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Flat knot 6.224

Min(phi) over symmetries of the knot is: [-4,-2,-1,1,2,4,0,2,1,4,5,0,0,1,2,0,2,3,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.224']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 4*K1*K2 - 4*K1*K3 - K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.114', '6.224', '6.363']
Outer characteristic polynomial of the knot is: t^7+107t^5+157t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.224']
2-strand cable arrow polynomial of the knot is: -272*K1**4 + 576*K1**3*K2*K3 - 640*K1**3*K3 - 128*K1**2*K2**4 + 512*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 - 4064*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 416*K1**2*K2*K4 + 5360*K1**2*K2 - 880*K1**2*K3**2 - 32*K1**2*K3*K5 - 4280*K1**2 + 2272*K1*K2**3*K3 + 416*K1*K2**2*K3*K4 - 1984*K1*K2**2*K3 - 160*K1*K2**2*K5 + 224*K1*K2*K3**3 - 544*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6664*K1*K2*K3 - 64*K1*K3**2*K5 + 1072*K1*K3*K4 + 32*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1920*K2**4 + 160*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 128*K2**2*K3**4 + 64*K2**2*K3**2*K4 + 32*K2**2*K3**2*K6 - 2496*K2**2*K3**2 - 32*K2**2*K3*K7 - 256*K2**2*K4**2 + 1872*K2**2*K4 - 16*K2**2*K5**2 - 16*K2**2*K6**2 - 2642*K2**2 + 32*K2*K3**3*K5 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1344*K2*K3*K5 + 72*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 160*K3**4 + 128*K3**2*K6 - 1908*K3**2 + 8*K3*K4*K7 - 444*K4**2 - 168*K5**2 - 30*K6**2 - 4*K7**2 - 2*K8**2 + 3228
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.224']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16341', 'vk6.16384', 'vk6.18063', 'vk6.18401', 'vk6.22680', 'vk6.22759', 'vk6.24506', 'vk6.24929', 'vk6.34620', 'vk6.34703', 'vk6.36639', 'vk6.37063', 'vk6.42311', 'vk6.42342', 'vk6.43925', 'vk6.44244', 'vk6.54604', 'vk6.54643', 'vk6.55891', 'vk6.56179', 'vk6.59090', 'vk6.59129', 'vk6.60415', 'vk6.60774', 'vk6.64633', 'vk6.64679', 'vk6.65521', 'vk6.65837', 'vk6.67990', 'vk6.68016', 'vk6.68607', 'vk6.68824']
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 O1O2O3O4O5U3O6U1U6U2U4U5
R3 orbit {'O1O2O3O4O5U3O6U1U6U2U4U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U2U4U6U5O6U3
Gauss code of K* O1O2O3O4O5U1U3U6U4U5O6U2
Gauss code of -K* O1O2O3O4O5U4O6U1U2U6U3U5
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 -1 -2 2 4 1],[ 4 0 2 0 4 5 1],[ 1 -2 0 0 2 3 0],[ 2 0 0 0 1 2 0],[-2 -4 -2 -1 0 1 0],[-4 -5 -3 -2 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 4 2 1 -1 -2 -4],[-4 0 -1 0 -3 -2 -5],[-2 1 0 0 -2 -1 -4],[-1 0 0 0 0 0 -1],[ 1 3 2 0 0 0 -2],[ 2 2 1 0 0 0 0],[ 4 5 4 1 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,-1,1,2,4,1,0,3,2,5,0,2,1,4,0,0,1,0,2,0]
Phi over symmetry [-4,-2,-1,1,2,4,0,2,1,4,5,0,0,1,2,0,2,3,0,0,1]
Phi of -K [-4,-2,-1,1,2,4,2,1,4,2,3,1,3,3,4,2,1,2,1,3,1]
Phi of K* [-4,-2,-1,1,2,4,1,3,2,4,3,1,1,3,2,2,3,4,1,1,2]
Phi of -K* [-4,-2,-1,1,2,4,0,2,1,4,5,0,0,1,2,0,2,3,0,0,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 7z^2+26z+25
Enhanced Jones-Krushkal polynomial 7w^3z^2+26w^2z+25w
Inner characteristic polynomial t^6+65t^4+49t^2+1
Outer characteristic polynomial t^7+107t^5+157t^3+6t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 4*K1*K2 - 4*K1*K3 - K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial -272*K1**4 + 576*K1**3*K2*K3 - 640*K1**3*K3 - 128*K1**2*K2**4 + 512*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 - 4064*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 416*K1**2*K2*K4 + 5360*K1**2*K2 - 880*K1**2*K3**2 - 32*K1**2*K3*K5 - 4280*K1**2 + 2272*K1*K2**3*K3 + 416*K1*K2**2*K3*K4 - 1984*K1*K2**2*K3 - 160*K1*K2**2*K5 + 224*K1*K2*K3**3 - 544*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6664*K1*K2*K3 - 64*K1*K3**2*K5 + 1072*K1*K3*K4 + 32*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1920*K2**4 + 160*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 128*K2**2*K3**4 + 64*K2**2*K3**2*K4 + 32*K2**2*K3**2*K6 - 2496*K2**2*K3**2 - 32*K2**2*K3*K7 - 256*K2**2*K4**2 + 1872*K2**2*K4 - 16*K2**2*K5**2 - 16*K2**2*K6**2 - 2642*K2**2 + 32*K2*K3**3*K5 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1344*K2*K3*K5 + 72*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 160*K3**4 + 128*K3**2*K6 - 1908*K3**2 + 8*K3*K4*K7 - 444*K4**2 - 168*K5**2 - 30*K6**2 - 4*K7**2 - 2*K8**2 + 3228
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}]]
If K is slice False
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