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Flat knot 6.1176

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,-1,1,1,3,2,0,1,0,0,0,1,1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1176', '7.24087']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 4*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.406', '6.410', '6.412', '6.1151', '6.1175', '6.1176']
Outer characteristic polynomial of the knot is: t^7+66t^5+133t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1176', '7.24087']
2-strand cable arrow polynomial of the knot is: -896*K1**4*K2**2 + 1568*K1**4*K2 - 3264*K1**4 + 736*K1**3*K2*K3 - 288*K1**3*K3 + 384*K1**2*K2**5 - 3520*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 6048*K1**2*K2**3 + 320*K1**2*K2**2*K4 - 12624*K1**2*K2**2 - 928*K1**2*K2*K4 + 8992*K1**2*K2 - 192*K1**2*K3**2 - 32*K1**2*K4**2 - 2564*K1**2 + 256*K1*K2**5*K3 - 384*K1*K2**4*K3 - 256*K1*K2**4*K5 + 3808*K1*K2**3*K3 + 320*K1*K2**2*K3*K4 - 2624*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 704*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7264*K1*K2*K3 - 32*K1*K2*K4*K5 + 528*K1*K3*K4 + 96*K1*K4*K5 + 8*K1*K5*K6 - 128*K2**8 + 256*K2**6*K4 - 1600*K2**6 - 128*K2**5*K6 - 192*K2**4*K3**2 - 64*K2**4*K4**2 + 1536*K2**4*K4 - 4096*K2**4 + 224*K2**3*K3*K5 + 64*K2**3*K4*K6 - 224*K2**3*K6 - 1104*K2**2*K3**2 - 32*K2**2*K3*K7 - 344*K2**2*K4**2 + 2896*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 562*K2**2 + 448*K2*K3*K5 + 120*K2*K4*K6 + 8*K2*K5*K7 - 972*K3**2 - 350*K4**2 - 64*K5**2 - 14*K6**2 + 2644
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1176']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.480', 'vk6.548', 'vk6.578', 'vk6.948', 'vk6.1046', 'vk6.1078', 'vk6.1637', 'vk6.1748', 'vk6.1806', 'vk6.2136', 'vk6.2233', 'vk6.2262', 'vk6.2552', 'vk6.2873', 'vk6.3040', 'vk6.3170', 'vk6.20417', 'vk6.20709', 'vk6.21782', 'vk6.22151', 'vk6.27770', 'vk6.28256', 'vk6.29293', 'vk6.29679', 'vk6.39198', 'vk6.39715', 'vk6.41959', 'vk6.46280', 'vk6.57279', 'vk6.57646', 'vk6.58520', 'vk6.61943']
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 O1O2O3O4U1U5U6O5O6U4U2U3
R3 orbit {'O1O2O3O4U1U5U6O5O6U4U2U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U3U1O5O6U5U6U4
Gauss code of K* O1O2O3U2U3O4O5O6U1U5U6U4
Gauss code of -K* O1O2O3U4U5O4O5O6U3U1U2U6
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 3 3],[ 0 -2 0 1 0 -1 1],[-2 -3 -1 0 0 -3 -1],[-1 -1 0 0 0 -2 0],[ 1 -3 1 3 2 0 1],[-1 -3 -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 -3],[ 0 1 0 1 0 -1 -2],[ 1 3 2 1 1 0 -3],[ 3 3 1 3 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,1,1,3,3,0,0,2,1,1,1,3,1,2,3]
Phi over symmetry [-3,-1,0,1,1,2,-1,1,1,3,2,0,1,0,0,0,1,1,0,0,1]
Phi of -K [-3,-1,0,1,1,2,-1,1,1,3,2,0,1,0,0,0,1,1,0,0,1]
Phi of K* [-2,-1,-1,0,1,3,0,1,1,0,2,0,0,1,1,1,0,3,0,1,-1]
Phi of -K* [-3,-1,0,1,1,2,3,2,1,3,3,1,2,1,3,0,1,1,0,0,1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 6z^2+25z+27
Enhanced Jones-Krushkal polynomial -2w^4z^2+8w^3z^2+25w^2z+27w
Inner characteristic polynomial t^6+50t^4+96t^2+1
Outer characteristic polynomial t^7+66t^5+133t^3+10t
Flat arrow polynomial -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 4*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial -896*K1**4*K2**2 + 1568*K1**4*K2 - 3264*K1**4 + 736*K1**3*K2*K3 - 288*K1**3*K3 + 384*K1**2*K2**5 - 3520*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 6048*K1**2*K2**3 + 320*K1**2*K2**2*K4 - 12624*K1**2*K2**2 - 928*K1**2*K2*K4 + 8992*K1**2*K2 - 192*K1**2*K3**2 - 32*K1**2*K4**2 - 2564*K1**2 + 256*K1*K2**5*K3 - 384*K1*K2**4*K3 - 256*K1*K2**4*K5 + 3808*K1*K2**3*K3 + 320*K1*K2**2*K3*K4 - 2624*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 704*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7264*K1*K2*K3 - 32*K1*K2*K4*K5 + 528*K1*K3*K4 + 96*K1*K4*K5 + 8*K1*K5*K6 - 128*K2**8 + 256*K2**6*K4 - 1600*K2**6 - 128*K2**5*K6 - 192*K2**4*K3**2 - 64*K2**4*K4**2 + 1536*K2**4*K4 - 4096*K2**4 + 224*K2**3*K3*K5 + 64*K2**3*K4*K6 - 224*K2**3*K6 - 1104*K2**2*K3**2 - 32*K2**2*K3*K7 - 344*K2**2*K4**2 + 2896*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 562*K2**2 + 448*K2*K3*K5 + 120*K2*K4*K6 + 8*K2*K5*K7 - 972*K3**2 - 350*K4**2 - 64*K5**2 - 14*K6**2 + 2644
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{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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