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Flat knot 6.126

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,-1,1,2,3,2,1,2,2,2,-1,-1,0,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.126']
Arrow polynomial of the knot is: -2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + 2*K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.126', '6.195', '6.367', '6.438', '6.869', '6.872', '6.896', '6.1147']
Outer characteristic polynomial of the knot is: t^7+60t^5+67t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.126']
2-strand cable arrow polynomial of the knot is: 384*K1**4*K2 - 1936*K1**4 + 608*K1**3*K2*K3 + 64*K1**3*K3*K4 - 704*K1**3*K3 + 384*K1**2*K2**3 + 96*K1**2*K2**2*K4 - 3520*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 7128*K1**2*K2 - 848*K1**2*K3**2 - 128*K1**2*K4**2 - 32*K1**2*K5**2 - 32*K1**2*K6**2 - 5884*K1**2 + 256*K1*K2**3*K3 - 1344*K1*K2**2*K3 - 224*K1*K2**2*K5 - 416*K1*K2*K3*K4 + 7080*K1*K2*K3 + 2088*K1*K3*K4 + 400*K1*K4*K5 + 184*K1*K5*K6 + 96*K1*K6*K7 - 480*K2**4 - 64*K2**3*K6 - 592*K2**2*K3**2 - 32*K2**2*K3*K7 - 136*K2**2*K4**2 + 1936*K2**2*K4 - 80*K2**2*K5**2 - 48*K2**2*K6**2 - 5462*K2**2 + 1160*K2*K3*K5 + 440*K2*K4*K6 + 120*K2*K5*K7 + 32*K2*K6*K8 + 40*K3**2*K6 - 2980*K3**2 + 40*K3*K4*K7 - 1444*K4**2 - 580*K5**2 - 298*K6**2 - 92*K7**2 - 4*K8**2 + 5526
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.126']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4870', 'vk6.5213', 'vk6.6452', 'vk6.6871', 'vk6.8413', 'vk6.8832', 'vk6.9765', 'vk6.10056', 'vk6.11673', 'vk6.12024', 'vk6.13015', 'vk6.20503', 'vk6.20778', 'vk6.21872', 'vk6.27915', 'vk6.29409', 'vk6.29741', 'vk6.32658', 'vk6.32999', 'vk6.39340', 'vk6.39810', 'vk6.46370', 'vk6.47608', 'vk6.47945', 'vk6.48828', 'vk6.49097', 'vk6.51347', 'vk6.51558', 'vk6.53276', 'vk6.57364', 'vk6.64337', 'vk6.66917']
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 O1O2O3O4O5O6U3U6U5U1U4U2
R3 orbit {'O1O2O3O4O5O6U3U6U5U1U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U3U6U2U1U4
Gauss code of K* O1O2O3O4O5O6U4U6U1U5U3U2
Gauss code of -K* O1O2O3O4O5O6U5U4U2U6U1U3
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 -2 1 -3 2 1 1],[ 2 0 2 -2 2 1 1],[-1 -2 0 -3 1 1 1],[ 3 2 3 0 3 2 1],[-2 -2 -1 -3 0 0 0],[-1 -1 -1 -2 0 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 2 1 1 1 -2 -3],[-2 0 0 0 -1 -2 -3],[-1 0 0 0 -1 -1 -1],[-1 0 0 0 -1 -1 -2],[-1 1 1 1 0 -2 -3],[ 2 2 1 1 2 0 -2],[ 3 3 1 2 3 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,-1,2,3,0,0,1,2,3,0,1,1,1,1,1,2,2,3,2]
Phi over symmetry [-3,-2,1,1,1,2,-1,1,2,3,2,1,2,2,2,-1,-1,0,0,1,1]
Phi of -K [-3,-2,1,1,1,2,-1,1,2,3,2,1,2,2,2,-1,-1,0,0,1,1]
Phi of K* [-2,-1,-1,-1,2,3,0,1,1,2,2,1,1,1,1,0,2,2,2,3,-1]
Phi of -K* [-3,-2,1,1,1,2,2,1,2,3,3,1,1,2,2,0,-1,0,-1,0,1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+40t^4+25t^2+1
Outer characteristic polynomial t^7+60t^5+67t^3+11t
Flat arrow polynomial -2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + 2*K4 + 1
2-strand cable arrow polynomial 384*K1**4*K2 - 1936*K1**4 + 608*K1**3*K2*K3 + 64*K1**3*K3*K4 - 704*K1**3*K3 + 384*K1**2*K2**3 + 96*K1**2*K2**2*K4 - 3520*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 7128*K1**2*K2 - 848*K1**2*K3**2 - 128*K1**2*K4**2 - 32*K1**2*K5**2 - 32*K1**2*K6**2 - 5884*K1**2 + 256*K1*K2**3*K3 - 1344*K1*K2**2*K3 - 224*K1*K2**2*K5 - 416*K1*K2*K3*K4 + 7080*K1*K2*K3 + 2088*K1*K3*K4 + 400*K1*K4*K5 + 184*K1*K5*K6 + 96*K1*K6*K7 - 480*K2**4 - 64*K2**3*K6 - 592*K2**2*K3**2 - 32*K2**2*K3*K7 - 136*K2**2*K4**2 + 1936*K2**2*K4 - 80*K2**2*K5**2 - 48*K2**2*K6**2 - 5462*K2**2 + 1160*K2*K3*K5 + 440*K2*K4*K6 + 120*K2*K5*K7 + 32*K2*K6*K8 + 40*K3**2*K6 - 2980*K3**2 + 40*K3*K4*K7 - 1444*K4**2 - 580*K5**2 - 298*K6**2 - 92*K7**2 - 4*K8**2 + 5526
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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