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Flat knot 6.1096

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,1,1,1,2,3,0,0,1,1,1,1,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1096']
Arrow polynomial of the knot is: -4*K1**2 - 6*K1*K2 + 3*K1 + 2*K2 + 3*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.586', '6.590', '6.958', '6.987', '6.991', '6.993', '6.999', '6.1054', '6.1065', '6.1096', '6.1168', '6.1182']
Outer characteristic polynomial of the knot is: t^7+64t^5+49t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1096']
2-strand cable arrow polynomial of the knot is: 224*K1**4*K2 - 912*K1**4 + 128*K1**3*K2*K3 - 1984*K1**3*K3 + 32*K1**2*K2**2*K4 - 976*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 544*K1**2*K2*K4 + 6304*K1**2*K2 - 1104*K1**2*K3**2 - 96*K1**2*K3*K5 - 128*K1**2*K4**2 - 96*K1**2*K4*K6 - 6592*K1**2 + 96*K1*K2**3*K3 - 608*K1*K2**2*K3 - 544*K1*K2*K3*K4 + 6848*K1*K2*K3 + 2320*K1*K3*K4 + 392*K1*K4*K5 + 72*K1*K5*K6 - 112*K2**4 - 336*K2**2*K3**2 - 88*K2**2*K4**2 + 880*K2**2*K4 - 4338*K2**2 + 488*K2*K3*K5 + 144*K2*K4*K6 - 2900*K3**2 - 1020*K4**2 - 180*K5**2 - 62*K6**2 + 4594
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1096']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81575', 'vk6.81578', 'vk6.81655', 'vk6.81659', 'vk6.81738', 'vk6.81741', 'vk6.81853', 'vk6.81856', 'vk6.82241', 'vk6.82243', 'vk6.82391', 'vk6.82400', 'vk6.82507', 'vk6.82514', 'vk6.82567', 'vk6.82574', 'vk6.83173', 'vk6.83184', 'vk6.83598', 'vk6.83606', 'vk6.84142', 'vk6.84149', 'vk6.84340', 'vk6.84351', 'vk6.84556', 'vk6.84561', 'vk6.86480', 'vk6.86490', 'vk6.88728', 'vk6.88743', 'vk6.88910', 'vk6.88917']
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 O1O2O3O4U1U5O6U2U4O5U3U6
R3 orbit {'O1O2O3O4U1U5O6U2U4O5U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U2O6U1U3O5U6U4
Gauss code of K* O1O2U3O4O5U2O6O3U1U4U6U5
Gauss code of -K* O1O2U3O4O5U1O3O6U4U2U5U6
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 -3 -1 1 2 -1 2],[ 3 0 1 3 2 1 3],[ 1 -1 0 1 1 0 2],[-1 -3 -1 0 1 -2 1],[-2 -2 -1 -1 0 -2 0],[ 1 -1 0 2 2 0 2],[-2 -3 -2 -1 0 -2 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 0 -1 -1 -2 -2],[-2 0 0 -1 -2 -2 -3],[-1 1 1 0 -1 -2 -3],[ 1 1 2 1 0 0 -1],[ 1 2 2 2 0 0 -1],[ 3 2 3 3 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,0,1,1,2,2,1,2,2,3,1,2,3,0,1,1]
Phi over symmetry [-3,-1,-1,1,2,2,1,1,1,2,3,0,0,1,1,1,1,2,0,0,0]
Phi of -K [-3,-1,-1,1,2,2,1,1,1,2,3,0,0,1,1,1,1,2,0,0,0]
Phi of K* [-2,-2,-1,1,1,3,0,0,1,1,2,0,1,2,3,0,1,1,0,1,1]
Phi of -K* [-3,-1,-1,1,2,2,1,1,3,2,3,0,1,1,2,2,2,2,1,1,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 3z^2+22z+33
Enhanced Jones-Krushkal polynomial 3w^3z^2+22w^2z+33w
Inner characteristic polynomial t^6+44t^4+23t^2+1
Outer characteristic polynomial t^7+64t^5+49t^3+6t
Flat arrow polynomial -4*K1**2 - 6*K1*K2 + 3*K1 + 2*K2 + 3*K3 + 3
2-strand cable arrow polynomial 224*K1**4*K2 - 912*K1**4 + 128*K1**3*K2*K3 - 1984*K1**3*K3 + 32*K1**2*K2**2*K4 - 976*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 544*K1**2*K2*K4 + 6304*K1**2*K2 - 1104*K1**2*K3**2 - 96*K1**2*K3*K5 - 128*K1**2*K4**2 - 96*K1**2*K4*K6 - 6592*K1**2 + 96*K1*K2**3*K3 - 608*K1*K2**2*K3 - 544*K1*K2*K3*K4 + 6848*K1*K2*K3 + 2320*K1*K3*K4 + 392*K1*K4*K5 + 72*K1*K5*K6 - 112*K2**4 - 336*K2**2*K3**2 - 88*K2**2*K4**2 + 880*K2**2*K4 - 4338*K2**2 + 488*K2*K3*K5 + 144*K2*K4*K6 - 2900*K3**2 - 1020*K4**2 - 180*K5**2 - 62*K6**2 + 4594
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {2, 5}, {1, 4}, {3}]]
If K is slice False
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