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Flat knot 6.751

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,-1,2,3,4,2,1,2,2,1,0,0,1,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.751']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 6*K1*K2 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.211', '6.557', '6.676', '6.685', '6.750', '6.751', '6.856', '6.919', '6.1093', '6.1371']
Outer characteristic polynomial of the knot is: t^7+70t^5+38t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.751']
2-strand cable arrow polynomial of the knot is: -64*K1**6 - 448*K1**4*K2**2 + 4032*K1**4*K2 - 6512*K1**4 + 512*K1**3*K2*K3 + 64*K1**3*K3*K4 - 2272*K1**3*K3 - 192*K1**2*K2**4 + 1248*K1**2*K2**3 + 224*K1**2*K2**2*K4 - 6768*K1**2*K2**2 - 992*K1**2*K2*K4 + 11864*K1**2*K2 - 880*K1**2*K3**2 - 160*K1**2*K3*K5 - 176*K1**2*K4**2 - 32*K1**2*K4*K6 - 6308*K1**2 + 128*K1*K2**3*K3 - 864*K1*K2**2*K3 - 128*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 7888*K1*K2*K3 - 32*K1*K2*K4*K5 + 2096*K1*K3*K4 + 496*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 640*K2**4 - 112*K2**2*K3**2 - 56*K2**2*K4**2 + 1080*K2**2*K4 - 4980*K2**2 + 280*K2*K3*K5 + 48*K2*K4*K6 - 2416*K3**2 - 960*K4**2 - 204*K5**2 - 20*K6**2 + 5526
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.751']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4083', 'vk6.4116', 'vk6.5325', 'vk6.5358', 'vk6.7445', 'vk6.7476', 'vk6.8948', 'vk6.8981', 'vk6.10111', 'vk6.10276', 'vk6.10301', 'vk6.14534', 'vk6.15285', 'vk6.15412', 'vk6.15754', 'vk6.16169', 'vk6.29851', 'vk6.29884', 'vk6.33923', 'vk6.34005', 'vk6.34209', 'vk6.34387', 'vk6.48467', 'vk6.49172', 'vk6.50213', 'vk6.50242', 'vk6.51587', 'vk6.53982', 'vk6.54038', 'vk6.54174', 'vk6.54483', 'vk6.63300']
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 O1O2O3O4U2O5U3U6U5O6U1U4
R3 orbit {'O1O2O3O4U2O5U3U6U5O6U1U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U4O5U6U5U2O6U3
Gauss code of K* O1O2O3U2O4O5U4U6U1U5O6U3
Gauss code of -K* O1O2O3U1O4U5U3U4U6O5O6U2
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 3 2 -1],[ 1 0 -1 0 3 2 0],[ 2 1 0 1 2 1 2],[ 1 0 -1 0 2 1 0],[-3 -3 -2 -2 0 1 -4],[-2 -2 -1 -1 -1 0 -2],[ 1 0 -2 0 4 2 0]]
Primitive based matrix [[ 0 3 2 -1 -1 -1 -2],[-3 0 1 -2 -3 -4 -2],[-2 -1 0 -1 -2 -2 -1],[ 1 2 1 0 0 0 -1],[ 1 3 2 0 0 0 -1],[ 1 4 2 0 0 0 -2],[ 2 2 1 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,1,1,1,2,-1,2,3,4,2,1,2,2,1,0,0,1,0,1,2]
Phi over symmetry [-3,-2,1,1,1,2,-1,2,3,4,2,1,2,2,1,0,0,1,0,1,2]
Phi of -K [-2,-1,-1,-1,2,3,-1,0,0,3,3,0,0,1,0,0,1,1,2,2,2]
Phi of K* [-3,-2,1,1,1,2,2,0,1,2,3,1,1,2,3,0,0,-1,0,0,0]
Phi of -K* [-2,-1,-1,-1,2,3,1,1,2,1,2,0,0,1,2,0,2,3,2,4,-1]
Symmetry type of based matrix c
u-polynomial -t^3+3t
Normalized Jones-Krushkal polynomial 3z^2+24z+37
Enhanced Jones-Krushkal polynomial 3w^3z^2+24w^2z+37w
Inner characteristic polynomial t^6+50t^4+20t^2
Outer characteristic polynomial t^7+70t^5+38t^3+4t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 6*K1*K2 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -64*K1**6 - 448*K1**4*K2**2 + 4032*K1**4*K2 - 6512*K1**4 + 512*K1**3*K2*K3 + 64*K1**3*K3*K4 - 2272*K1**3*K3 - 192*K1**2*K2**4 + 1248*K1**2*K2**3 + 224*K1**2*K2**2*K4 - 6768*K1**2*K2**2 - 992*K1**2*K2*K4 + 11864*K1**2*K2 - 880*K1**2*K3**2 - 160*K1**2*K3*K5 - 176*K1**2*K4**2 - 32*K1**2*K4*K6 - 6308*K1**2 + 128*K1*K2**3*K3 - 864*K1*K2**2*K3 - 128*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 7888*K1*K2*K3 - 32*K1*K2*K4*K5 + 2096*K1*K3*K4 + 496*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 640*K2**4 - 112*K2**2*K3**2 - 56*K2**2*K4**2 + 1080*K2**2*K4 - 4980*K2**2 + 280*K2*K3*K5 + 48*K2*K4*K6 - 2416*K3**2 - 960*K4**2 - 204*K5**2 - 20*K6**2 + 5526
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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