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Flat knot 6.1918

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,0,1,1,2,0,0,0,1,0,0,1,1,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.1918']
Arrow polynomial of the knot is: 4*K1**2*K2 - 2*K1**2 - 2*K1*K3 - 2*K2**2 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1191', '6.1918']
Outer characteristic polynomial of the knot is: t^7+25t^5+68t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1786', '6.1918']
2-strand cable arrow polynomial of the knot is: -1408*K1**2*K2**2 - 320*K1**2*K2*K4 + 2872*K1**2*K2 - 64*K1**2*K3**2 - 3176*K1**2 + 288*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 864*K1*K2**2*K3 - 32*K1*K2**2*K5 + 32*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 3576*K1*K2*K3 + 1184*K1*K3*K4 + 80*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 760*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 736*K2**2*K3**2 + 32*K2**2*K4**3 - 352*K2**2*K4**2 + 1720*K2**2*K4 - 8*K2**2*K6**2 - 2820*K2**2 - 96*K2*K3**2*K4 + 832*K2*K3*K5 - 32*K2*K4**2*K6 + 280*K2*K4*K6 + 8*K2*K6*K8 - 64*K3**4 + 112*K3**2*K6 - 1688*K3**2 - 8*K4**4 + 8*K4**2*K8 - 912*K4**2 - 208*K5**2 - 100*K6**2 - 2*K8**2 + 2848
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1918']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4228', 'vk6.4307', 'vk6.5495', 'vk6.5609', 'vk6.7600', 'vk6.7691', 'vk6.9097', 'vk6.9176', 'vk6.18381', 'vk6.18719', 'vk6.24838', 'vk6.25295', 'vk6.37030', 'vk6.37478', 'vk6.44195', 'vk6.44514', 'vk6.48540', 'vk6.48595', 'vk6.49239', 'vk6.49351', 'vk6.50327', 'vk6.50386', 'vk6.51070', 'vk6.51101', 'vk6.56154', 'vk6.56381', 'vk6.60683', 'vk6.61032', 'vk6.65822', 'vk6.66074', 'vk6.68815', 'vk6.69023']
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 O1O2O3U1U3U4O5O6O4U2U6U5
R3 orbit {'O1O2O3U1U3U4O5O6O4U2U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4U5O4O6O5U3U2U6
Gauss code of K* O1O2O3U4U1U5O4O5O6U3U2U6
Gauss code of -K* O1O2O3U2U4U3O5O6O4U1U6U5
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 -2 -1 1 2 0 0],[ 2 0 2 1 2 1 1],[ 1 -2 0 0 1 1 0],[-1 -1 0 0 -1 0 0],[-2 -2 -1 1 0 -1 0],[ 0 -1 -1 0 1 0 0],[ 0 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 1 0 -1 -1 -2],[-1 -1 0 0 0 0 -1],[ 0 0 0 0 0 0 -1],[ 0 1 0 0 0 -1 -1],[ 1 1 0 0 1 0 -2],[ 2 2 1 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,-1,0,1,1,2,0,0,0,1,0,0,1,1,1,2]
Phi over symmetry [-2,-1,0,0,1,2,-1,0,1,1,2,0,0,0,1,0,0,1,1,1,2]
Phi of -K [-2,-1,0,0,1,2,-1,1,1,2,2,0,1,2,2,0,1,1,1,2,2]
Phi of K* [-2,-1,0,0,1,2,2,1,2,2,2,1,1,2,2,0,0,1,1,1,-1]
Phi of -K* [-2,-1,0,0,1,2,2,1,1,1,2,0,1,0,1,0,0,0,0,1,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial 7w^3z^2+24w^2z+21w
Inner characteristic polynomial t^6+15t^4+28t^2+1
Outer characteristic polynomial t^7+25t^5+68t^3+5t
Flat arrow polynomial 4*K1**2*K2 - 2*K1**2 - 2*K1*K3 - 2*K2**2 + K4 + 2
2-strand cable arrow polynomial -1408*K1**2*K2**2 - 320*K1**2*K2*K4 + 2872*K1**2*K2 - 64*K1**2*K3**2 - 3176*K1**2 + 288*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 864*K1*K2**2*K3 - 32*K1*K2**2*K5 + 32*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 3576*K1*K2*K3 + 1184*K1*K3*K4 + 80*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 760*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 736*K2**2*K3**2 + 32*K2**2*K4**3 - 352*K2**2*K4**2 + 1720*K2**2*K4 - 8*K2**2*K6**2 - 2820*K2**2 - 96*K2*K3**2*K4 + 832*K2*K3*K5 - 32*K2*K4**2*K6 + 280*K2*K4*K6 + 8*K2*K6*K8 - 64*K3**4 + 112*K3**2*K6 - 1688*K3**2 - 8*K4**4 + 8*K4**2*K8 - 912*K4**2 - 208*K5**2 - 100*K6**2 - 2*K8**2 + 2848
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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