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Flat knot 6.866

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,0,1,3,1,1,0,1,1,0,1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.866', '7.28075']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.121', '6.125', '6.866', '6.894', '6.936', '6.937']
Outer characteristic polynomial of the knot is: t^7+36t^5+53t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.866', '7.28075']
2-strand cable arrow polynomial of the knot is: 1024*K1**4*K2**3 - 2560*K1**4*K2**2 + 2816*K1**4*K2 - 3552*K1**4 - 128*K1**3*K2**2*K3 + 768*K1**3*K2*K3 + 128*K1**3*K3*K4 - 320*K1**3*K3 + 1152*K1**2*K2**5 - 4992*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 6720*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 10928*K1**2*K2**2 - 832*K1**2*K2*K4 + 7200*K1**2*K2 - 352*K1**2*K3**2 - 32*K1**2*K3*K5 - 128*K1**2*K4**2 - 1892*K1**2 + 256*K1*K2**5*K3 - 768*K1*K2**4*K3 - 128*K1*K2**3*K3*K4 + 3904*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 1888*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 512*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5968*K1*K2*K3 - 32*K1*K2*K4*K5 + 592*K1*K3*K4 + 120*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1472*K2**6 - 320*K2**4*K3**2 - 192*K2**4*K4**2 + 1280*K2**4*K4 - 3248*K2**4 + 256*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 1008*K2**2*K3**2 - 472*K2**2*K4**2 + 2032*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 178*K2**2 - 32*K2*K3**2*K4 + 408*K2*K3*K5 + 144*K2*K4*K6 + 8*K2*K5*K7 - 824*K3**2 - 306*K4**2 - 44*K5**2 - 6*K6**2 + 2264
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.866']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.41', 'vk6.82', 'vk6.177', 'vk6.240', 'vk6.285', 'vk6.662', 'vk6.671', 'vk6.1234', 'vk6.1323', 'vk6.1384', 'vk6.1431', 'vk6.1915', 'vk6.2369', 'vk6.2423', 'vk6.2620', 'vk6.2971', 'vk6.10081', 'vk6.10094', 'vk6.14582', 'vk6.15802', 'vk6.16209', 'vk6.17769', 'vk6.24271', 'vk6.29818', 'vk6.33405', 'vk6.33477', 'vk6.33551', 'vk6.36577', 'vk6.43689', 'vk6.53732', 'vk6.53779', 'vk6.63286']
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 O1O2O3O4U1U4O5O6U5U6U2U3
R3 orbit {'O1O2O3O4U1U4O5O6U5U6U2U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U3U5U6O5O6U1U4
Gauss code of K* O1O2O3O4U5U3U4U6O5O6U1U2
Gauss code of -K* O1O2O3O4U3U4O5O6U5U1U2U6
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 0 2 1 -1 1],[ 3 0 2 3 1 0 0],[ 0 -2 0 1 0 -1 1],[-2 -3 -1 0 0 -1 1],[-1 -1 0 0 0 0 0],[ 1 0 1 1 0 0 1],[-1 0 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 1 0 -1 -1 -3],[-1 -1 0 0 -1 -1 0],[-1 0 0 0 0 0 -1],[ 0 1 1 0 0 -1 -2],[ 1 1 1 0 1 0 0],[ 3 3 0 1 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,-1,0,1,1,3,0,1,1,0,0,0,1,1,2,0]
Phi over symmetry [-3,-1,0,1,1,2,0,2,0,1,3,1,1,0,1,1,0,1,0,-1,0]
Phi of -K [-3,-1,0,1,1,2,2,1,3,4,2,0,2,1,2,1,0,1,0,1,2]
Phi of K* [-2,-1,-1,0,1,3,1,2,1,2,2,0,1,2,3,0,1,4,0,1,2]
Phi of -K* [-3,-1,0,1,1,2,0,2,0,1,3,1,1,0,1,1,0,1,0,-1,0]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 3z^2+16z+21
Enhanced Jones-Krushkal polynomial -2w^4z^2+5w^3z^2-6w^3z+22w^2z+21w
Inner characteristic polynomial t^6+20t^4+22t^2+1
Outer characteristic polynomial t^7+36t^5+53t^3+8t
Flat arrow polynomial -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial 1024*K1**4*K2**3 - 2560*K1**4*K2**2 + 2816*K1**4*K2 - 3552*K1**4 - 128*K1**3*K2**2*K3 + 768*K1**3*K2*K3 + 128*K1**3*K3*K4 - 320*K1**3*K3 + 1152*K1**2*K2**5 - 4992*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 6720*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 10928*K1**2*K2**2 - 832*K1**2*K2*K4 + 7200*K1**2*K2 - 352*K1**2*K3**2 - 32*K1**2*K3*K5 - 128*K1**2*K4**2 - 1892*K1**2 + 256*K1*K2**5*K3 - 768*K1*K2**4*K3 - 128*K1*K2**3*K3*K4 + 3904*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 1888*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 512*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5968*K1*K2*K3 - 32*K1*K2*K4*K5 + 592*K1*K3*K4 + 120*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1472*K2**6 - 320*K2**4*K3**2 - 192*K2**4*K4**2 + 1280*K2**4*K4 - 3248*K2**4 + 256*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 1008*K2**2*K3**2 - 472*K2**2*K4**2 + 2032*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 178*K2**2 - 32*K2*K3**2*K4 + 408*K2*K3*K5 + 144*K2*K4*K6 + 8*K2*K5*K7 - 824*K3**2 - 306*K4**2 - 44*K5**2 - 6*K6**2 + 2264
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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