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Flat knot 6.190

Min(phi) over symmetries of the knot is: [-4,-3,1,1,2,3,0,2,3,5,3,1,2,3,2,0,1,1,1,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.190']
Arrow polynomial of the knot is: -2*K1**2 - 4*K1*K2 + 2*K1 - 2*K2**2 + K2 + 2*K3 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.83', '6.151', '6.160', '6.190', '6.247', '6.262', '6.491', '6.514']
Outer characteristic polynomial of the knot is: t^7+116t^5+112t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.190']
2-strand cable arrow polynomial of the knot is: -320*K1**3*K3 + 1280*K1**2*K2**3 + 32*K1**2*K2**2*K4 - 3664*K1**2*K2**2 - 512*K1**2*K2*K4 + 5064*K1**2*K2 - 320*K1**2*K3**2 - 80*K1**2*K4**2 - 5692*K1**2 + 448*K1*K2**3*K3 - 1216*K1*K2**2*K3 - 32*K1*K2**2*K5 - 640*K1*K2*K3*K4 + 6224*K1*K2*K3 - 32*K1*K2*K4*K5 + 1984*K1*K3*K4 + 432*K1*K4*K5 + 56*K1*K5*K6 + 8*K1*K6*K7 - 904*K2**4 - 672*K2**2*K3**2 - 48*K2**2*K4**2 + 1416*K2**2*K4 - 3692*K2**2 - 96*K2*K3**2*K4 + 704*K2*K3*K5 + 120*K2*K4*K6 - 96*K3**4 - 32*K3**2*K4**2 + 168*K3**2*K6 - 2728*K3**2 + 40*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1202*K4**2 - 292*K5**2 - 108*K6**2 - 16*K7**2 - 2*K8**2 + 4418
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.190']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73747', 'vk6.73863', 'vk6.74203', 'vk6.74825', 'vk6.75664', 'vk6.75866', 'vk6.76382', 'vk6.76880', 'vk6.78675', 'vk6.78862', 'vk6.79236', 'vk6.79713', 'vk6.80289', 'vk6.80416', 'vk6.80722', 'vk6.81075', 'vk6.81629', 'vk6.81815', 'vk6.81996', 'vk6.82324', 'vk6.82364', 'vk6.82727', 'vk6.83230', 'vk6.84249', 'vk6.84319', 'vk6.84413', 'vk6.84496', 'vk6.85652', 'vk6.86537', 'vk6.87566', 'vk6.88278', 'vk6.89408']
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 O1O2O3O4O5U2O6U1U5U4U6U3
R3 orbit {'O1O2O3O4O5U2O6U1U5U4U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U6U2U1U5O6U4
Gauss code of K* O1O2O3O4O5U1U6U5U3U2O6U4
Gauss code of -K* O1O2O3O4O5U2O6U4U3U1U6U5
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 -4 -3 2 1 1 3],[ 4 0 0 5 3 2 3],[ 3 0 0 3 2 1 2],[-2 -5 -3 0 -1 -1 2],[-1 -3 -2 1 0 0 2],[-1 -2 -1 1 0 0 1],[-3 -3 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 3 2 1 1 -3 -4],[-3 0 -2 -1 -2 -2 -3],[-2 2 0 -1 -1 -3 -5],[-1 1 1 0 0 -1 -2],[-1 2 1 0 0 -2 -3],[ 3 2 3 1 2 0 0],[ 4 3 5 2 3 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,-1,3,4,2,1,2,2,3,1,1,3,5,0,1,2,2,3,0]
Phi over symmetry [-4,-3,1,1,2,3,0,2,3,5,3,1,2,3,2,0,1,1,1,2,2]
Phi of -K [-4,-3,1,1,2,3,1,2,3,1,4,2,3,2,4,0,0,0,0,1,-1]
Phi of K* [-3,-2,-1,-1,3,4,-1,0,1,4,4,0,0,2,1,0,2,2,3,3,1]
Phi of -K* [-4,-3,1,1,2,3,0,2,3,5,3,1,2,3,2,0,1,1,1,2,2]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial -2w^4z^2+7w^3z^2-2w^3z+24w^2z+25w
Inner characteristic polynomial t^6+76t^4+15t^2
Outer characteristic polynomial t^7+116t^5+112t^3+8t
Flat arrow polynomial -2*K1**2 - 4*K1*K2 + 2*K1 - 2*K2**2 + K2 + 2*K3 + K4 + 3
2-strand cable arrow polynomial -320*K1**3*K3 + 1280*K1**2*K2**3 + 32*K1**2*K2**2*K4 - 3664*K1**2*K2**2 - 512*K1**2*K2*K4 + 5064*K1**2*K2 - 320*K1**2*K3**2 - 80*K1**2*K4**2 - 5692*K1**2 + 448*K1*K2**3*K3 - 1216*K1*K2**2*K3 - 32*K1*K2**2*K5 - 640*K1*K2*K3*K4 + 6224*K1*K2*K3 - 32*K1*K2*K4*K5 + 1984*K1*K3*K4 + 432*K1*K4*K5 + 56*K1*K5*K6 + 8*K1*K6*K7 - 904*K2**4 - 672*K2**2*K3**2 - 48*K2**2*K4**2 + 1416*K2**2*K4 - 3692*K2**2 - 96*K2*K3**2*K4 + 704*K2*K3*K5 + 120*K2*K4*K6 - 96*K3**4 - 32*K3**2*K4**2 + 168*K3**2*K6 - 2728*K3**2 + 40*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1202*K4**2 - 292*K5**2 - 108*K6**2 - 16*K7**2 - 2*K8**2 + 4418
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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