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Flat knot 6.378

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,0,3,1,3,4,1,0,1,1,0,1,2,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.378']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 6*K1*K2 - 2*K1*K3 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.271', '6.378']
Outer characteristic polynomial of the knot is: t^7+72t^5+44t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.378']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 768*K1**4*K2**2 + 1056*K1**4*K2 - 1328*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 - 384*K1**2*K2**4 + 832*K1**2*K2**3 - 2224*K1**2*K2**2 + 2232*K1**2*K2 - 176*K1**2*K3**2 - 32*K1**2*K4**2 - 1216*K1**2 + 288*K1*K2**3*K3 + 1608*K1*K2*K3 + 424*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 712*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 320*K2**2*K3**2 - 216*K2**2*K4**2 + 432*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 856*K2**2 + 304*K2*K3*K5 + 96*K2*K4*K6 + 16*K2*K5*K7 - 616*K3**2 - 344*K4**2 - 112*K5**2 - 16*K6**2 + 1462
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.378']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16549', 'vk6.16642', 'vk6.16965', 'vk6.17208', 'vk6.17531', 'vk6.17588', 'vk6.18854', 'vk6.18933', 'vk6.19199', 'vk6.19492', 'vk6.22244', 'vk6.23071', 'vk6.24134', 'vk6.25480', 'vk6.26004', 'vk6.26388', 'vk6.28306', 'vk6.34942', 'vk6.35062', 'vk6.35420', 'vk6.35849', 'vk6.35851', 'vk6.36312', 'vk6.36385', 'vk6.37581', 'vk6.39912', 'vk6.39916', 'vk6.42514', 'vk6.42625', 'vk6.43166', 'vk6.43492', 'vk6.44593', 'vk6.46466', 'vk6.54780', 'vk6.55119', 'vk6.55380', 'vk6.56547', 'vk6.59831', 'vk6.60194', 'vk6.66094']
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 O1O2O3O4O5U4U1O6U3U6U2U5
R3 orbit {'O1O2O3O4O5U4U1U2O6U3U6U5', 'O1O2O3O4O5U4U1O6U3U6U2U5'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5U1U4U6U3O6U5U2
Gauss code of K* O1O2O3O4U5U3U1U6U4O6O5U2
Gauss code of -K* O1O2O3O4U3O5O6U1U6U4U2U5
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 0 -1 -1 4 1],[ 3 0 2 1 0 4 1],[ 0 -2 0 -1 0 3 1],[ 1 -1 1 0 0 3 1],[ 1 0 0 0 0 1 0],[-4 -4 -3 -3 -1 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 4 1 0 -1 -1 -3],[-4 0 0 -3 -1 -3 -4],[-1 0 0 -1 0 -1 -1],[ 0 3 1 0 0 -1 -2],[ 1 1 0 0 0 0 0],[ 1 3 1 1 0 0 -1],[ 3 4 1 2 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,0,1,1,3,0,3,1,3,4,1,0,1,1,0,1,2,0,0,1]
Phi over symmetry [-4,-1,0,1,1,3,0,3,1,3,4,1,0,1,1,0,1,2,0,0,1]
Phi of -K [-3,-1,-1,0,1,4,1,2,1,3,3,0,0,1,2,1,2,4,0,1,3]
Phi of K* [-4,-1,0,1,1,3,3,1,2,4,3,0,1,2,3,0,1,1,0,1,2]
Phi of -K* [-3,-1,-1,0,1,4,0,1,2,1,4,0,0,0,1,1,1,3,1,3,0]
Symmetry type of based matrix c
u-polynomial -t^4+t^3+t
Normalized Jones-Krushkal polynomial 9z+19
Enhanced Jones-Krushkal polynomial -4w^3z+13w^2z+19w
Inner characteristic polynomial t^6+44t^4+12t^2
Outer characteristic polynomial t^7+72t^5+44t^3
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 6*K1*K2 - 2*K1*K3 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial 256*K1**4*K2**3 - 768*K1**4*K2**2 + 1056*K1**4*K2 - 1328*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 - 384*K1**2*K2**4 + 832*K1**2*K2**3 - 2224*K1**2*K2**2 + 2232*K1**2*K2 - 176*K1**2*K3**2 - 32*K1**2*K4**2 - 1216*K1**2 + 288*K1*K2**3*K3 + 1608*K1*K2*K3 + 424*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 712*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 320*K2**2*K3**2 - 216*K2**2*K4**2 + 432*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 856*K2**2 + 304*K2*K3*K5 + 96*K2*K4*K6 + 16*K2*K5*K7 - 616*K3**2 - 344*K4**2 - 112*K5**2 - 16*K6**2 + 1462
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}]]
If K is slice False
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