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Flat knot 6.934

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,0,2,3,0,1,0,0,0,1,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.934', '7.16069']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.115', '6.407', '6.413', '6.448', '6.844', '6.879', '6.888', '6.926', '6.934', '6.1140', '6.1143', '6.1161', '6.1177']
Outer characteristic polynomial of the knot is: t^7+84t^5+254t^3+21t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.934', '7.16069']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2 - 1280*K1**4 - 256*K1**3*K2**2*K3 + 2304*K1**3*K2*K3 - 704*K1**3*K3 - 512*K1**2*K2**4 + 832*K1**2*K2**3 - 1280*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 6928*K1**2*K2**2 + 384*K1**2*K2*K3**2 + 128*K1**2*K2*K3*K5 - 1120*K1**2*K2*K4 + 5024*K1**2*K2 - 1600*K1**2*K3**2 - 32*K1**2*K4**2 - 2124*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 4704*K1*K2**3*K3 + 864*K1*K2**2*K3*K4 - 1888*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 992*K1*K2**2*K5 + 128*K1*K2*K3**3 - 672*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6624*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1312*K1*K3*K4 + 64*K1*K4*K5 + 16*K1*K5*K6 + 8*K1*K6*K7 - 128*K2**6 - 1024*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 2336*K2**4 + 704*K2**3*K3*K5 + 64*K2**3*K4*K6 + 128*K2**2*K3**2*K4 - 2784*K2**2*K3**2 - 64*K2**2*K3*K7 - 264*K2**2*K4**2 - 32*K2**2*K4*K8 + 1664*K2**2*K4 - 80*K2**2*K5**2 - 16*K2**2*K6**2 - 918*K2**2 - 32*K2*K3**2*K4 + 1296*K2*K3*K5 + 120*K2*K4*K6 + 16*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1408*K3**2 - 288*K4**2 - 112*K5**2 - 26*K6**2 - 4*K7**2 - 2*K8**2 + 1984
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.934']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.469', 'vk6.529', 'vk6.567', 'vk6.928', 'vk6.979', 'vk6.1027', 'vk6.1069', 'vk6.1710', 'vk6.1787', 'vk6.2108', 'vk6.2214', 'vk6.2249', 'vk6.2544', 'vk6.2825', 'vk6.2858', 'vk6.3163', 'vk6.20312', 'vk6.20641', 'vk6.21651', 'vk6.22072', 'vk6.27612', 'vk6.28128', 'vk6.29160', 'vk6.39035', 'vk6.39561', 'vk6.41292', 'vk6.41791', 'vk6.46176', 'vk6.57178', 'vk6.57549', 'vk6.58385', 'vk6.66790']
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 O1O2O3O4U5U6O5O6U1U2U4U3
R3 orbit {'O1O2O3O4U5U6O5O6U1U2U4U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U1U3U4O5O6U5U6
Gauss code of K* O1O2O3O4U1U2U4U3O5O6U5U6
Gauss code of -K* O1O2O3O4U5U6O5O6U2U1U3U4
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 2 2 -1 1],[ 3 0 1 3 2 2 4],[ 1 -1 0 2 1 0 2],[-2 -3 -2 0 0 -3 -1],[-2 -2 -1 0 0 -3 -1],[ 1 -2 0 3 3 0 1],[-1 -4 -2 1 1 -1 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 0 -1 -1 -3 -2],[-2 0 0 -1 -2 -3 -3],[-1 1 1 0 -2 -1 -4],[ 1 1 2 2 0 0 -1],[ 1 3 3 1 0 0 -2],[ 3 2 3 4 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,0,1,1,3,2,1,2,3,3,2,1,4,0,1,2]
Phi over symmetry [-3,-1,-1,1,2,2,0,1,0,2,3,0,1,0,0,0,1,2,0,0,0]
Phi of -K [-3,-1,-1,1,2,2,0,1,0,2,3,0,1,0,0,0,1,2,0,0,0]
Phi of K* [-2,-2,-1,1,1,3,0,0,0,1,2,0,0,2,3,1,0,0,0,0,1]
Phi of -K* [-3,-1,-1,1,2,2,1,2,4,2,3,0,2,1,2,1,3,3,1,1,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial -4w^4z^2+11w^3z^2+24w^2z+21w
Inner characteristic polynomial t^6+64t^4+188t^2+9
Outer characteristic polynomial t^7+84t^5+254t^3+21t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial 256*K1**4*K2 - 1280*K1**4 - 256*K1**3*K2**2*K3 + 2304*K1**3*K2*K3 - 704*K1**3*K3 - 512*K1**2*K2**4 + 832*K1**2*K2**3 - 1280*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 6928*K1**2*K2**2 + 384*K1**2*K2*K3**2 + 128*K1**2*K2*K3*K5 - 1120*K1**2*K2*K4 + 5024*K1**2*K2 - 1600*K1**2*K3**2 - 32*K1**2*K4**2 - 2124*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 4704*K1*K2**3*K3 + 864*K1*K2**2*K3*K4 - 1888*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 992*K1*K2**2*K5 + 128*K1*K2*K3**3 - 672*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6624*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1312*K1*K3*K4 + 64*K1*K4*K5 + 16*K1*K5*K6 + 8*K1*K6*K7 - 128*K2**6 - 1024*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 2336*K2**4 + 704*K2**3*K3*K5 + 64*K2**3*K4*K6 + 128*K2**2*K3**2*K4 - 2784*K2**2*K3**2 - 64*K2**2*K3*K7 - 264*K2**2*K4**2 - 32*K2**2*K4*K8 + 1664*K2**2*K4 - 80*K2**2*K5**2 - 16*K2**2*K6**2 - 918*K2**2 - 32*K2*K3**2*K4 + 1296*K2*K3*K5 + 120*K2*K4*K6 + 16*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1408*K3**2 - 288*K4**2 - 112*K5**2 - 26*K6**2 - 4*K7**2 - 2*K8**2 + 1984
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}]]
If K is slice False
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