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Flat knot 6.226

Min(phi) over symmetries of the knot is: [-4,-2,0,1,1,4,0,3,1,3,5,1,0,1,2,0,1,3,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.226']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 4*K1*K2 - 4*K1*K3 - K1 + 5*K2 + K3 + K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.226']
Outer characteristic polynomial of the knot is: t^7+99t^5+83t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.226']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 352*K1**4*K2 - 1552*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 384*K1**3*K2*K3 - 480*K1**3*K3 - 384*K1**2*K2**4 + 1632*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 5408*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 512*K1**2*K2*K4 + 7448*K1**2*K2 - 736*K1**2*K3**2 - 32*K1**2*K3*K5 - 64*K1**2*K4**2 - 6060*K1**2 + 1088*K1*K2**3*K3 - 1216*K1*K2**2*K3 - 32*K1*K2**2*K5 + 96*K1*K2*K3**3 - 288*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7328*K1*K2*K3 - 32*K1*K3**2*K5 + 1576*K1*K3*K4 + 320*K1*K4*K5 + 64*K1*K5*K6 + 16*K1*K6*K7 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 320*K2**4*K4 - 1752*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1152*K2**2*K3**2 - 336*K2**2*K4**2 + 1728*K2**2*K4 - 48*K2**2*K5**2 - 16*K2**2*K6**2 - 4066*K2**2 + 808*K2*K3*K5 + 296*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 48*K3**4 + 72*K3**2*K6 - 2652*K3**2 + 32*K3*K4*K7 + 8*K3*K5*K8 + 8*K4**2*K8 - 1074*K4**2 - 320*K5**2 - 142*K6**2 - 24*K7**2 - 10*K8**2 + 5130
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.226']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16521', 'vk6.16614', 'vk6.18083', 'vk6.18419', 'vk6.22948', 'vk6.23045', 'vk6.24530', 'vk6.24947', 'vk6.34919', 'vk6.35030', 'vk6.36665', 'vk6.37087', 'vk6.42486', 'vk6.42599', 'vk6.43941', 'vk6.44256', 'vk6.54748', 'vk6.54845', 'vk6.55911', 'vk6.56199', 'vk6.59208', 'vk6.59273', 'vk6.60437', 'vk6.60794', 'vk6.64758', 'vk6.64819', 'vk6.65549', 'vk6.65859', 'vk6.68052', 'vk6.68117', 'vk6.68627', 'vk6.68840']
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 O1O2O3O4O5U3O6U1U6U4U2U5
R3 orbit {'O1O2O3O4O5U3O6U1U6U4U2U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U4U2U6U5O6U3
Gauss code of K* O1O2O3O4O5U1U4U6U3U5O6U2
Gauss code of -K* O1O2O3O4O5U4O6U1U3U6U2U5
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 0 -2 1 4 1],[ 4 0 3 0 3 5 1],[ 0 -3 0 -1 1 3 0],[ 2 0 1 0 1 2 0],[-1 -3 -1 -1 0 1 0],[-4 -5 -3 -2 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 4 1 1 0 -2 -4],[-4 0 0 -1 -3 -2 -5],[-1 0 0 0 0 0 -1],[-1 1 0 0 -1 -1 -3],[ 0 3 0 1 0 -1 -3],[ 2 2 0 1 1 0 0],[ 4 5 1 3 3 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,-1,0,2,4,0,1,3,2,5,0,0,0,1,1,1,3,1,3,0]
Phi over symmetry [-4,-2,0,1,1,4,0,3,1,3,5,1,0,1,2,0,1,3,0,0,1]
Phi of -K [-4,-2,0,1,1,4,2,1,2,4,3,1,2,3,4,0,1,1,0,2,3]
Phi of K* [-4,-1,-1,0,2,4,2,3,1,4,3,0,0,2,2,1,3,4,1,1,2]
Phi of -K* [-4,-2,0,1,1,4,0,3,1,3,5,1,0,1,2,0,1,3,0,0,1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial z^2+18z+33
Enhanced Jones-Krushkal polynomial w^3z^2-4w^3z+22w^2z+33w
Inner characteristic polynomial t^6+61t^4+20t^2
Outer characteristic polynomial t^7+99t^5+83t^3+8t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 4*K1*K2 - 4*K1*K3 - K1 + 5*K2 + K3 + K4 + 5
2-strand cable arrow polynomial -192*K1**4*K2**2 + 352*K1**4*K2 - 1552*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 384*K1**3*K2*K3 - 480*K1**3*K3 - 384*K1**2*K2**4 + 1632*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 5408*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 512*K1**2*K2*K4 + 7448*K1**2*K2 - 736*K1**2*K3**2 - 32*K1**2*K3*K5 - 64*K1**2*K4**2 - 6060*K1**2 + 1088*K1*K2**3*K3 - 1216*K1*K2**2*K3 - 32*K1*K2**2*K5 + 96*K1*K2*K3**3 - 288*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7328*K1*K2*K3 - 32*K1*K3**2*K5 + 1576*K1*K3*K4 + 320*K1*K4*K5 + 64*K1*K5*K6 + 16*K1*K6*K7 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 320*K2**4*K4 - 1752*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1152*K2**2*K3**2 - 336*K2**2*K4**2 + 1728*K2**2*K4 - 48*K2**2*K5**2 - 16*K2**2*K6**2 - 4066*K2**2 + 808*K2*K3*K5 + 296*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 48*K3**4 + 72*K3**2*K6 - 2652*K3**2 + 32*K3*K4*K7 + 8*K3*K5*K8 + 8*K4**2*K8 - 1074*K4**2 - 320*K5**2 - 142*K6**2 - 24*K7**2 - 10*K8**2 + 5130
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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