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Flat knot 6.29

Min(phi) over symmetries of the knot is: [-5,-2,1,2,2,2,1,5,2,3,4,3,1,2,3,1,1,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.29']
Arrow polynomial of the knot is: 4*K1*K2**2 - 4*K1*K2 + K1 - 2*K2*K3 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.25', '6.29']
Outer characteristic polynomial of the knot is: t^7+123t^5+148t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.29']
2-strand cable arrow polynomial of the knot is: -144*K1**4 + 96*K1**3*K3*K4 + 192*K1**2*K2 - 208*K1**2*K3**2 - 656*K1**2*K4**2 - 1016*K1**2 + 736*K1*K2*K3 + 96*K1*K3*K4**3 + 1576*K1*K3*K4 + 448*K1*K4*K5 - 32*K2**2*K4**4 + 64*K2**2*K4**3 - 592*K2**2*K4**2 + 752*K2**2*K4 - 1002*K2**2 + 48*K2*K3*K5 + 32*K2*K4**3*K6 + 456*K2*K4*K6 - 256*K3**2*K4**2 + 48*K3**2*K6 - 972*K3**2 + 136*K3*K4*K7 - 80*K4**4 - 8*K4**2*K6**2 - 912*K4**2 - 108*K5**2 - 118*K6**2 - 16*K7**2 + 1438
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.29']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19958', 'vk6.20095', 'vk6.21213', 'vk6.21375', 'vk6.26953', 'vk6.27160', 'vk6.28697', 'vk6.28847', 'vk6.38363', 'vk6.38564', 'vk6.40517', 'vk6.40759', 'vk6.45226', 'vk6.45457', 'vk6.47043', 'vk6.47197', 'vk6.56750', 'vk6.56908', 'vk6.57857', 'vk6.58044', 'vk6.61205', 'vk6.61437', 'vk6.62433', 'vk6.62592', 'vk6.66458', 'vk6.66612', 'vk6.67235', 'vk6.67401', 'vk6.69104', 'vk6.69260', 'vk6.69881', 'vk6.69999']
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 O1O2O3O4O5O6U1U3U6U5U4U2
R3 orbit {'O1O2O3O4O5O6U1U3U6U5U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U3U2U1U4U6
Gauss code of K* O1O2O3O4O5O6U1U6U2U5U4U3
Gauss code of -K* O1O2O3O4O5O6U4U3U2U5U1U6
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -5 1 -2 2 2 2],[ 5 0 5 1 4 3 2],[-1 -5 0 -3 1 1 1],[ 2 -1 3 0 3 2 1],[-2 -4 -1 -3 0 0 0],[-2 -3 -1 -2 0 0 0],[-2 -2 -1 -1 0 0 0]]
Primitive based matrix [[ 0 2 2 2 1 -2 -5],[-2 0 0 0 -1 -1 -2],[-2 0 0 0 -1 -2 -3],[-2 0 0 0 -1 -3 -4],[-1 1 1 1 0 -3 -5],[ 2 1 2 3 3 0 -1],[ 5 2 3 4 5 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-2,-1,2,5,0,0,1,1,2,0,1,2,3,1,3,4,3,5,1]
Phi over symmetry [-5,-2,1,2,2,2,1,5,2,3,4,3,1,2,3,1,1,1,0,0,0]
Phi of -K [-5,-2,1,2,2,2,2,1,3,4,5,0,1,2,3,0,0,0,0,0,0]
Phi of K* [-2,-2,-2,-1,2,5,0,0,0,1,3,0,0,2,4,0,3,5,0,1,2]
Phi of -K* [-5,-2,1,2,2,2,1,5,2,3,4,3,1,2,3,1,1,1,0,0,0]
Symmetry type of based matrix c
u-polynomial t^5-2t^2-t
Normalized Jones-Krushkal polynomial 2z+5
Enhanced Jones-Krushkal polynomial 4w^4z-16w^3z+14w^2z+5w
Inner characteristic polynomial t^6+81t^4+26t^2
Outer characteristic polynomial t^7+123t^5+148t^3
Flat arrow polynomial 4*K1*K2**2 - 4*K1*K2 + K1 - 2*K2*K3 + K3 + 1
2-strand cable arrow polynomial -144*K1**4 + 96*K1**3*K3*K4 + 192*K1**2*K2 - 208*K1**2*K3**2 - 656*K1**2*K4**2 - 1016*K1**2 + 736*K1*K2*K3 + 96*K1*K3*K4**3 + 1576*K1*K3*K4 + 448*K1*K4*K5 - 32*K2**2*K4**4 + 64*K2**2*K4**3 - 592*K2**2*K4**2 + 752*K2**2*K4 - 1002*K2**2 + 48*K2*K3*K5 + 32*K2*K4**3*K6 + 456*K2*K4*K6 - 256*K3**2*K4**2 + 48*K3**2*K6 - 972*K3**2 + 136*K3*K4*K7 - 80*K4**4 - 8*K4**2*K6**2 - 912*K4**2 - 108*K5**2 - 118*K6**2 - 16*K7**2 + 1438
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {5}, {3, 4}, {1}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {5}, {2, 3}, {1}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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