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Flat knot 6.171

Min(phi) over symmetries of the knot is: [-4,-2,0,0,3,3,0,1,3,3,4,1,2,2,2,0,1,1,2,2,-1]
Flat knots (up to 7 crossings) with same phi are :['6.171']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 - 4*K1 + 3*K2 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.73', '6.171', '6.183']
Outer characteristic polynomial of the knot is: t^7+107t^5+46t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.171']
2-strand cable arrow polynomial of the knot is: -768*K1**4 + 256*K1**3*K2*K3 + 32*K1**3*K3*K4 - 288*K1**3*K3 - 128*K1**2*K2**4 + 480*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 3040*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 288*K1**2*K2*K4 + 3960*K1**2*K2 - 384*K1**2*K3**2 - 32*K1**2*K3*K5 - 16*K1**2*K4**2 - 2544*K1**2 + 992*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 672*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 224*K1*K2**2*K5 + 32*K1*K2*K3**3 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3496*K1*K2*K3 + 472*K1*K3*K4 + 32*K1*K4*K5 - 64*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 1008*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 672*K2**2*K3**2 - 128*K2**2*K4**2 + 840*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 1452*K2**2 + 304*K2*K3*K5 + 32*K2*K4*K6 - 884*K3**2 - 186*K4**2 - 20*K5**2 - 4*K6**2 + 1880
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.171']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73273', 'vk6.73416', 'vk6.73707', 'vk6.73824', 'vk6.74547', 'vk6.74813', 'vk6.75181', 'vk6.75632', 'vk6.76023', 'vk6.76366', 'vk6.76759', 'vk6.76874', 'vk6.78138', 'vk6.78611', 'vk6.78992', 'vk6.79228', 'vk6.79547', 'vk6.79701', 'vk6.79971', 'vk6.80249', 'vk6.80512', 'vk6.80707', 'vk6.80982', 'vk6.81069', 'vk6.81617', 'vk6.81787', 'vk6.82162', 'vk6.82164', 'vk6.82651', 'vk6.84041', 'vk6.84056', 'vk6.84225', 'vk6.84595', 'vk6.84927', 'vk6.85932', 'vk6.86730', 'vk6.87653', 'vk6.87768', 'vk6.88215', 'vk6.89970']
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 O1O2O3O4O5U1O6U5U2U3U6U4
R3 orbit {'O1O2O3O4O5U1O6U5U2U3U6U4', 'O1O2O3O4O5U1U4O6U2U3U5U6'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5U2U6U3U4U1O6U5
Gauss code of K* O1O2O3O4O5U6U2U3U5U1O6U4
Gauss code of -K* O1O2O3O4O5U2O6U5U1U3U4U6
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 -4 -2 0 3 0 3],[ 4 0 2 3 4 1 3],[ 2 -2 0 1 3 0 3],[ 0 -3 -1 0 2 0 2],[-3 -4 -3 -2 0 -1 1],[ 0 -1 0 0 1 0 1],[-3 -3 -3 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 3 0 0 -2 -4],[-3 0 1 -1 -2 -3 -4],[-3 -1 0 -1 -2 -3 -3],[ 0 1 1 0 0 0 -1],[ 0 2 2 0 0 -1 -3],[ 2 3 3 0 1 0 -2],[ 4 4 3 1 3 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,0,0,2,4,-1,1,2,3,4,1,2,3,3,0,0,1,1,3,2]
Phi over symmetry [-4,-2,0,0,3,3,0,1,3,3,4,1,2,2,2,0,1,1,2,2,-1]
Phi of -K [-4,-2,0,0,3,3,0,1,3,3,4,1,2,2,2,0,1,1,2,2,-1]
Phi of K* [-3,-3,0,0,2,4,-1,1,2,2,4,1,2,2,3,0,1,1,2,3,0]
Phi of -K* [-4,-2,0,0,3,3,2,1,3,3,4,0,1,3,3,0,1,1,2,2,-1]
Symmetry type of based matrix c
u-polynomial t^4-2t^3+t^2
Normalized Jones-Krushkal polynomial 3z^2+16z+21
Enhanced Jones-Krushkal polynomial 3w^3z^2+16w^2z+21w
Inner characteristic polynomial t^6+69t^4+15t^2
Outer characteristic polynomial t^7+107t^5+46t^3+3t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 - 4*K1 + 3*K2 + 4
2-strand cable arrow polynomial -768*K1**4 + 256*K1**3*K2*K3 + 32*K1**3*K3*K4 - 288*K1**3*K3 - 128*K1**2*K2**4 + 480*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 3040*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 288*K1**2*K2*K4 + 3960*K1**2*K2 - 384*K1**2*K3**2 - 32*K1**2*K3*K5 - 16*K1**2*K4**2 - 2544*K1**2 + 992*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 672*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 224*K1*K2**2*K5 + 32*K1*K2*K3**3 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3496*K1*K2*K3 + 472*K1*K3*K4 + 32*K1*K4*K5 - 64*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 1008*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 672*K2**2*K3**2 - 128*K2**2*K4**2 + 840*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 1452*K2**2 + 304*K2*K3*K5 + 32*K2*K4*K6 - 884*K3**2 - 186*K4**2 - 20*K5**2 - 4*K6**2 + 1880
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {3}, {1, 2}]]
If K is slice False
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