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Flat knot 6.1158

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,-1,2,2,3,4,0,2,2,1,1,1,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1158']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 6*K1*K2 + K2 + 2*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.362', '6.624', '6.789', '6.859', '6.882', '6.975', '6.989', '6.1048', '6.1057', '6.1158']
Outer characteristic polynomial of the knot is: t^7+38t^5+107t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1158']
2-strand cable arrow polynomial of the knot is: -768*K1**2*K2**4 + 1984*K1**2*K2**3 - 4496*K1**2*K2**2 - 384*K1**2*K2*K4 + 3576*K1**2*K2 - 16*K1**2*K3**2 - 3132*K1**2 - 640*K1*K2**4*K3 + 1952*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 1280*K1*K2**2*K3 - 96*K1*K2**2*K5 - 128*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4568*K1*K2*K3 - 32*K1*K2*K4*K5 + 744*K1*K3*K4 + 48*K1*K4*K5 + 48*K1*K5*K6 - 288*K2**6 + 832*K2**4*K4 - 2296*K2**4 - 32*K2**3*K6 - 1056*K2**2*K3**2 - 536*K2**2*K4**2 + 1976*K2**2*K4 - 1752*K2**2 - 32*K2*K3**2*K4 + 344*K2*K3*K5 + 192*K2*K4*K6 - 16*K3**4 + 24*K3**2*K6 - 1476*K3**2 - 754*K4**2 - 88*K5**2 - 56*K6**2 + 2712
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1158']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11547', 'vk6.11882', 'vk6.12899', 'vk6.13203', 'vk6.20688', 'vk6.22126', 'vk6.28206', 'vk6.29629', 'vk6.31325', 'vk6.31726', 'vk6.32487', 'vk6.32898', 'vk6.39664', 'vk6.41903', 'vk6.46252', 'vk6.47857', 'vk6.52327', 'vk6.52588', 'vk6.53173', 'vk6.53468', 'vk6.57618', 'vk6.58775', 'vk6.62290', 'vk6.63223', 'vk6.63826', 'vk6.63959', 'vk6.64271', 'vk6.64467', 'vk6.67084', 'vk6.67947', 'vk6.69692', 'vk6.70373']
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 O1O2O3O4U1U4U3O5O6U2U5U6
R3 orbit {'O1O2O3O4U1U4U3O5O6U2U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U3O5O6U2U1U4
Gauss code of K* O1O2O3U4U5O6O4O5U1U6U3U2
Gauss code of -K* O1O2O3U1U2O4O5O6U5U4U3U6
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 1 1 0 2],[ 3 0 3 2 1 1 1],[ 1 -3 0 0 0 1 2],[-1 -2 0 0 0 0 0],[-1 -1 0 0 0 0 0],[ 0 -1 -1 0 0 0 1],[-2 -1 -2 0 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 0 -1 -2 -1],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 0 -2],[ 0 1 0 0 0 -1 -1],[ 1 2 0 0 1 0 -3],[ 3 1 1 2 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,0,1,2,1,0,0,0,1,0,0,2,1,1,3]
Phi over symmetry [-3,-1,0,1,1,2,-1,2,2,3,4,0,2,2,1,1,1,1,0,1,1]
Phi of -K [-3,-1,0,1,1,2,-1,2,2,3,4,0,2,2,1,1,1,1,0,1,1]
Phi of K* [-2,-1,-1,0,1,3,1,1,1,1,4,0,1,2,2,1,2,3,0,2,-1]
Phi of -K* [-3,-1,0,1,1,2,3,1,1,2,1,1,0,0,2,0,0,1,0,0,0]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 5z^2+18z+17
Enhanced Jones-Krushkal polynomial -4w^4z^2+9w^3z^2-4w^3z+22w^2z+17w
Inner characteristic polynomial t^6+22t^4+34t^2
Outer characteristic polynomial t^7+38t^5+107t^3+7t
Flat arrow polynomial 4*K1**3 - 2*K1**2 - 6*K1*K2 + K2 + 2*K3 + 2
2-strand cable arrow polynomial -768*K1**2*K2**4 + 1984*K1**2*K2**3 - 4496*K1**2*K2**2 - 384*K1**2*K2*K4 + 3576*K1**2*K2 - 16*K1**2*K3**2 - 3132*K1**2 - 640*K1*K2**4*K3 + 1952*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 1280*K1*K2**2*K3 - 96*K1*K2**2*K5 - 128*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4568*K1*K2*K3 - 32*K1*K2*K4*K5 + 744*K1*K3*K4 + 48*K1*K4*K5 + 48*K1*K5*K6 - 288*K2**6 + 832*K2**4*K4 - 2296*K2**4 - 32*K2**3*K6 - 1056*K2**2*K3**2 - 536*K2**2*K4**2 + 1976*K2**2*K4 - 1752*K2**2 - 32*K2*K3**2*K4 + 344*K2*K3*K5 + 192*K2*K4*K6 - 16*K3**4 + 24*K3**2*K6 - 1476*K3**2 - 754*K4**2 - 88*K5**2 - 56*K6**2 + 2712
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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