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Flat knot 6.844

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,0,2,3,0,1,1,1,0,1,2,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.844', '7.27992']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.115', '6.407', '6.413', '6.448', '6.844', '6.879', '6.888', '6.926', '6.934', '6.1140', '6.1143', '6.1161', '6.1177']
Outer characteristic polynomial of the knot is: t^7+44t^5+62t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.844', '7.27992']
2-strand cable arrow polynomial of the knot is: -1024*K1**4*K2**2 + 768*K1**4*K2 - 1088*K1**4 + 1664*K1**3*K2*K3 - 704*K1**3*K3 - 1792*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 2560*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6832*K1**2*K2**2 - 992*K1**2*K2*K4 + 4928*K1**2*K2 - 1024*K1**2*K3**2 - 32*K1**2*K4**2 - 2380*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**4*K5 + 3936*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 1664*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 736*K1*K2**2*K5 - 480*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 5936*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1056*K1*K3*K4 + 128*K1*K4*K5 + 16*K1*K5*K6 + 8*K1*K6*K7 - 128*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 2336*K2**4 + 320*K2**3*K3*K5 + 64*K2**3*K4*K6 - 2144*K2**2*K3**2 - 64*K2**2*K3*K7 - 328*K2**2*K4**2 - 32*K2**2*K4*K8 + 1584*K2**2*K4 - 80*K2**2*K5**2 - 16*K2**2*K6**2 - 742*K2**2 - 32*K2*K3**2*K4 + 1008*K2*K3*K5 + 152*K2*K4*K6 + 32*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1216*K3**2 - 288*K4**2 - 112*K5**2 - 26*K6**2 - 4*K7**2 - 2*K8**2 + 1920
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.844']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.470', 'vk6.530', 'vk6.569', 'vk6.927', 'vk6.983', 'vk6.1026', 'vk6.1071', 'vk6.1711', 'vk6.1789', 'vk6.2107', 'vk6.2213', 'vk6.2251', 'vk6.2540', 'vk6.2824', 'vk6.2856', 'vk6.3162', 'vk6.20313', 'vk6.20643', 'vk6.21652', 'vk6.22074', 'vk6.27611', 'vk6.28132', 'vk6.29159', 'vk6.39036', 'vk6.39563', 'vk6.41293', 'vk6.41793', 'vk6.46180', 'vk6.57179', 'vk6.57547', 'vk6.58386', 'vk6.66791']
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 O1O2O3O4U1U2O5O6U5U6U4U3
R3 orbit {'O1O2O3O4U1U2O5O6U5U6U4U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U1U5U6O5O6U3U4
Gauss code of K* O1O2O3O4U5U6U4U3O5O6U1U2
Gauss code of -K* O1O2O3O4U3U4O5O6U2U1U5U6
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 1 3 2 0 0],[ 1 -1 0 2 1 0 0],[-2 -3 -2 0 0 -1 1],[-2 -2 -1 0 0 -1 1],[ 1 0 0 1 1 0 1],[-1 0 0 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 0 1 -1 -1 -2],[-2 0 0 1 -1 -2 -3],[-1 -1 -1 0 -1 0 0],[ 1 1 1 1 0 0 0],[ 1 1 2 0 0 0 -1],[ 3 2 3 0 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,0,-1,1,1,2,-1,1,2,3,1,0,0,0,0,1]
Phi over symmetry [-3,-1,-1,1,2,2,0,1,0,2,3,0,1,1,1,0,1,2,-1,-1,0]
Phi of -K [-3,-1,-1,1,2,2,1,2,4,2,3,0,2,1,2,1,2,2,2,2,0]
Phi of K* [-2,-2,-1,1,1,3,0,2,1,2,2,2,2,2,3,2,1,4,0,1,2]
Phi of -K* [-3,-1,-1,1,2,2,0,1,0,2,3,0,1,1,1,0,1,2,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial -4w^4z^2+11w^3z^2+24w^2z+21w
Inner characteristic polynomial t^6+24t^4+28t^2+1
Outer characteristic polynomial t^7+44t^5+62t^3+13t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial -1024*K1**4*K2**2 + 768*K1**4*K2 - 1088*K1**4 + 1664*K1**3*K2*K3 - 704*K1**3*K3 - 1792*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 2560*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6832*K1**2*K2**2 - 992*K1**2*K2*K4 + 4928*K1**2*K2 - 1024*K1**2*K3**2 - 32*K1**2*K4**2 - 2380*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**4*K5 + 3936*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 1664*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 736*K1*K2**2*K5 - 480*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 5936*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1056*K1*K3*K4 + 128*K1*K4*K5 + 16*K1*K5*K6 + 8*K1*K6*K7 - 128*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 2336*K2**4 + 320*K2**3*K3*K5 + 64*K2**3*K4*K6 - 2144*K2**2*K3**2 - 64*K2**2*K3*K7 - 328*K2**2*K4**2 - 32*K2**2*K4*K8 + 1584*K2**2*K4 - 80*K2**2*K5**2 - 16*K2**2*K6**2 - 742*K2**2 - 32*K2*K3**2*K4 + 1008*K2*K3*K5 + 152*K2*K4*K6 + 32*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1216*K3**2 - 288*K4**2 - 112*K5**2 - 26*K6**2 - 4*K7**2 - 2*K8**2 + 1920
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}]]
If K is slice False
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