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Flat knot 6.105

Min(phi) over symmetries of the knot is: [-4,-2,1,1,1,3,0,2,3,4,3,1,2,2,2,0,0,1,0,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.105']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.105', '6.144', '6.261', '6.285', '6.392', '6.480']
Outer characteristic polynomial of the knot is: t^7+82t^5+41t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.105']
2-strand cable arrow polynomial of the knot is: 544*K1**2*K2**3 - 1344*K1**2*K2**2 - 224*K1**2*K2*K4 + 1424*K1**2*K2 - 1328*K1**2 + 128*K1*K2**2*K3*K4 - 448*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 192*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 1560*K1*K2*K3 + 296*K1*K3*K4 + 224*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 752*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 224*K2**2*K3**2 - 248*K2**2*K4**2 + 1000*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 926*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 440*K2*K3*K5 + 88*K2*K4*K6 + 24*K2*K5*K7 - 520*K3**2 - 394*K4**2 - 208*K5**2 - 18*K6**2 + 1160
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.105']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16364', 'vk6.16405', 'vk6.18123', 'vk6.18457', 'vk6.22694', 'vk6.22793', 'vk6.24580', 'vk6.24992', 'vk6.34659', 'vk6.34744', 'vk6.35521', 'vk6.35967', 'vk6.36713', 'vk6.37133', 'vk6.39436', 'vk6.41634', 'vk6.42320', 'vk6.42365', 'vk6.42937', 'vk6.43230', 'vk6.43985', 'vk6.44297', 'vk6.46020', 'vk6.47688', 'vk6.54650', 'vk6.55208', 'vk6.56226', 'vk6.57440', 'vk6.59165', 'vk6.59603', 'vk6.60827', 'vk6.62112', 'vk6.64646', 'vk6.64692', 'vk6.65581', 'vk6.65891', 'vk6.67999', 'vk6.68023', 'vk6.68658', 'vk6.68870']
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 O1O2O3O4O5O6U2U6U5U1U3U4
R3 orbit {'O1O2O3O4O5U1U5U6U3U2O6U4', 'O1O2O3O4O5O6U2U6U5U1U3U4'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5O6U3U4U6U2U1U5
Gauss code of K* O1O2O3O4O5O6U4U1U5U6U3U2
Gauss code of -K* O1O2O3O4O5O6U5U4U1U2U6U3
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 -2 -4 1 3 1 1],[ 2 0 -2 2 3 1 1],[ 4 2 0 3 4 2 1],[-1 -2 -3 0 1 0 0],[-3 -3 -4 -1 0 0 0],[-1 -1 -2 0 0 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 3 1 1 1 -2 -4],[-3 0 0 0 -1 -3 -4],[-1 0 0 0 0 -1 -1],[-1 0 0 0 0 -1 -2],[-1 1 0 0 0 -2 -3],[ 2 3 1 1 2 0 -2],[ 4 4 1 2 3 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,-1,2,4,0,0,1,3,4,0,0,1,1,0,1,2,2,3,2]
Phi over symmetry [-4,-2,1,1,1,3,0,2,3,4,3,1,2,2,2,0,0,1,0,2,2]
Phi of -K [-4,-2,1,1,1,3,0,2,3,4,3,1,2,2,2,0,0,1,0,2,2]
Phi of K* [-3,-1,-1,-1,2,4,1,2,2,2,3,0,0,1,2,0,2,3,2,4,0]
Phi of -K* [-4,-2,1,1,1,3,2,1,2,3,4,1,1,2,3,0,0,0,0,0,1]
Symmetry type of based matrix c
u-polynomial t^4-t^3+t^2-3t
Normalized Jones-Krushkal polynomial 6z^2+19z+15
Enhanced Jones-Krushkal polynomial 6w^3z^2+19w^2z+15w
Inner characteristic polynomial t^6+50t^4+16t^2+1
Outer characteristic polynomial t^7+82t^5+41t^3+4t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + 2
2-strand cable arrow polynomial 544*K1**2*K2**3 - 1344*K1**2*K2**2 - 224*K1**2*K2*K4 + 1424*K1**2*K2 - 1328*K1**2 + 128*K1*K2**2*K3*K4 - 448*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 192*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 1560*K1*K2*K3 + 296*K1*K3*K4 + 224*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 752*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 224*K2**2*K3**2 - 248*K2**2*K4**2 + 1000*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 926*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 440*K2*K3*K5 + 88*K2*K4*K6 + 24*K2*K5*K7 - 520*K3**2 - 394*K4**2 - 208*K5**2 - 18*K6**2 + 1160
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {5}, {1, 4}, {2}]]
If K is slice False
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