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Flat knot 6.376

Min(phi) over symmetries of the knot is: [-3,-2,-1,2,2,2,0,2,2,2,3,1,1,2,2,2,2,2,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.376']
Arrow polynomial of the knot is: -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.136', '6.207', '6.342', '6.370', '6.376', '6.442', '6.456', '6.539', '6.631', '6.636', '6.674', '6.679', '6.705', '6.740', '6.760', '6.794', '6.795', '6.1369']
Outer characteristic polynomial of the knot is: t^7+71t^5+39t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.376']
2-strand cable arrow polynomial of the knot is: -1280*K1**4 + 32*K1**3*K2*K3 - 96*K1**3*K3 - 1632*K1**2*K2**2 - 128*K1**2*K2*K4 + 3760*K1**2*K2 - 256*K1**2*K3**2 - 96*K1**2*K3*K5 - 64*K1**2*K4**2 - 2512*K1**2 - 384*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 2672*K1*K2*K3 + 904*K1*K3*K4 + 192*K1*K4*K5 - 240*K2**4 - 8*K2**2*K4**2 + 632*K2**2*K4 - 2094*K2**2 + 80*K2*K3*K5 + 8*K2*K4*K6 - 1072*K3**2 - 528*K4**2 - 88*K5**2 - 2*K6**2 + 2230
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.376']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16502', 'vk6.16593', 'vk6.18106', 'vk6.18444', 'vk6.22929', 'vk6.23024', 'vk6.24553', 'vk6.24972', 'vk6.34910', 'vk6.35017', 'vk6.36688', 'vk6.37112', 'vk6.42475', 'vk6.42586', 'vk6.43964', 'vk6.44281', 'vk6.54729', 'vk6.54824', 'vk6.55917', 'vk6.56208', 'vk6.59189', 'vk6.59252', 'vk6.60444', 'vk6.60803', 'vk6.64749', 'vk6.64806', 'vk6.65556', 'vk6.65868', 'vk6.68041', 'vk6.68104', 'vk6.68634', 'vk6.68849', 'vk6.73733', 'vk6.73850', 'vk6.78320', 'vk6.78480', 'vk6.78649', 'vk6.78842', 'vk6.85164', 'vk6.89446']
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 O1O2O3O4O5U4U1O6U2U5U6U3
R3 orbit {'O1O2O3O4U5U2O6U1U4U6O5U3', 'O1O2O3O4O5U4U1O6U2U5U6U3'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5U3U6U1U4O6U5U2
Gauss code of K* O1O2O3O4U5U1U4U6U2O6O5U3
Gauss code of -K* O1O2O3O4U2O5O6U3U6U1U4U5
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 -2 2 -1 2 2],[ 3 0 1 3 0 3 2],[ 2 -1 0 3 0 2 2],[-2 -3 -3 0 -1 0 1],[ 1 0 0 1 0 1 1],[-2 -3 -2 0 -1 0 1],[-2 -2 -2 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 2 2 -1 -2 -3],[-2 0 1 0 -1 -2 -3],[-2 -1 0 -1 -1 -2 -2],[-2 0 1 0 -1 -3 -3],[ 1 1 1 1 0 0 0],[ 2 2 2 3 0 0 -1],[ 3 3 2 3 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-2,1,2,3,-1,0,1,2,3,1,1,2,2,1,3,3,0,0,1]
Phi over symmetry [-3,-2,-1,2,2,2,0,2,2,2,3,1,1,2,2,2,2,2,0,-1,-1]
Phi of -K [-3,-2,-1,2,2,2,0,2,2,2,3,1,1,2,2,2,2,2,0,-1,-1]
Phi of K* [-2,-2,-2,1,2,3,-1,-1,2,2,3,0,2,1,2,2,2,2,1,2,0]
Phi of -K* [-3,-2,-1,2,2,2,1,0,2,3,3,0,2,2,3,1,1,1,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial z^2+14z+25
Enhanced Jones-Krushkal polynomial w^3z^2+14w^2z+25w
Inner characteristic polynomial t^6+45t^4+19t^2+1
Outer characteristic polynomial t^7+71t^5+39t^3+4t
Flat arrow polynomial -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -1280*K1**4 + 32*K1**3*K2*K3 - 96*K1**3*K3 - 1632*K1**2*K2**2 - 128*K1**2*K2*K4 + 3760*K1**2*K2 - 256*K1**2*K3**2 - 96*K1**2*K3*K5 - 64*K1**2*K4**2 - 2512*K1**2 - 384*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 2672*K1*K2*K3 + 904*K1*K3*K4 + 192*K1*K4*K5 - 240*K2**4 - 8*K2**2*K4**2 + 632*K2**2*K4 - 2094*K2**2 + 80*K2*K3*K5 + 8*K2*K4*K6 - 1072*K3**2 - 528*K4**2 - 88*K5**2 - 2*K6**2 + 2230
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {1, 3}, {2}]]
If K is slice False
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