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Flat knot 6.723

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,0,2,4,2,0,1,1,1,0,0,-1,-1,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.723']
Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 2*K1*K2 - 2*K1 + 3*K2 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.573', '6.628', '6.703', '6.704', '6.723', '6.796', '6.1083', '6.1099', '6.1315']
Outer characteristic polynomial of the knot is: t^7+60t^5+42t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.723']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 256*K1**4*K2**3 - 640*K1**4*K2**2 + 1024*K1**4*K2 - 2448*K1**4 + 160*K1**3*K2*K3 - 416*K1**3*K3 - 1216*K1**2*K2**4 + 3232*K1**2*K2**3 - 6560*K1**2*K2**2 - 352*K1**2*K2*K4 + 6728*K1**2*K2 - 48*K1**2*K3**2 - 3116*K1**2 + 928*K1*K2**3*K3 - 480*K1*K2**2*K3 - 32*K1*K2**2*K5 + 3856*K1*K2*K3 + 120*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 1928*K2**4 - 208*K2**2*K3**2 - 8*K2**2*K4**2 + 712*K2**2*K4 - 1152*K2**2 + 24*K2*K3*K5 - 684*K3**2 - 86*K4**2 + 2460
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.723']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17008', 'vk6.17251', 'vk6.20238', 'vk6.21538', 'vk6.23416', 'vk6.23723', 'vk6.27455', 'vk6.29056', 'vk6.35489', 'vk6.35940', 'vk6.38871', 'vk6.41069', 'vk6.42918', 'vk6.43217', 'vk6.45634', 'vk6.47376', 'vk6.55193', 'vk6.55433', 'vk6.57071', 'vk6.58219', 'vk6.59576', 'vk6.59906', 'vk6.61603', 'vk6.62780', 'vk6.64992', 'vk6.65199', 'vk6.66699', 'vk6.67556', 'vk6.68276', 'vk6.68429', 'vk6.69354', 'vk6.70097']
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 O1O2O3O4U1O5U3U4U2O6U5U6
R3 orbit {'O1O2O3O4U1O5U3U4U2O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6O5U3U1U2O6U4
Gauss code of K* O1O2O3U4O5O4U6U3U1U2O6U5
Gauss code of -K* O1O2O3U4O5U2U3U1U5O6O4U6
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 0 -1 1 2 1],[ 3 0 3 1 2 3 0],[ 0 -3 0 -1 1 3 1],[ 1 -1 1 0 1 2 1],[-1 -2 -1 -1 0 1 1],[-2 -3 -3 -2 -1 0 1],[-1 0 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 1 -1 -3 -2 -3],[-1 -1 0 -1 -1 -1 0],[-1 1 1 0 -1 -1 -2],[ 0 3 1 1 0 -1 -3],[ 1 2 1 1 1 0 -1],[ 3 3 0 2 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,-1,1,3,2,3,1,1,1,0,1,1,2,1,3,1]
Phi over symmetry [-3,-1,0,1,1,2,1,0,2,4,2,0,1,1,1,0,0,-1,-1,0,2]
Phi of -K [-3,-1,0,1,1,2,1,0,2,4,2,0,1,1,1,0,0,-1,-1,0,2]
Phi of K* [-2,-1,-1,0,1,3,0,2,-1,1,2,1,0,1,2,0,1,4,0,0,1]
Phi of -K* [-3,-1,0,1,1,2,1,3,0,2,3,1,1,1,2,1,1,3,-1,-1,1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 2z^2+15z+23
Enhanced Jones-Krushkal polynomial -2w^4z^2+4w^3z^2-6w^3z+21w^2z+23w
Inner characteristic polynomial t^6+44t^4+17t^2
Outer characteristic polynomial t^7+60t^5+42t^3+7t
Flat arrow polynomial 4*K1**3 - 6*K1**2 - 2*K1*K2 - 2*K1 + 3*K2 + 4
2-strand cable arrow polynomial -128*K1**6 + 256*K1**4*K2**3 - 640*K1**4*K2**2 + 1024*K1**4*K2 - 2448*K1**4 + 160*K1**3*K2*K3 - 416*K1**3*K3 - 1216*K1**2*K2**4 + 3232*K1**2*K2**3 - 6560*K1**2*K2**2 - 352*K1**2*K2*K4 + 6728*K1**2*K2 - 48*K1**2*K3**2 - 3116*K1**2 + 928*K1*K2**3*K3 - 480*K1*K2**2*K3 - 32*K1*K2**2*K5 + 3856*K1*K2*K3 + 120*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 1928*K2**4 - 208*K2**2*K3**2 - 8*K2**2*K4**2 + 712*K2**2*K4 - 1152*K2**2 + 24*K2*K3*K5 - 684*K3**2 - 86*K4**2 + 2460
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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