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Flat knot 6.455

Min(phi) over symmetries of the knot is: [-4,-1,1,1,1,2,1,1,4,4,3,0,1,2,2,-1,0,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.455']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 14*K1**2 - 8*K1*K2 + K1 - 2*K2**2 + 5*K2 + 3*K3 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.455', '6.486']
Outer characteristic polynomial of the knot is: t^7+80t^5+203t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.455']
2-strand cable arrow polynomial of the knot is: -192*K1**6 - 128*K1**4*K2**2 + 1024*K1**4*K2 - 3344*K1**4 + 672*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1056*K1**3*K3 - 128*K1**2*K2**4 + 576*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5552*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 544*K1**2*K2*K4 + 9392*K1**2*K2 - 1424*K1**2*K3**2 - 32*K1**2*K3*K5 - 208*K1**2*K4**2 - 5536*K1**2 + 608*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1664*K1*K2**2*K3 - 128*K1*K2**2*K5 + 192*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 480*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8480*K1*K2*K3 - 32*K1*K3**2*K5 + 2064*K1*K3*K4 + 240*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1352*K2**4 + 32*K2**3*K3*K5 - 32*K2**3*K6 + 64*K2**2*K3**2*K4 - 1344*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 288*K2**2*K4**2 + 1888*K2**2*K4 - 4538*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 928*K2*K3*K5 + 152*K2*K4*K6 - 160*K3**4 - 80*K3**2*K4**2 + 88*K3**2*K6 - 2532*K3**2 + 48*K3*K4*K7 - 8*K4**4 - 874*K4**2 - 172*K5**2 - 30*K6**2 + 4952
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.455']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11474', 'vk6.11778', 'vk6.12795', 'vk6.13131', 'vk6.17043', 'vk6.17285', 'vk6.20870', 'vk6.20937', 'vk6.22277', 'vk6.22349', 'vk6.23768', 'vk6.28344', 'vk6.31237', 'vk6.31588', 'vk6.32812', 'vk6.35547', 'vk6.35998', 'vk6.39964', 'vk6.40103', 'vk6.42037', 'vk6.42954', 'vk6.43251', 'vk6.46505', 'vk6.46625', 'vk6.52239', 'vk6.53075', 'vk6.53396', 'vk6.55451', 'vk6.58859', 'vk6.59933', 'vk6.64416', 'vk6.69727']
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 O1O2O3O4O5U6U3O6U2U4U1U5
R3 orbit {'O1O2O3O4O5U6U3O6U2U4U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U5U2U4O6U3U6
Gauss code of K* O1O2O3O4U3U1U5U2U4O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U1U3U6U4U2
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 -2 -1 1 4 -1],[ 1 0 -1 0 2 4 0],[ 2 1 0 1 2 3 1],[ 1 0 -1 0 0 1 1],[-1 -2 -2 0 0 1 -1],[-4 -4 -3 -1 -1 0 -4],[ 1 0 -1 -1 1 4 0]]
Primitive based matrix [[ 0 4 1 -1 -1 -1 -2],[-4 0 -1 -1 -4 -4 -3],[-1 1 0 0 -1 -2 -2],[ 1 1 0 0 1 0 -1],[ 1 4 1 -1 0 0 -1],[ 1 4 2 0 0 0 -1],[ 2 3 2 1 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,1,1,1,2,1,1,4,4,3,0,1,2,2,-1,0,1,0,1,1]
Phi over symmetry [-4,-1,1,1,1,2,1,1,4,4,3,0,1,2,2,-1,0,1,0,1,1]
Phi of -K [-2,-1,-1,-1,1,4,0,0,0,1,3,-1,0,2,4,0,1,1,0,1,2]
Phi of K* [-4,-1,1,1,1,2,2,1,1,4,3,0,1,2,1,0,0,0,-1,0,0]
Phi of -K* [-2,-1,-1,-1,1,4,1,1,1,2,3,-1,0,1,4,0,0,1,2,4,1]
Symmetry type of based matrix c
u-polynomial -t^4+t^2+2t
Normalized Jones-Krushkal polynomial 3z^2+22z+33
Enhanced Jones-Krushkal polynomial 3w^3z^2+22w^2z+33w
Inner characteristic polynomial t^6+56t^4+122t^2+4
Outer characteristic polynomial t^7+80t^5+203t^3+9t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 14*K1**2 - 8*K1*K2 + K1 - 2*K2**2 + 5*K2 + 3*K3 + 8
2-strand cable arrow polynomial -192*K1**6 - 128*K1**4*K2**2 + 1024*K1**4*K2 - 3344*K1**4 + 672*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1056*K1**3*K3 - 128*K1**2*K2**4 + 576*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5552*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 544*K1**2*K2*K4 + 9392*K1**2*K2 - 1424*K1**2*K3**2 - 32*K1**2*K3*K5 - 208*K1**2*K4**2 - 5536*K1**2 + 608*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1664*K1*K2**2*K3 - 128*K1*K2**2*K5 + 192*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 480*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8480*K1*K2*K3 - 32*K1*K3**2*K5 + 2064*K1*K3*K4 + 240*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1352*K2**4 + 32*K2**3*K3*K5 - 32*K2**3*K6 + 64*K2**2*K3**2*K4 - 1344*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 288*K2**2*K4**2 + 1888*K2**2*K4 - 4538*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 928*K2*K3*K5 + 152*K2*K4*K6 - 160*K3**4 - 80*K3**2*K4**2 + 88*K3**2*K6 - 2532*K3**2 + 48*K3*K4*K7 - 8*K4**4 - 874*K4**2 - 172*K5**2 - 30*K6**2 + 4952
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}]]
If K is slice False
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