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Flat knot 6.850

Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,-1,2,1,3,4,1,1,3,2,1,1,2,0,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.850']
Arrow polynomial of the knot is: -4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + K2 + 2*K3 + 2*K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.850']
Outer characteristic polynomial of the knot is: t^7+62t^5+76t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.850']
2-strand cable arrow polynomial of the knot is: 32*K1**2*K2 - 2064*K1**2*K3**2 - 48*K1**2*K4**2 - 112*K1**2*K6**2 - 1584*K1**2 + 64*K1*K2*K3**3 + 2544*K1*K2*K3 + 64*K1*K3**3*K4 + 2448*K1*K3*K4 + 144*K1*K4*K5 + 112*K1*K5*K6 + 144*K1*K6*K7 - 224*K2**2*K3**2 - 16*K2**2*K4**2 + 192*K2**2*K4 - 8*K2**2*K6**2 - 1044*K2**2 + 264*K2*K3*K5 + 48*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 - 112*K3**4 - 96*K3**2*K4**2 + 32*K3**2*K6 - 1444*K3**2 + 72*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 894*K4**2 - 184*K5**2 - 100*K6**2 - 76*K7**2 - 12*K8**2 + 1808
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.850']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11010', 'vk6.11091', 'vk6.12180', 'vk6.12289', 'vk6.18201', 'vk6.18537', 'vk6.24658', 'vk6.25083', 'vk6.30575', 'vk6.30672', 'vk6.31849', 'vk6.31898', 'vk6.36795', 'vk6.37248', 'vk6.44035', 'vk6.44376', 'vk6.51809', 'vk6.51878', 'vk6.52677', 'vk6.52773', 'vk6.56007', 'vk6.56281', 'vk6.60545', 'vk6.60885', 'vk6.63489', 'vk6.63535', 'vk6.63971', 'vk6.64017', 'vk6.65672', 'vk6.65957', 'vk6.68717', 'vk6.68926']
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 O1O2O3O4U1U3O5O6U5U2U4U6
R3 orbit {'O1O2O3O4U1U3O5O6U5U2U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U1U3U6O5O6U2U4
Gauss code of K* O1O2O3O4U5U2U6U3O5O6U1U4
Gauss code of -K* O1O2O3O4U1U4O5O6U2U5U3U6
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 0 2 -1 3],[ 3 0 2 1 3 0 2],[ 1 -2 0 0 2 0 3],[ 0 -1 0 0 1 0 1],[-2 -3 -2 -1 0 0 2],[ 1 0 0 0 0 0 1],[-3 -2 -3 -1 -2 -1 0]]
Primitive based matrix [[ 0 3 2 0 -1 -1 -3],[-3 0 -2 -1 -1 -3 -2],[-2 2 0 -1 0 -2 -3],[ 0 1 1 0 0 0 -1],[ 1 1 0 0 0 0 0],[ 1 3 2 0 0 0 -2],[ 3 2 3 1 0 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,0,1,1,3,2,1,1,3,2,1,0,2,3,0,0,1,0,0,2]
Phi over symmetry [-3,-2,0,1,1,3,-1,2,1,3,4,1,1,3,2,1,1,2,0,0,2]
Phi of -K [-3,-1,-1,0,2,3,0,2,2,2,4,0,1,1,1,1,3,3,1,2,-1]
Phi of K* [-3,-2,0,1,1,3,-1,2,1,3,4,1,1,3,2,1,1,2,0,0,2]
Phi of -K* [-3,-1,-1,0,2,3,0,2,1,3,2,0,0,0,1,0,2,3,1,1,2]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 11z+23
Enhanced Jones-Krushkal polynomial -4w^3z+15w^2z+23w
Inner characteristic polynomial t^6+38t^4+23t^2
Outer characteristic polynomial t^7+62t^5+76t^3
Flat arrow polynomial -4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + K2 + 2*K3 + 2*K4 + 2
2-strand cable arrow polynomial 32*K1**2*K2 - 2064*K1**2*K3**2 - 48*K1**2*K4**2 - 112*K1**2*K6**2 - 1584*K1**2 + 64*K1*K2*K3**3 + 2544*K1*K2*K3 + 64*K1*K3**3*K4 + 2448*K1*K3*K4 + 144*K1*K4*K5 + 112*K1*K5*K6 + 144*K1*K6*K7 - 224*K2**2*K3**2 - 16*K2**2*K4**2 + 192*K2**2*K4 - 8*K2**2*K6**2 - 1044*K2**2 + 264*K2*K3*K5 + 48*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 - 112*K3**4 - 96*K3**2*K4**2 + 32*K3**2*K6 - 1444*K3**2 + 72*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 894*K4**2 - 184*K5**2 - 100*K6**2 - 76*K7**2 - 12*K8**2 + 1808
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {4, 5}, {1, 3}, {2}]]
If K is slice False
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