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Flat knot 6.776

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,-1,1,3,4,3,0,2,2,1,0,-1,0,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.776']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 6*K1*K2 + 2*K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.552', '6.652', '6.764', '6.776', '6.784', '6.839', '6.903', '6.1010', '6.1166']
Outer characteristic polynomial of the knot is: t^7+68t^5+60t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.776']
2-strand cable arrow polynomial of the knot is: 1952*K1**4*K2 - 4208*K1**4 + 1120*K1**3*K2*K3 + 32*K1**3*K3*K4 - 2528*K1**3*K3 - 128*K1**2*K2**4 + 544*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 5664*K1**2*K2**2 - 1088*K1**2*K2*K4 + 9712*K1**2*K2 - 1040*K1**2*K3**2 - 96*K1**2*K3*K5 - 128*K1**2*K4**2 - 5368*K1**2 + 288*K1*K2**3*K3 - 800*K1*K2**2*K3 - 160*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8064*K1*K2*K3 - 32*K1*K3**2*K5 + 1632*K1*K3*K4 + 240*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 752*K2**4 - 288*K2**2*K3**2 - 56*K2**2*K4**2 + 1256*K2**2*K4 - 4284*K2**2 + 432*K2*K3*K5 + 40*K2*K4*K6 + 16*K3**2*K6 - 2384*K3**2 - 752*K4**2 - 184*K5**2 - 20*K6**2 + 4534
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.776']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4371', 'vk6.4404', 'vk6.5693', 'vk6.5726', 'vk6.7754', 'vk6.7787', 'vk6.9236', 'vk6.9269', 'vk6.10487', 'vk6.10566', 'vk6.10663', 'vk6.10704', 'vk6.10737', 'vk6.10848', 'vk6.14598', 'vk6.15317', 'vk6.15442', 'vk6.16217', 'vk6.17970', 'vk6.24410', 'vk6.30166', 'vk6.30245', 'vk6.30342', 'vk6.30467', 'vk6.33955', 'vk6.34356', 'vk6.34410', 'vk6.43837', 'vk6.50440', 'vk6.50473', 'vk6.54202', 'vk6.63426']
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 O1O2O3O4U3O5U2U6U5O6U1U4
R3 orbit {'O1O2O3O4U3O5U2U6U5O6U1U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U4O5U6U5U3O6U2
Gauss code of K* O1O2O3U2O4O5U4U1U6U5O6U3
Gauss code of -K* O1O2O3U1O4U5U4U3U6O5O6U2
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 3 2 -1],[ 1 0 -1 0 3 2 0],[ 2 1 0 0 3 1 1],[ 1 0 0 0 1 0 1],[-3 -3 -3 -1 0 1 -4],[-2 -2 -1 0 -1 0 -2],[ 1 0 -1 -1 4 2 0]]
Primitive based matrix [[ 0 3 2 -1 -1 -1 -2],[-3 0 1 -1 -3 -4 -3],[-2 -1 0 0 -2 -2 -1],[ 1 1 0 0 0 1 0],[ 1 3 2 0 0 0 -1],[ 1 4 2 -1 0 0 -1],[ 2 3 1 0 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,1,1,1,2,-1,1,3,4,3,0,2,2,1,0,-1,0,0,1,1]
Phi over symmetry [-3,-2,1,1,1,2,-1,1,3,4,3,0,2,2,1,0,-1,0,0,1,1]
Phi of -K [-2,-1,-1,-1,2,3,0,0,1,3,2,0,0,1,1,1,1,0,3,3,2]
Phi of K* [-3,-2,1,1,1,2,2,0,1,3,2,1,1,3,3,0,-1,0,0,0,1]
Phi of -K* [-2,-1,-1,-1,2,3,0,1,1,1,3,0,1,0,1,0,2,3,2,4,-1]
Symmetry type of based matrix c
u-polynomial -t^3+3t
Normalized Jones-Krushkal polynomial 6z^2+26z+29
Enhanced Jones-Krushkal polynomial 6w^3z^2+26w^2z+29w
Inner characteristic polynomial t^6+48t^4+36t^2+4
Outer characteristic polynomial t^7+68t^5+60t^3+10t
Flat arrow polynomial 4*K1**3 - 4*K1**2 - 6*K1*K2 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial 1952*K1**4*K2 - 4208*K1**4 + 1120*K1**3*K2*K3 + 32*K1**3*K3*K4 - 2528*K1**3*K3 - 128*K1**2*K2**4 + 544*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 5664*K1**2*K2**2 - 1088*K1**2*K2*K4 + 9712*K1**2*K2 - 1040*K1**2*K3**2 - 96*K1**2*K3*K5 - 128*K1**2*K4**2 - 5368*K1**2 + 288*K1*K2**3*K3 - 800*K1*K2**2*K3 - 160*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8064*K1*K2*K3 - 32*K1*K3**2*K5 + 1632*K1*K3*K4 + 240*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 752*K2**4 - 288*K2**2*K3**2 - 56*K2**2*K4**2 + 1256*K2**2*K4 - 4284*K2**2 + 432*K2*K3*K5 + 40*K2*K4*K6 + 16*K3**2*K6 - 2384*K3**2 - 752*K4**2 - 184*K5**2 - 20*K6**2 + 4534
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {5}, {1, 4}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {5}, {1, 3}, {2}], [{6}, {2, 5}, {3, 4}, {1}]]
If K is slice False
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