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Flat knot 6.853

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,1,0,3,2,0,1,1,1,0,1,1,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.853', '7.28026']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 6*K1**2 - 6*K1*K2 - 3*K1 - 2*K2**2 + 3*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.853', '6.915']
Outer characteristic polynomial of the knot is: t^7+38t^5+52t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.853', '7.28026']
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 576*K1**4*K2**2 + 2880*K1**4*K2 - 5840*K1**4 - 384*K1**3*K2**2*K3 + 1472*K1**3*K2*K3 - 1280*K1**3*K3 + 384*K1**2*K2**5 - 1664*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 4192*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 1024*K1**2*K2**2*K4 - 13216*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 96*K1**2*K2*K4**2 - 1376*K1**2*K2*K4 + 10872*K1**2*K2 - 1520*K1**2*K3**2 - 64*K1**2*K3*K5 - 256*K1**2*K4**2 - 2116*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 2464*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 2080*K1*K2**2*K3 - 512*K1*K2**2*K5 + 192*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 672*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8944*K1*K2*K3 - 32*K1*K3**2*K5 + 1344*K1*K3*K4 + 208*K1*K4*K5 + 8*K1*K5*K6 - 128*K2**8 + 256*K2**6*K4 - 1088*K2**6 - 192*K2**4*K3**2 - 192*K2**4*K4**2 + 1376*K2**4*K4 - 3832*K2**4 + 32*K2**3*K3*K5 - 128*K2**3*K6 + 192*K2**2*K3**2*K4 - 1328*K2**2*K3**2 + 64*K2**2*K4**3 - 536*K2**2*K4**2 + 2552*K2**2*K4 - 1110*K2**2 - 32*K2*K3*K4*K5 + 624*K2*K3*K5 + 144*K2*K4*K6 - 64*K3**4 - 48*K3**2*K4**2 + 16*K3**2*K6 - 1200*K3**2 + 8*K3*K4*K7 - 8*K4**4 - 402*K4**2 - 52*K5**2 - 10*K6**2 + 3040
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.853']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.312', 'vk6.349', 'vk6.424', 'vk6.701', 'vk6.746', 'vk6.832', 'vk6.873', 'vk6.1499', 'vk6.1574', 'vk6.1946', 'vk6.1983', 'vk6.2051', 'vk6.2479', 'vk6.2649', 'vk6.2725', 'vk6.3118', 'vk6.10270', 'vk6.10413', 'vk6.18311', 'vk6.18648', 'vk6.19408', 'vk6.19703', 'vk6.25199', 'vk6.25856', 'vk6.26192', 'vk6.36930', 'vk6.37394', 'vk6.37963', 'vk6.38026', 'vk6.44861', 'vk6.56101', 'vk6.65748']
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 O1O2O3O4U1U3O5O6U5U6U4U2
R3 orbit {'O1O2O3O4U1U3O5O6U5U6U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U1U5U6O5O6U2U4
Gauss code of K* O1O2O3O4U5U4U6U3O5O6U1U2
Gauss code of -K* O1O2O3O4U3U4O5O6U2U5U1U6
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 2 -1 1],[ 3 0 3 1 2 0 0],[-1 -3 0 -1 1 -1 1],[ 0 -1 1 0 1 0 0],[-2 -2 -1 -1 0 -1 1],[ 1 0 1 0 1 0 1],[-1 0 -1 0 -1 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 1 -1 -1 -1 -2],[-1 -1 0 -1 0 -1 0],[-1 1 1 0 -1 -1 -3],[ 0 1 0 1 0 0 -1],[ 1 1 1 1 0 0 0],[ 3 2 0 3 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,-1,1,1,1,2,1,0,1,0,1,1,3,0,1,0]
Phi over symmetry [-3,-1,0,1,1,2,0,1,0,3,2,0,1,1,1,0,1,1,-1,-1,1]
Phi of -K [-3,-1,0,1,1,2,2,2,1,4,3,1,1,1,2,0,1,1,-1,0,2]
Phi of K* [-2,-1,-1,0,1,3,0,2,1,2,3,1,0,1,1,1,1,4,1,2,2]
Phi of -K* [-3,-1,0,1,1,2,0,1,0,3,2,0,1,1,1,0,1,1,-1,-1,1]
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+22t^4+23t^2
Outer characteristic polynomial t^7+38t^5+52t^3+9t
Flat arrow polynomial -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 6*K1**2 - 6*K1*K2 - 3*K1 - 2*K2**2 + 3*K2 + K3 + 6
2-strand cable arrow polynomial -256*K1**6 - 576*K1**4*K2**2 + 2880*K1**4*K2 - 5840*K1**4 - 384*K1**3*K2**2*K3 + 1472*K1**3*K2*K3 - 1280*K1**3*K3 + 384*K1**2*K2**5 - 1664*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 4192*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 1024*K1**2*K2**2*K4 - 13216*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 96*K1**2*K2*K4**2 - 1376*K1**2*K2*K4 + 10872*K1**2*K2 - 1520*K1**2*K3**2 - 64*K1**2*K3*K5 - 256*K1**2*K4**2 - 2116*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 2464*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 2080*K1*K2**2*K3 - 512*K1*K2**2*K5 + 192*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 672*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8944*K1*K2*K3 - 32*K1*K3**2*K5 + 1344*K1*K3*K4 + 208*K1*K4*K5 + 8*K1*K5*K6 - 128*K2**8 + 256*K2**6*K4 - 1088*K2**6 - 192*K2**4*K3**2 - 192*K2**4*K4**2 + 1376*K2**4*K4 - 3832*K2**4 + 32*K2**3*K3*K5 - 128*K2**3*K6 + 192*K2**2*K3**2*K4 - 1328*K2**2*K3**2 + 64*K2**2*K4**3 - 536*K2**2*K4**2 + 2552*K2**2*K4 - 1110*K2**2 - 32*K2*K3*K4*K5 + 624*K2*K3*K5 + 144*K2*K4*K6 - 64*K3**4 - 48*K3**2*K4**2 + 16*K3**2*K6 - 1200*K3**2 + 8*K3*K4*K7 - 8*K4**4 - 402*K4**2 - 52*K5**2 - 10*K6**2 + 3040
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {2, 3}, {1}]]
If K is slice False
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