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Flat knot 6.1172

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,0,3,2,4,0,1,0,1,1,1,2,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1172']
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+60t^5+38t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1172']
2-strand cable arrow polynomial of the knot is: -64*K1**6 - 64*K1**4*K2**2 + 608*K1**4*K2 - 3024*K1**4 + 288*K1**3*K2*K3 - 768*K1**3*K3 + 256*K1**2*K2**3 - 4032*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 736*K1**2*K2*K4 + 7296*K1**2*K2 - 1328*K1**2*K3**2 - 224*K1**2*K3*K5 - 96*K1**2*K4**2 - 64*K1**2*K5**2 - 4380*K1**2 + 128*K1*K2**3*K3 - 384*K1*K2**2*K3 - 160*K1*K2**2*K5 + 64*K1*K2*K3**3 - 288*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6696*K1*K2*K3 - 64*K1*K3**2*K5 + 1976*K1*K3*K4 + 456*K1*K4*K5 + 80*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 720*K2**4 - 576*K2**2*K3**2 - 72*K2**2*K4**2 + 1192*K2**2*K4 - 3712*K2**2 + 824*K2*K3*K5 + 64*K2*K4*K6 - 64*K3**4 + 64*K3**2*K6 - 2188*K3**2 - 860*K4**2 - 328*K5**2 - 40*K6**2 + 4114
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1172']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4080', 'vk6.4111', 'vk6.5322', 'vk6.5353', 'vk6.7448', 'vk6.7479', 'vk6.8953', 'vk6.8984', 'vk6.10120', 'vk6.10291', 'vk6.10314', 'vk6.14540', 'vk6.15275', 'vk6.15402', 'vk6.15764', 'vk6.16179', 'vk6.29870', 'vk6.29901', 'vk6.33917', 'vk6.34000', 'vk6.34215', 'vk6.34385', 'vk6.48463', 'vk6.49167', 'vk6.50216', 'vk6.50245', 'vk6.51596', 'vk6.53952', 'vk6.54015', 'vk6.54168', 'vk6.54457', 'vk6.63315']
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 O1O2O3O4U1U5U4O5O6U2U6U3
R3 orbit {'O1O2O3O4U1U5U4O5O6U2U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5U3O5O6U1U6U4
Gauss code of K* O1O2O3U2U4O5O4O6U1U5U6U3
Gauss code of -K* O1O2O3U2U4O5O4O6U5U1U3U6
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 -1 2 2 -1 1],[ 3 0 2 3 1 2 1],[ 1 -2 0 2 1 0 1],[-2 -3 -2 0 1 -3 0],[-2 -1 -1 -1 0 -2 0],[ 1 -2 0 3 2 0 1],[-1 -1 -1 0 0 -1 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 1 0 -2 -3 -3],[-2 -1 0 0 -1 -2 -1],[-1 0 0 0 -1 -1 -1],[ 1 2 1 1 0 0 -2],[ 1 3 2 1 0 0 -2],[ 3 3 1 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,-1,0,2,3,3,0,1,2,1,1,1,1,0,2,2]
Phi over symmetry [-3,-1,-1,1,2,2,0,0,3,2,4,0,1,0,1,1,1,2,1,1,-1]
Phi of -K [-3,-1,-1,1,2,2,0,0,3,2,4,0,1,0,1,1,1,2,1,1,-1]
Phi of K* [-2,-2,-1,1,1,3,-1,1,1,2,4,1,0,1,2,1,1,3,0,0,0]
Phi of -K* [-3,-1,-1,1,2,2,2,2,1,1,3,0,1,1,2,1,2,3,0,0,-1]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial z^2+18z+33
Enhanced Jones-Krushkal polynomial w^3z^2+18w^2z+33w
Inner characteristic polynomial t^6+40t^4+18t^2+1
Outer characteristic polynomial t^7+60t^5+38t^3+5t
Flat arrow polynomial 4*K1**3 - 12*K1**2 - 10*K1*K2 + 2*K1 + 6*K2 + 4*K3 + 7
2-strand cable arrow polynomial -64*K1**6 - 64*K1**4*K2**2 + 608*K1**4*K2 - 3024*K1**4 + 288*K1**3*K2*K3 - 768*K1**3*K3 + 256*K1**2*K2**3 - 4032*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 736*K1**2*K2*K4 + 7296*K1**2*K2 - 1328*K1**2*K3**2 - 224*K1**2*K3*K5 - 96*K1**2*K4**2 - 64*K1**2*K5**2 - 4380*K1**2 + 128*K1*K2**3*K3 - 384*K1*K2**2*K3 - 160*K1*K2**2*K5 + 64*K1*K2*K3**3 - 288*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6696*K1*K2*K3 - 64*K1*K3**2*K5 + 1976*K1*K3*K4 + 456*K1*K4*K5 + 80*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 720*K2**4 - 576*K2**2*K3**2 - 72*K2**2*K4**2 + 1192*K2**2*K4 - 3712*K2**2 + 824*K2*K3*K5 + 64*K2*K4*K6 - 64*K3**4 + 64*K3**2*K6 - 2188*K3**2 - 860*K4**2 - 328*K5**2 - 40*K6**2 + 4114
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{6}, {2, 5}, {3, 4}, {1}]]
If K is slice False
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