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Flat knot 6.811

Min(phi) over symmetries of the knot is: [-3,-1,1,3,0,2,4,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.811']
Arrow polynomial of the knot is: -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.235', '6.379', '6.411', '6.547', '6.811', '6.818', '6.823', '6.897', '6.898', '6.1008', '6.1053', '6.1109', '6.1110', '6.1130', '6.1222', '6.1239', '6.1303', '6.1307', '6.1342', '6.1491', '6.1495', '6.1496', '6.1519', '6.1592', '6.1593', '6.1642', '6.1652', '6.1653', '6.1671', '6.1673', '6.1717']
Outer characteristic polynomial of the knot is: t^5+46t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.811']
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 128*K1**4*K2**2 + 640*K1**4*K2 - 1760*K1**4 + 128*K1**3*K2*K3 - 1136*K1**2*K2**2 + 2472*K1**2*K2 - 608*K1**2*K3**2 - 144*K1**2*K4**2 - 1216*K1**2 + 1952*K1*K2*K3 + 792*K1*K3*K4 + 144*K1*K4*K5 + 8*K1*K5*K6 - 96*K2**4 - 80*K2**2*K3**2 - 16*K2**2*K4**2 + 184*K2**2*K4 - 1372*K2**2 + 104*K2*K3*K5 + 16*K2*K4*K6 + 8*K3**2*K6 - 848*K3**2 - 320*K4**2 - 72*K5**2 - 12*K6**2 + 1614
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.811']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10945', 'vk6.10968', 'vk6.10976', 'vk6.11001', 'vk6.12111', 'vk6.12134', 'vk6.12142', 'vk6.12167', 'vk6.13785', 'vk6.13810', 'vk6.14219', 'vk6.14242', 'vk6.14666', 'vk6.14691', 'vk6.14856', 'vk6.14881', 'vk6.15822', 'vk6.15845', 'vk6.31809', 'vk6.31834', 'vk6.33617', 'vk6.33640', 'vk6.33648', 'vk6.33673', 'vk6.51789', 'vk6.51798', 'vk6.52652', 'vk6.52659', 'vk6.53811', 'vk6.53820', 'vk6.54237', 'vk6.54244']
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 O1O2O3O4U1O5U3U5U6U4O6U2
R3 orbit {'O1O2O3O4U1O5U3U5U6U4O6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3O5U1U5U6U2O6U4
Gauss code of K* O1O2O3O4U3O5U6U5U1U4O6U2
Gauss code of -K* O1O2O3O4U3O5U1U4U6U5O6U2
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 1 -1 3 1 -1],[ 3 0 3 1 2 1 2],[-1 -3 0 -2 2 1 -2],[ 1 -1 2 0 2 1 0],[-3 -2 -2 -2 0 0 -3],[-1 -1 -1 -1 0 0 -1],[ 1 -2 2 0 3 1 0]]
Primitive based matrix [[ 0 3 1 -1 -3],[-3 0 -2 -2 -2],[-1 2 0 -2 -3],[ 1 2 2 0 -1],[ 3 2 3 1 0]]
If based matrix primitive False
Phi of primitive based matrix [-3,-1,1,3,2,2,2,2,3,1]
Phi over symmetry [-3,-1,1,3,0,2,4,0,1,1]
Phi of -K [-3,-1,1,3,1,1,4,0,2,0]
Phi of K* [-3,-1,1,3,0,2,4,0,1,1]
Phi of -K* [-3,-1,1,3,1,3,2,2,2,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 13z+27
Enhanced Jones-Krushkal polynomial 13w^2z+27w
Inner characteristic polynomial t^4+26t^2
Outer characteristic polynomial t^5+46t^3+10t
Flat arrow polynomial -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -256*K1**6 - 128*K1**4*K2**2 + 640*K1**4*K2 - 1760*K1**4 + 128*K1**3*K2*K3 - 1136*K1**2*K2**2 + 2472*K1**2*K2 - 608*K1**2*K3**2 - 144*K1**2*K4**2 - 1216*K1**2 + 1952*K1*K2*K3 + 792*K1*K3*K4 + 144*K1*K4*K5 + 8*K1*K5*K6 - 96*K2**4 - 80*K2**2*K3**2 - 16*K2**2*K4**2 + 184*K2**2*K4 - 1372*K2**2 + 104*K2*K3*K5 + 16*K2*K4*K6 + 8*K3**2*K6 - 848*K3**2 - 320*K4**2 - 72*K5**2 - 12*K6**2 + 1614
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice True
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