 'vk6.37183', 'vk6.39390', 'vk6.41581', 'vk6.42698', 'vk6.42710', 'vk6.44335', 'vk6.44338', 'vk6.45967', 'vk6.47642', 'vk6.54977', 'vk6.55012', 'vk6.55962', 'vk6.57394', 'vk6.59363', 'vk6.59376', 'vk6.59548', 'vk6.62059', 'vk6.65182', 'vk6.65625', 'vk6.68164', 'vk6.68173']
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 O1O2O3O4O5U1U5O6U3U4U6U2
R3 orbit {'O1O2O3O4O5U1U5O6U3U4U6U2', 'O1O2O3O4O5U1U5U2O6U4U3U6'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5U4U6U2U3O6U1U5
Gauss code of K* O1O2O3O4U5U4U1U2U6O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U5U3U4U1U6
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 -1 1 1 2],[ 4 0 4 2 3 1 2],[-1 -4 0 -2 0 0 2],[ 1 -2 2 0 1 0 2],[-1 -3 0 -1 0 0 1],[-1 -1 0 0 0 0 0],[-2 -2 -2 -2 -1 0 0]]
Primitive based matrix [[ 0 2 1 1 1 -1 -4],[-2 0 0 -1 -2 -2 -2],[-1 0 0 0 0 0 -1],[-1 1 0 0 0 -1 -3],[-1 2 0 0 0 -2 -4],[ 1 2 0 1 2 0 -2],[ 4 2 1 3 4 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,-1,1,4,0,1,2,2,2,0,0,0,1,0,1,3,2,4,2]
Phi over symmetry [-4,-1,1,1,1,2,1,1,2,4,4,0,1,2,1,0,0,-1,0,0,1]
Phi of -K [-4,-1,1,1,1,2,1,1,2,4,4,0,1,2,1,0,0,-1,0,0,1]
Phi of K* [-2,-1,-1,-1,1,4,-1,0,1,1,4,0,0,0,1,0,1,2,2,4,1]
Phi of -K* [-4,-1,1,1,1,2,2,1,3,4,2,0,1,2,2,0,0,0,0,1,2]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 6z^2+19z+15
Enhanced Jones-Krushkal polynomial 6w^3z^2+19w^2z+15w
Inner characteristic polynomial t^6+48t^4+26t^2
Outer characteristic polynomial t^7+72t^5+79t^3+3t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + 2
2-strand cable arrow polynomial -192*K1**2*K2**4 + 640*K1**2*K2**3 - 2736*K1**2*K2**2 - 320*K1**2*K2*K4 + 2368*K1**2*K2 - 1728*K1**2 + 512*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 672*K1*K2**2*K3 - 384*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 2944*K1*K2*K3 - 32*K1*K2*K4*K5 + 296*K1*K3*K4 + 56*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1200*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 560*K2**2*K3**2 - 192*K2**2*K4**2 + 1216*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 1118*K2**2 + 576*K2*K3*K5 + 80*K2*K4*K6 + 24*K2*K5*K7 - 760*K3**2 - 266*K4**2 - 136*K5**2 - 10*K6**2 + 1400
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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