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Flat knot 6.1054

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,0,2,2,3,4,-1,1,1,0,1,1,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1054']
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+50t^5+169t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1054']
2-strand cable arrow polynomial of the knot is: 96*K1**4*K2 - 1648*K1**4 + 352*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1344*K1**3*K3 + 96*K1**2*K2**2*K4 - 1360*K1**2*K2**2 - 736*K1**2*K2*K4 + 5192*K1**2*K2 - 608*K1**2*K3**2 - 128*K1**2*K4**2 - 4096*K1**2 - 256*K1*K2**2*K3 - 96*K1*K2**2*K5 + 32*K1*K2*K3**3 - 128*K1*K2*K3*K4 + 4768*K1*K2*K3 - 32*K1*K3**2*K5 + 1528*K1*K3*K4 + 152*K1*K4*K5 + 8*K1*K5*K6 - 80*K2**4 - 80*K2**2*K3**2 - 24*K2**2*K4**2 + 712*K2**2*K4 - 3142*K2**2 + 288*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 16*K3**2*K6 - 1932*K3**2 - 760*K4**2 - 132*K5**2 - 18*K6**2 + 3270
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1054']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4863', 'vk6.5208', 'vk6.6441', 'vk6.6862', 'vk6.8406', 'vk6.8827', 'vk6.9762', 'vk6.10055', 'vk6.11670', 'vk6.12023', 'vk6.13016', 'vk6.20496', 'vk6.20773', 'vk6.21859', 'vk6.27904', 'vk6.29404', 'vk6.29738', 'vk6.32659', 'vk6.33002', 'vk6.39337', 'vk6.39813', 'vk6.46373', 'vk6.47607', 'vk6.47950', 'vk6.48829', 'vk6.49100', 'vk6.51352', 'vk6.51565', 'vk6.53277', 'vk6.57365', 'vk6.64338', 'vk6.66922']
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 O1O2O3O4U5U6O5U3O6U2U1U4
R3 orbit {'O1O2O3O4U5U6O5U3O6U2U1U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U4U3O5U2O6U5U6
Gauss code of K* O1O2O3U2U1U4U3O5O6U5O4U6
Gauss code of -K* O1O2O3U4O5U6O4O6U1U5U3U2
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 -1 -1 0 3 -1 0],[ 1 0 0 1 3 0 1],[ 1 0 0 1 2 0 1],[ 0 -1 -1 0 0 0 1],[-3 -3 -2 0 0 -4 -2],[ 1 0 0 0 4 0 0],[ 0 -1 -1 -1 2 0 0]]
Primitive based matrix [[ 0 3 0 0 -1 -1 -1],[-3 0 0 -2 -2 -3 -4],[ 0 0 0 1 -1 -1 0],[ 0 2 -1 0 -1 -1 0],[ 1 2 1 1 0 0 0],[ 1 3 1 1 0 0 0],[ 1 4 0 0 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,0,0,1,1,1,0,2,2,3,4,-1,1,1,0,1,1,0,0,0,0]
Phi over symmetry [-3,0,0,1,1,1,0,2,2,3,4,-1,1,1,0,1,1,0,0,0,0]
Phi of -K [-1,-1,-1,0,0,3,0,0,0,0,1,0,0,0,2,1,1,0,-1,3,1]
Phi of K* [-3,0,0,1,1,1,1,3,0,1,2,-1,1,0,0,1,0,0,0,0,0]
Phi of -K* [-1,-1,-1,0,0,3,0,0,0,0,4,0,1,1,2,1,1,3,-1,2,0]
Symmetry type of based matrix c
u-polynomial -t^3+3t
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial 4w^3z^2+21w^2z+27w
Inner characteristic polynomial t^6+38t^4+123t^2
Outer characteristic polynomial t^7+50t^5+169t^3+4t
Flat arrow polynomial -4*K1**2 - 6*K1*K2 + 3*K1 + 2*K2 + 3*K3 + 3
2-strand cable arrow polynomial 96*K1**4*K2 - 1648*K1**4 + 352*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1344*K1**3*K3 + 96*K1**2*K2**2*K4 - 1360*K1**2*K2**2 - 736*K1**2*K2*K4 + 5192*K1**2*K2 - 608*K1**2*K3**2 - 128*K1**2*K4**2 - 4096*K1**2 - 256*K1*K2**2*K3 - 96*K1*K2**2*K5 + 32*K1*K2*K3**3 - 128*K1*K2*K3*K4 + 4768*K1*K2*K3 - 32*K1*K3**2*K5 + 1528*K1*K3*K4 + 152*K1*K4*K5 + 8*K1*K5*K6 - 80*K2**4 - 80*K2**2*K3**2 - 24*K2**2*K4**2 + 712*K2**2*K4 - 3142*K2**2 + 288*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 16*K3**2*K6 - 1932*K3**2 - 760*K4**2 - 132*K5**2 - 18*K6**2 + 3270
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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