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Flat knot 6.942

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,1,2,0,2,0,0,1,0,0,0,1,0,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.942', '7.16077']
Arrow polynomial of the knot is: 8*K1**3 - 4*K1*K2 - 4*K1 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.134', '6.409', '6.424', '6.534', '6.942', '6.969', '6.1192', '6.1280', '6.1310', '6.1325', '6.1858', '6.1925']
Outer characteristic polynomial of the knot is: t^7+41t^5+134t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.942', '7.16077']
2-strand cable arrow polynomial of the knot is: -896*K1**2*K2**4 + 512*K1**2*K2**3 - 1888*K1**2*K2**2 + 928*K1**2*K2 - 160*K1**2 + 640*K1*K2**3*K3 + 1056*K1*K2*K3 - 704*K2**6 + 448*K2**4*K4 - 1088*K2**4 - 96*K2**2*K3**2 - 48*K2**2*K4**2 + 704*K2**2*K4 + 560*K2**2 - 96*K3**2 - 48*K4**2 + 174
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.942']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.322', 'vk6.361', 'vk6.716', 'vk6.763', 'vk6.1454', 'vk6.1513', 'vk6.1821', 'vk6.1956', 'vk6.1995', 'vk6.2269', 'vk6.2461', 'vk6.2513', 'vk6.2664', 'vk6.2778', 'vk6.3004', 'vk6.3126', 'vk6.18392', 'vk6.18732', 'vk6.24851', 'vk6.25312', 'vk6.37051', 'vk6.39745', 'vk6.42005', 'vk6.44206', 'vk6.46307', 'vk6.47886']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The 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 O1O2O3O4U5U6O5O6U4U1U2U3
R3 orbit {'O1O2O3O4U5U6O5O6U4U1U2U3', 'O1O2O3U4U5O4O5U6U2U1O6U3'}
R3 orbit length 2
Gauss code of -K O1O2O3O4U2U3U4U1O5O6U5U6
Gauss code of K* O1O2O3O4U2U3U4U1O5O6U5U6
Gauss code of -K* Same
Diagrammatic symmetry type -
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 0 2 0 -1 1],[ 2 0 1 2 0 1 3],[ 0 -1 0 1 0 -1 1],[-2 -2 -1 0 0 -3 -1],[ 0 0 0 0 0 -1 1],[ 1 -1 1 3 1 0 1],[-1 -3 -1 1 -1 -1 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -1 0 -1 -3 -2],[-1 1 0 -1 -1 -1 -3],[ 0 0 1 0 0 -1 0],[ 0 1 1 0 0 -1 -1],[ 1 3 1 1 1 0 -1],[ 2 2 3 0 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,1,0,1,3,2,1,1,1,3,0,1,0,1,1,1]
Phi over symmetry [-2,-1,0,0,1,2,0,1,2,0,2,0,0,1,0,0,0,1,0,2,0]
Phi of -K [-2,-1,0,0,1,2,0,1,2,0,2,0,0,1,0,0,0,1,0,2,0]
Phi of K* [-2,-1,0,0,1,2,0,1,2,0,2,0,0,1,0,0,0,1,0,2,0]
Phi of -K* [-2,-1,0,0,1,2,1,0,1,3,2,1,1,1,3,0,1,0,1,1,1]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial -z-1
Enhanced Jones-Krushkal polynomial -10w^3z+9w^2z-w
Inner characteristic polynomial t^6+31t^4+98t^2
Outer characteristic polynomial t^7+41t^5+134t^3
Flat arrow polynomial 8*K1**3 - 4*K1*K2 - 4*K1 + 1
2-strand cable arrow polynomial -896*K1**2*K2**4 + 512*K1**2*K2**3 - 1888*K1**2*K2**2 + 928*K1**2*K2 - 160*K1**2 + 640*K1*K2**3*K3 + 1056*K1*K2*K3 - 704*K2**6 + 448*K2**4*K4 - 1088*K2**4 - 96*K2**2*K3**2 - 48*K2**2*K4**2 + 704*K2**2*K4 + 560*K2**2 - 96*K3**2 - 48*K4**2 + 174
Genus of based matrix 0
Fillings of based matrix [[{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {4}, {1, 3}, {2}]]
If K is slice True
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