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Flat knot 6.179

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,1,3,2,4,3,2,1,1,2,0,0,2,-1,1,3]
Flat knots (up to 7 crossings) with same phi are :['6.179']
Arrow polynomial of the knot is: -8*K1**4 + 4*K1**3 + 4*K1**2*K2 + 4*K1**2 - 2*K1*K2 - 2*K1 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.179', '6.278', '6.415']
Outer characteristic polynomial of the knot is: t^7+72t^5+137t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.179']
2-strand cable arrow polynomial of the knot is: -336*K1**4 - 32*K1**3*K3 - 768*K1**2*K2**6 + 2304*K1**2*K2**5 - 4544*K1**2*K2**4 + 4128*K1**2*K2**3 - 4704*K1**2*K2**2 - 64*K1**2*K2*K4 + 3680*K1**2*K2 - 48*K1**2*K3**2 - 2528*K1**2 + 768*K1*K2**5*K3 - 1664*K1*K2**4*K3 + 3104*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 704*K1*K2**2*K3 + 2672*K1*K2*K3 + 176*K1*K3*K4 - 128*K2**8 + 128*K2**6*K4 - 1888*K2**6 - 448*K2**4*K3**2 - 32*K2**4*K4**2 + 1216*K2**4*K4 - 1200*K2**4 + 128*K2**3*K3*K5 - 576*K2**2*K3**2 - 24*K2**2*K4**2 + 624*K2**2*K4 - 280*K2**2 + 64*K2*K3*K5 - 632*K3**2 - 96*K4**2 - 8*K5**2 + 1678
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.179']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4632', 'vk6.4897', 'vk6.6044', 'vk6.6561', 'vk6.8083', 'vk6.8456', 'vk6.9459', 'vk6.9838', 'vk6.20286', 'vk6.21617', 'vk6.27566', 'vk6.29130', 'vk6.38983', 'vk6.41230', 'vk6.45750', 'vk6.47445', 'vk6.48670', 'vk6.48853', 'vk6.49394', 'vk6.49639', 'vk6.50676', 'vk6.50849', 'vk6.51149', 'vk6.51372', 'vk6.57131', 'vk6.58323', 'vk6.61737', 'vk6.62878', 'vk6.66756', 'vk6.67640', 'vk6.69410', 'vk6.70134']
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 O1O2O3O4O5U1O6U5U6U2U3U4
R3 orbit {'O1O2O3O4O5U1O6U5U6U2U3U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U3U4U6U1O6U5
Gauss code of K* O1O2O3O4O5U6U3U4U5U1O6U2
Gauss code of -K* O1O2O3O4O5U4O6U5U1U2U3U6
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 -4 -1 1 3 0 1],[ 4 0 2 3 4 1 1],[ 1 -2 0 1 2 -1 1],[-1 -3 -1 0 1 -1 1],[-3 -4 -2 -1 0 -1 1],[ 0 -1 1 1 1 0 1],[-1 -1 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 3 1 1 0 -1 -4],[-3 0 1 -1 -1 -2 -4],[-1 -1 0 -1 -1 -1 -1],[-1 1 1 0 -1 -1 -3],[ 0 1 1 1 0 1 -1],[ 1 2 1 1 -1 0 -2],[ 4 4 1 3 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,0,1,4,-1,1,1,2,4,1,1,1,1,1,1,3,-1,1,2]
Phi over symmetry [-4,-1,0,1,1,3,1,3,2,4,3,2,1,1,2,0,0,2,-1,1,3]
Phi of -K [-4,-1,0,1,1,3,1,3,2,4,3,2,1,1,2,0,0,2,-1,1,3]
Phi of K* [-3,-1,-1,0,1,4,1,3,2,2,3,1,0,1,2,0,1,4,2,3,1]
Phi of -K* [-4,-1,0,1,1,3,2,1,1,3,4,-1,1,1,2,1,1,1,-1,-1,1]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t
Normalized Jones-Krushkal polynomial 5z+11
Enhanced Jones-Krushkal polynomial -6w^4z^2+6w^3z^2-10w^3z+15w^2z+11w
Inner characteristic polynomial t^6+44t^4+35t^2
Outer characteristic polynomial t^7+72t^5+137t^3+6t
Flat arrow polynomial -8*K1**4 + 4*K1**3 + 4*K1**2*K2 + 4*K1**2 - 2*K1*K2 - 2*K1 + 1
2-strand cable arrow polynomial -336*K1**4 - 32*K1**3*K3 - 768*K1**2*K2**6 + 2304*K1**2*K2**5 - 4544*K1**2*K2**4 + 4128*K1**2*K2**3 - 4704*K1**2*K2**2 - 64*K1**2*K2*K4 + 3680*K1**2*K2 - 48*K1**2*K3**2 - 2528*K1**2 + 768*K1*K2**5*K3 - 1664*K1*K2**4*K3 + 3104*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 704*K1*K2**2*K3 + 2672*K1*K2*K3 + 176*K1*K3*K4 - 128*K2**8 + 128*K2**6*K4 - 1888*K2**6 - 448*K2**4*K3**2 - 32*K2**4*K4**2 + 1216*K2**4*K4 - 1200*K2**4 + 128*K2**3*K3*K5 - 576*K2**2*K3**2 - 24*K2**2*K4**2 + 624*K2**2*K4 - 280*K2**2 + 64*K2*K3*K5 - 632*K3**2 - 96*K4**2 - 8*K5**2 + 1678
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}]]
If K is slice False
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