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Flat knot 6.405

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,-1,1,1,3,2,1,1,1,0,1,1,1,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.405', '7.19810']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 2*K1**2 - 6*K1*K2 - 3*K1 - 2*K2**2 + K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.405', '6.1157']
Outer characteristic polynomial of the knot is: t^7+40t^5+58t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.405', '7.19810']
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 576*K1**4*K2**2 + 3904*K1**4*K2 - 6320*K1**4 - 384*K1**3*K2**2*K3 + 2112*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1760*K1**3*K3 + 384*K1**2*K2**5 - 1664*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 4032*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 1792*K1**2*K2**2*K4 - 12352*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 96*K1**2*K2*K4**2 - 1856*K1**2*K2*K4 + 10000*K1**2*K2 - 1904*K1**2*K3**2 - 32*K1**2*K3*K5 - 416*K1**2*K4**2 - 1692*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 2336*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 2016*K1*K2**2*K3 - 608*K1*K2**2*K5 + 192*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 960*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8128*K1*K2*K3 - 32*K1*K2*K4*K5 + 1448*K1*K3*K4 + 224*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1088*K2**6 - 192*K2**4*K3**2 - 192*K2**4*K4**2 + 1728*K2**4*K4 - 3560*K2**4 + 32*K2**3*K3*K5 - 288*K2**3*K6 + 192*K2**2*K3**2*K4 - 1296*K2**2*K3**2 - 32*K2**2*K3*K7 + 64*K2**2*K4**3 - 632*K2**2*K4**2 + 2256*K2**2*K4 - 974*K2**2 + 592*K2*K3*K5 + 168*K2*K4*K6 - 64*K3**4 - 48*K3**2*K4**2 + 16*K3**2*K6 - 1016*K3**2 + 8*K3*K4*K7 - 8*K4**4 - 334*K4**2 - 12*K5**2 - 2*K6**2 + 2732
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.405']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.315', 'vk6.353', 'vk6.426', 'vk6.705', 'vk6.750', 'vk6.830', 'vk6.871', 'vk6.1495', 'vk6.1576', 'vk6.1942', 'vk6.1980', 'vk6.2049', 'vk6.2480', 'vk6.2653', 'vk6.2723', 'vk6.3115', 'vk6.10254', 'vk6.10397', 'vk6.18314', 'vk6.18652', 'vk6.19410', 'vk6.19705', 'vk6.25202', 'vk6.25872', 'vk6.26190', 'vk6.36927', 'vk6.37390', 'vk6.37979', 'vk6.38037', 'vk6.44859', 'vk6.56104', 'vk6.65747']
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 O1O2O3O4O5U4U5O6U1U3U6U2
R3 orbit {'O1O2O3O4O5U4U5O6U1U3U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U6U3U5O6U1U2
Gauss code of K* O1O2O3O4U1U4U2U5U6O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U5U6U3U1U4
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 1 0 -1 1 2],[ 3 0 3 1 -1 1 2],[-1 -3 0 -1 -1 1 1],[ 0 -1 1 0 -1 1 1],[ 1 1 1 1 0 1 0],[-1 -1 -1 -1 -1 0 0],[-2 -2 -1 -1 0 0 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 -1 -1 0 -2],[-1 0 0 -1 -1 -1 -1],[-1 1 1 0 -1 -1 -3],[ 0 1 1 1 0 -1 -1],[ 1 0 1 1 1 0 1],[ 3 2 1 3 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,1,1,0,2,1,1,1,1,1,1,3,1,1,-1]
Phi over symmetry [-3,-1,0,1,1,2,-1,1,1,3,2,1,1,1,0,1,1,1,-1,0,1]
Phi of -K [-3,-1,0,1,1,2,3,2,1,3,3,0,1,1,3,0,0,1,-1,0,1]
Phi of K* [-2,-1,-1,0,1,3,0,1,1,3,3,1,0,1,1,0,1,3,0,2,3]
Phi of -K* [-3,-1,0,1,1,2,-1,1,1,3,2,1,1,1,0,1,1,1,-1,0,1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 6z^2+25z+27
Enhanced Jones-Krushkal polynomial -2w^4z^2+8w^3z^2+25w^2z+27w
Inner characteristic polynomial t^6+24t^4+23t^2
Outer characteristic polynomial t^7+40t^5+58t^3+9t
Flat arrow polynomial -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 2*K1**2 - 6*K1*K2 - 3*K1 - 2*K2**2 + K2 + K3 + 4
2-strand cable arrow polynomial -256*K1**6 - 576*K1**4*K2**2 + 3904*K1**4*K2 - 6320*K1**4 - 384*K1**3*K2**2*K3 + 2112*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1760*K1**3*K3 + 384*K1**2*K2**5 - 1664*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 4032*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 1792*K1**2*K2**2*K4 - 12352*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 96*K1**2*K2*K4**2 - 1856*K1**2*K2*K4 + 10000*K1**2*K2 - 1904*K1**2*K3**2 - 32*K1**2*K3*K5 - 416*K1**2*K4**2 - 1692*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 2336*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 2016*K1*K2**2*K3 - 608*K1*K2**2*K5 + 192*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 960*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8128*K1*K2*K3 - 32*K1*K2*K4*K5 + 1448*K1*K3*K4 + 224*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1088*K2**6 - 192*K2**4*K3**2 - 192*K2**4*K4**2 + 1728*K2**4*K4 - 3560*K2**4 + 32*K2**3*K3*K5 - 288*K2**3*K6 + 192*K2**2*K3**2*K4 - 1296*K2**2*K3**2 - 32*K2**2*K3*K7 + 64*K2**2*K4**3 - 632*K2**2*K4**2 + 2256*K2**2*K4 - 974*K2**2 + 592*K2*K3*K5 + 168*K2*K4*K6 - 64*K3**4 - 48*K3**2*K4**2 + 16*K3**2*K6 - 1016*K3**2 + 8*K3*K4*K7 - 8*K4**4 - 334*K4**2 - 12*K5**2 - 2*K6**2 + 2732
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{6}, {4, 5}, {2, 3}, {1}]]
If K is slice False
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