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Flat knot 6.1359

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,2,2,1,4,1,0,0,1,2,2,3,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1359']
Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 10*K1*K2 + 2*K1 + 6*K2 + 4*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.383', '6.922', '6.1172', '6.1356', '6.1359']
Outer characteristic polynomial of the knot is: t^7+66t^5+69t^3+14t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1359']
2-strand cable arrow polynomial of the knot is: -512*K1**4*K2**2 + 2176*K1**4*K2 - 4864*K1**4 + 1152*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1152*K1**3*K3 - 256*K1**2*K2**4 + 1664*K1**2*K2**3 + 512*K1**2*K2**2*K4 - 9440*K1**2*K2**2 - 1952*K1**2*K2*K4 + 12640*K1**2*K2 - 384*K1**2*K3**2 - 192*K1**2*K4**2 - 6968*K1**2 + 768*K1*K2**3*K3 - 1856*K1*K2**2*K3 - 896*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 10256*K1*K2*K3 - 128*K1*K2*K4*K5 + 1920*K1*K3*K4 + 576*K1*K4*K5 + 64*K1*K5*K6 - 32*K2**6 + 224*K2**4*K4 - 2048*K2**4 - 128*K2**3*K6 - 416*K2**2*K3**2 - 168*K2**2*K4**2 + 3344*K2**2*K4 - 6288*K2**2 + 880*K2*K3*K5 + 208*K2*K4*K6 - 2816*K3**2 - 1456*K4**2 - 392*K5**2 - 56*K6**2 + 6398
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1359']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4858', 'vk6.5203', 'vk6.6431', 'vk6.6857', 'vk6.8400', 'vk6.8821', 'vk6.9757', 'vk6.10051', 'vk6.20780', 'vk6.22182', 'vk6.29745', 'vk6.39822', 'vk6.46382', 'vk6.47957', 'vk6.49085', 'vk6.49919', 'vk6.51337', 'vk6.51554', 'vk6.58809', 'vk6.63273']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3U2O4O5U1U6U5O6U4U3
R3 orbit {'O1O2O3U2O4O5U1U6U5O6U4U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4O5U6U5U3O6O4U2
Gauss code of K* Same
Gauss code of -K* O1O2O3U1U4O5U6U5U3O6O4U2
Diagrammatic symmetry type +
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 1 2 -1],[ 3 0 0 4 2 1 2],[ 1 0 0 1 0 0 1],[-2 -4 -1 0 0 1 -3],[-1 -2 0 0 0 1 -2],[-2 -1 0 -1 -1 0 -2],[ 1 -2 -1 3 2 2 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 1 0 -1 -3 -4],[-2 -1 0 -1 0 -2 -1],[-1 0 1 0 0 -2 -2],[ 1 1 0 0 0 1 0],[ 1 3 2 2 -1 0 -2],[ 3 4 1 2 0 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,-1,0,1,3,4,1,0,2,1,0,2,2,-1,0,2]
Phi over symmetry [-3,-1,-1,1,2,2,0,2,2,1,4,1,0,0,1,2,2,3,1,0,-1]
Phi of -K [-3,-1,-1,1,2,2,0,2,2,1,4,1,0,0,1,2,2,3,1,0,-1]
Phi of K* [-2,-2,-1,1,1,3,-1,0,1,3,4,1,0,2,1,0,2,2,-1,0,2]
Phi of -K* [-3,-1,-1,1,2,2,0,2,2,1,4,1,0,0,1,2,2,3,1,0,-1]
Symmetry type of based matrix +
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+46t^4+39t^2+4
Outer characteristic polynomial t^7+66t^5+69t^3+14t
Flat arrow polynomial 4*K1**3 - 12*K1**2 - 10*K1*K2 + 2*K1 + 6*K2 + 4*K3 + 7
2-strand cable arrow polynomial -512*K1**4*K2**2 + 2176*K1**4*K2 - 4864*K1**4 + 1152*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1152*K1**3*K3 - 256*K1**2*K2**4 + 1664*K1**2*K2**3 + 512*K1**2*K2**2*K4 - 9440*K1**2*K2**2 - 1952*K1**2*K2*K4 + 12640*K1**2*K2 - 384*K1**2*K3**2 - 192*K1**2*K4**2 - 6968*K1**2 + 768*K1*K2**3*K3 - 1856*K1*K2**2*K3 - 896*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 10256*K1*K2*K3 - 128*K1*K2*K4*K5 + 1920*K1*K3*K4 + 576*K1*K4*K5 + 64*K1*K5*K6 - 32*K2**6 + 224*K2**4*K4 - 2048*K2**4 - 128*K2**3*K6 - 416*K2**2*K3**2 - 168*K2**2*K4**2 + 3344*K2**2*K4 - 6288*K2**2 + 880*K2*K3*K5 + 208*K2*K4*K6 - 2816*K3**2 - 1456*K4**2 - 392*K5**2 - 56*K6**2 + 6398
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{3, 6}, {2, 5}, {1, 4}]]
If K is slice False
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