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Flat knot 6.318

Min(phi) over symmetries of the knot is: [-3,-3,0,1,2,3,-1,1,1,4,4,1,0,3,2,0,1,1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.318']
Arrow polynomial of the knot is: 12*K1**3 - 10*K1**2 - 10*K1*K2 - 4*K1 + 5*K2 + 2*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.318', '6.1178', '6.1184']
Outer characteristic polynomial of the knot is: t^7+84t^5+73t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.318']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 672*K1**4*K2 - 2240*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 768*K1**3*K2*K3 - 640*K1**3*K3 - 448*K1**2*K2**4 + 1216*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 7040*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 704*K1**2*K2*K4 + 10448*K1**2*K2 - 1216*K1**2*K3**2 - 32*K1**2*K4**2 - 8144*K1**2 + 928*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1856*K1*K2**2*K3 - 224*K1*K2**2*K5 + 96*K1*K2*K3**3 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 10720*K1*K2*K3 - 64*K1*K2*K4*K5 + 2080*K1*K3*K4 + 160*K1*K4*K5 + 32*K1*K5*K6 - 96*K2**6 + 128*K2**4*K4 - 1912*K2**4 - 1152*K2**2*K3**2 - 152*K2**2*K4**2 + 2472*K2**2*K4 - 6160*K2**2 - 160*K2*K3**2*K4 + 872*K2*K3*K5 + 192*K2*K4*K6 - 64*K3**4 + 120*K3**2*K6 - 3620*K3**2 - 1146*K4**2 - 220*K5**2 - 80*K6**2 + 6752
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.318']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11239', 'vk6.11319', 'vk6.12504', 'vk6.12617', 'vk6.18213', 'vk6.18548', 'vk6.24674', 'vk6.25095', 'vk6.30913', 'vk6.31038', 'vk6.32101', 'vk6.32222', 'vk6.36807', 'vk6.37264', 'vk6.44048', 'vk6.44388', 'vk6.51989', 'vk6.52086', 'vk6.52870', 'vk6.52919', 'vk6.56018', 'vk6.56292', 'vk6.60560', 'vk6.60899', 'vk6.63641', 'vk6.63688', 'vk6.64073', 'vk6.64120', 'vk6.65683', 'vk6.65973', 'vk6.68729', 'vk6.68937']
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 O1O2O3O4O5U2U4O6U1U6U5U3
R3 orbit {'O1O2O3O4O5U2U4O6U1U6U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U1U6U5O6U2U4
Gauss code of K* O1O2O3O4U1U5U4U6U3O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U2U5U1U6U4
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 -3 2 0 3 1],[ 3 0 -1 4 1 4 1],[ 3 1 0 3 1 2 0],[-2 -4 -3 0 -1 1 0],[ 0 -1 -1 1 0 1 0],[-3 -4 -2 -1 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 3 2 1 0 -3 -3],[-3 0 -1 0 -1 -2 -4],[-2 1 0 0 -1 -3 -4],[-1 0 0 0 0 0 -1],[ 0 1 1 0 0 -1 -1],[ 3 2 3 0 1 0 1],[ 3 4 4 1 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,0,3,3,1,0,1,2,4,0,1,3,4,0,0,1,1,1,-1]
Phi over symmetry [-3,-3,0,1,2,3,-1,1,1,4,4,1,0,3,2,0,1,1,0,0,1]
Phi of -K [-3,-3,0,1,2,3,-1,2,4,2,4,2,3,1,2,1,1,2,1,2,0]
Phi of K* [-3,-2,-1,0,3,3,0,2,2,2,4,1,1,1,2,1,3,4,2,2,-1]
Phi of -K* [-3,-3,0,1,2,3,-1,1,1,4,4,1,0,3,2,0,1,1,0,0,1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 4z^2+25z+35
Enhanced Jones-Krushkal polynomial 4w^3z^2+25w^2z+35w
Inner characteristic polynomial t^6+52t^4+28t^2
Outer characteristic polynomial t^7+84t^5+73t^3+4t
Flat arrow polynomial 12*K1**3 - 10*K1**2 - 10*K1*K2 - 4*K1 + 5*K2 + 2*K3 + 6
2-strand cable arrow polynomial -192*K1**4*K2**2 + 672*K1**4*K2 - 2240*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 768*K1**3*K2*K3 - 640*K1**3*K3 - 448*K1**2*K2**4 + 1216*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 7040*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 704*K1**2*K2*K4 + 10448*K1**2*K2 - 1216*K1**2*K3**2 - 32*K1**2*K4**2 - 8144*K1**2 + 928*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1856*K1*K2**2*K3 - 224*K1*K2**2*K5 + 96*K1*K2*K3**3 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 10720*K1*K2*K3 - 64*K1*K2*K4*K5 + 2080*K1*K3*K4 + 160*K1*K4*K5 + 32*K1*K5*K6 - 96*K2**6 + 128*K2**4*K4 - 1912*K2**4 - 1152*K2**2*K3**2 - 152*K2**2*K4**2 + 2472*K2**2*K4 - 6160*K2**2 - 160*K2*K3**2*K4 + 872*K2*K3*K5 + 192*K2*K4*K6 - 64*K3**4 + 120*K3**2*K6 - 3620*K3**2 - 1146*K4**2 - 220*K5**2 - 80*K6**2 + 6752
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {5}, {4}, {1, 2}]]
If K is slice False
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