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Flat knot 6.1388

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,1,2,1,2,3,0,0,0,1,0,1,1,-1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1388']
Arrow polynomial of the knot is: 4*K1**3 - 6*K1*K2 + 2*K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.426', '6.861', '6.1388']
Outer characteristic polynomial of the knot is: t^7+36t^5+75t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1370', '6.1388']
2-strand cable arrow polynomial of the knot is: 1728*K1**4*K2 - 2816*K1**4 + 512*K1**3*K2*K3 - 384*K1**3*K3 - 128*K1**2*K2**4 + 448*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4640*K1**2*K2**2 - 640*K1**2*K2*K4 + 4480*K1**2*K2 - 640*K1**2*K3**2 - 32*K1**2*K4**2 - 1464*K1**2 + 320*K1*K2**3*K3 - 768*K1*K2**2*K3 - 576*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 4336*K1*K2*K3 + 944*K1*K3*K4 + 192*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 592*K2**4 - 96*K2**3*K6 - 160*K2**2*K3**2 - 24*K2**2*K4**2 + 1136*K2**2*K4 - 2092*K2**2 + 496*K2*K3*K5 + 64*K2*K4*K6 - 1040*K3**2 - 444*K4**2 - 168*K5**2 - 12*K6**2 + 1994
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1388']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20151', 'vk6.20161', 'vk6.21440', 'vk6.21447', 'vk6.27271', 'vk6.27289', 'vk6.28931', 'vk6.28949', 'vk6.38690', 'vk6.38710', 'vk6.40888', 'vk6.47270', 'vk6.47283', 'vk6.56984', 'vk6.56989', 'vk6.58133', 'vk6.62689', 'vk6.67461', 'vk6.70041', 'vk6.70045']
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 O1O2O3U4O5O6U5U3U2O4U1U6
R3 orbit {'O1O2O3U4O5O6U5U3U2O4U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U3O5U2U1U6O4O6U5
Gauss code of K* Same
Gauss code of -K* O1O2O3U4U3O5U2U1U6O4O6U5
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 -1 0 0 -1 -1 3],[ 1 0 1 1 -1 0 3],[ 0 -1 0 0 -1 0 2],[ 0 -1 0 0 0 0 1],[ 1 1 1 0 0 -1 2],[ 1 0 0 0 1 0 1],[-3 -3 -2 -1 -2 -1 0]]
Primitive based matrix [[ 0 3 0 0 -1 -1 -1],[-3 0 -1 -2 -1 -2 -3],[ 0 1 0 0 0 0 -1],[ 0 2 0 0 0 -1 -1],[ 1 1 0 0 0 1 0],[ 1 2 0 1 -1 0 1],[ 1 3 1 1 0 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,0,0,1,1,1,1,2,1,2,3,0,0,0,1,0,1,1,-1,0,-1]
Phi over symmetry [-3,0,0,1,1,1,1,2,1,2,3,0,0,0,1,0,1,1,-1,0,-1]
Phi of -K [-1,-1,-1,0,0,3,-1,0,1,1,3,-1,0,1,2,0,0,1,0,1,2]
Phi of K* [-3,0,0,1,1,1,1,2,1,2,3,0,0,0,1,0,1,1,-1,0,-1]
Phi of -K* [-1,-1,-1,0,0,3,-1,0,1,1,3,-1,0,1,2,0,0,1,0,1,2]
Symmetry type of based matrix +
u-polynomial -t^3+3t
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial 7w^3z^2+24w^2z+21w
Inner characteristic polynomial t^6+24t^4+43t^2+4
Outer characteristic polynomial t^7+36t^5+75t^3+8t
Flat arrow polynomial 4*K1**3 - 6*K1*K2 + 2*K3 + 1
2-strand cable arrow polynomial 1728*K1**4*K2 - 2816*K1**4 + 512*K1**3*K2*K3 - 384*K1**3*K3 - 128*K1**2*K2**4 + 448*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4640*K1**2*K2**2 - 640*K1**2*K2*K4 + 4480*K1**2*K2 - 640*K1**2*K3**2 - 32*K1**2*K4**2 - 1464*K1**2 + 320*K1*K2**3*K3 - 768*K1*K2**2*K3 - 576*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 4336*K1*K2*K3 + 944*K1*K3*K4 + 192*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 592*K2**4 - 96*K2**3*K6 - 160*K2**2*K3**2 - 24*K2**2*K4**2 + 1136*K2**2*K4 - 2092*K2**2 + 496*K2*K3*K5 + 64*K2*K4*K6 - 1040*K3**2 - 444*K4**2 - 168*K5**2 - 12*K6**2 + 1994
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{6}, {1, 5}, {4}, {2, 3}]]
If K is slice False
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