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Flat knot 6.937

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,1,0,3,2,0,1,0,0,0,1,1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.937', '7.16072']
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+68t^5+213t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.937', '7.16072']
2-strand cable arrow polynomial of the knot is: -896*K1**4*K2**2 + 1152*K1**4*K2 - 2720*K1**4 + 512*K1**3*K2**3*K3 + 1024*K1**3*K2*K3 - 128*K1**3*K3 + 384*K1**2*K2**5 - 3904*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 5632*K1**2*K2**3 - 512*K1**2*K2**2*K3**2 - 11056*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 608*K1**2*K2*K4 + 7464*K1**2*K2 - 512*K1**2*K3**2 - 160*K1**2*K4**2 - 2172*K1**2 + 640*K1*K2**5*K3 - 640*K1*K2**4*K3 - 128*K1*K2**4*K5 - 128*K1*K2**3*K3*K4 + 4896*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 2240*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 416*K1*K2**2*K5 + 96*K1*K2*K3**3 - 224*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6312*K1*K2*K3 - 32*K1*K2*K4*K5 + 688*K1*K3*K4 + 168*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1472*K2**6 - 704*K2**4*K3**2 - 192*K2**4*K4**2 + 1312*K2**4*K4 - 3824*K2**4 + 320*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 + 64*K2**2*K3**2*K4 - 1680*K2**2*K3**2 - 32*K2**2*K3*K7 - 408*K2**2*K4**2 + 2328*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 26*K2**2 + 512*K2*K3*K5 + 152*K2*K4*K6 - 972*K3**2 - 338*K4**2 - 48*K5**2 - 14*K6**2 + 2360
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.937']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.477', 'vk6.542', 'vk6.579', 'vk6.943', 'vk6.1042', 'vk6.1079', 'vk6.1633', 'vk6.1740', 'vk6.1807', 'vk6.2128', 'vk6.2229', 'vk6.2263', 'vk6.2553', 'vk6.2874', 'vk6.3045', 'vk6.3173', 'vk6.20414', 'vk6.20708', 'vk6.21776', 'vk6.22152', 'vk6.27765', 'vk6.28255', 'vk6.29289', 'vk6.29680', 'vk6.39194', 'vk6.39714', 'vk6.41960', 'vk6.46279', 'vk6.57282', 'vk6.57645', 'vk6.58528', 'vk6.61948']
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 O1O2O3O4U5U6O5O6U1U4U2U3
R3 orbit {'O1O2O3O4U5U6O5O6U1U4U2U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U3U1U4O5O6U5U6
Gauss code of K* O1O2O3O4U1U3U4U2O5O6U5U6
Gauss code of -K* O1O2O3O4U5U6O5O6U3U1U2U4
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 2 4],[ 0 -2 0 1 0 -1 1],[-2 -3 -1 0 0 -3 -1],[-1 -1 0 0 0 -2 0],[ 1 -2 1 3 2 0 1],[-1 -4 -1 1 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 -1 -1 -3 -3],[-1 0 0 0 0 -2 -1],[-1 1 0 0 -1 -1 -4],[ 0 1 0 1 0 -1 -2],[ 1 3 2 1 1 0 -2],[ 3 3 1 4 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,1,1,3,3,0,0,2,1,1,1,4,1,2,2]
Phi over symmetry [-3,-1,0,1,1,2,0,1,0,3,2,0,1,0,0,0,1,1,0,0,1]
Phi of -K [-3,-1,0,1,1,2,0,1,0,3,2,0,1,0,0,0,1,1,0,0,1]
Phi of K* [-2,-1,-1,0,1,3,0,1,1,0,2,0,0,1,0,1,0,3,0,1,0]
Phi of -K* [-3,-1,0,1,1,2,2,2,1,4,3,1,2,1,3,0,1,1,0,0,1]
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+52t^4+158t^2+1
Outer characteristic polynomial t^7+68t^5+213t^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 -896*K1**4*K2**2 + 1152*K1**4*K2 - 2720*K1**4 + 512*K1**3*K2**3*K3 + 1024*K1**3*K2*K3 - 128*K1**3*K3 + 384*K1**2*K2**5 - 3904*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 5632*K1**2*K2**3 - 512*K1**2*K2**2*K3**2 - 11056*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 608*K1**2*K2*K4 + 7464*K1**2*K2 - 512*K1**2*K3**2 - 160*K1**2*K4**2 - 2172*K1**2 + 640*K1*K2**5*K3 - 640*K1*K2**4*K3 - 128*K1*K2**4*K5 - 128*K1*K2**3*K3*K4 + 4896*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 2240*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 416*K1*K2**2*K5 + 96*K1*K2*K3**3 - 224*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6312*K1*K2*K3 - 32*K1*K2*K4*K5 + 688*K1*K3*K4 + 168*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1472*K2**6 - 704*K2**4*K3**2 - 192*K2**4*K4**2 + 1312*K2**4*K4 - 3824*K2**4 + 320*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 + 64*K2**2*K3**2*K4 - 1680*K2**2*K3**2 - 32*K2**2*K3*K7 - 408*K2**2*K4**2 + 2328*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 26*K2**2 + 512*K2*K3*K5 + 152*K2*K4*K6 - 972*K3**2 - 338*K4**2 - 48*K5**2 - 14*K6**2 + 2360
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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