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Flat knot 6.1372

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,0,1,1,1,2,1,2,2,2,-1,-1,0,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1372']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1*K2 + K1 + 3*K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1231', '6.1372', '6.1722', '6.1817', '6.1862', '6.2082']
Outer characteristic polynomial of the knot is: t^7+36t^5+84t^3+2t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1372']
2-strand cable arrow polynomial of the knot is: 2304*K1**4*K2 - 4288*K1**4 + 1856*K1**3*K2*K3 - 512*K1**3*K3 - 128*K1**2*K2**4 + 1024*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 7520*K1**2*K2**2 - 768*K1**2*K2*K4 + 7088*K1**2*K2 - 2624*K1**2*K3**2 - 96*K1**2*K4**2 - 2176*K1**2 + 960*K1*K2**3*K3 - 2368*K1*K2**2*K3 - 512*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 7424*K1*K2*K3 + 2384*K1*K3*K4 + 160*K1*K4*K5 - 32*K2**6 + 224*K2**4*K4 - 1392*K2**4 - 96*K2**3*K6 - 1248*K2**2*K3**2 - 128*K2**2*K4**2 + 1720*K2**2*K4 - 2826*K2**2 + 976*K2*K3*K5 + 72*K2*K4*K6 - 1816*K3**2 - 648*K4**2 - 184*K5**2 - 6*K6**2 + 3110
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1372']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20141', 'vk6.20165', 'vk6.21431', 'vk6.21451', 'vk6.27253', 'vk6.27293', 'vk6.28913', 'vk6.28952', 'vk6.38678', 'vk6.38722', 'vk6.40900', 'vk6.47264', 'vk6.47291', 'vk6.56974', 'vk6.57007', 'vk6.58126', 'vk6.62675', 'vk6.67471', 'vk6.70037', 'vk6.70054']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3U4O5O4U2U1U3O6U5U6
R3 orbit {'O1O2O3U4O5O4U2U1U3O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O4U1U3U2O6O5U6
Gauss code of K* Same
Gauss code of -K* O1O2O3U4U5O4U1U3U2O6O5U6
Diagrammatic symmetry type +
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 -2 1 1 1 1],[ 2 0 0 2 2 2 1],[ 2 0 0 1 2 1 1],[-1 -2 -1 0 -1 0 1],[-1 -2 -2 1 0 1 0],[-1 -2 -1 0 -1 0 1],[-1 -1 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 1 -2 -2],[-1 0 1 1 0 -2 -2],[-1 -1 0 0 1 -1 -2],[-1 -1 0 0 1 -1 -2],[-1 0 -1 -1 0 -1 -1],[ 2 2 1 1 1 0 0],[ 2 2 2 2 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,-1,2,2,-1,-1,0,2,2,0,-1,1,2,-1,1,2,1,1,0]
Phi over symmetry [-2,-2,1,1,1,1,0,1,1,1,2,1,2,2,2,-1,-1,0,0,-1,-1]
Phi of -K [-2,-2,1,1,1,1,0,1,1,1,2,1,2,2,2,-1,-1,0,0,-1,-1]
Phi of K* [-1,-1,-1,-1,2,2,-1,-1,0,2,2,0,-1,1,2,-1,1,2,1,1,0]
Phi of -K* [-2,-2,1,1,1,1,0,1,1,1,2,1,2,2,2,-1,-1,0,0,-1,-1]
Symmetry type of based matrix +
u-polynomial 2t^2-4t
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+24t^4+46t^2
Outer characteristic polynomial t^7+36t^5+84t^3+2t
Flat arrow polynomial 4*K1**3 - 8*K1*K2 + K1 + 3*K3 + 1
2-strand cable arrow polynomial 2304*K1**4*K2 - 4288*K1**4 + 1856*K1**3*K2*K3 - 512*K1**3*K3 - 128*K1**2*K2**4 + 1024*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 7520*K1**2*K2**2 - 768*K1**2*K2*K4 + 7088*K1**2*K2 - 2624*K1**2*K3**2 - 96*K1**2*K4**2 - 2176*K1**2 + 960*K1*K2**3*K3 - 2368*K1*K2**2*K3 - 512*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 7424*K1*K2*K3 + 2384*K1*K3*K4 + 160*K1*K4*K5 - 32*K2**6 + 224*K2**4*K4 - 1392*K2**4 - 96*K2**3*K6 - 1248*K2**2*K3**2 - 128*K2**2*K4**2 + 1720*K2**2*K4 - 2826*K2**2 + 976*K2*K3*K5 + 72*K2*K4*K6 - 1816*K3**2 - 648*K4**2 - 184*K5**2 - 6*K6**2 + 3110
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {5}, {2, 4}, {3}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {3}, {1, 2}]]
If K is slice False
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