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Flat knot 6.55

Min(phi) over symmetries of the knot is: [-5,0,1,1,1,2,2,1,4,5,3,0,1,1,1,0,0,0,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.55']
Arrow polynomial of the knot is: -2*K1**2 + 4*K1*K2**2 - 6*K1*K2 - 2*K1*K4 + K1 + K2 + 3*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.16', '6.51', '6.55']
Outer characteristic polynomial of the knot is: t^7+92t^5+69t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.55']
2-strand cable arrow polynomial of the knot is: -1440*K1**4 - 448*K1**3*K3 - 2032*K1**2*K2**2 - 1120*K1**2*K2*K4 + 4600*K1**2*K2 - 192*K1**2*K3**2 - 64*K1**2*K3*K5 - 160*K1**2*K4**2 - 3356*K1**2 + 608*K1*K2**2*K3*K4 - 736*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 416*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 4672*K1*K2*K3 - 192*K1*K2*K4*K5 + 1784*K1*K3*K4 + 560*K1*K4*K5 + 72*K1*K5*K6 - 264*K2**4 + 192*K2**3*K3*K5 - 640*K2**2*K3**2 - 32*K2**2*K3*K7 - 32*K2**2*K4**4 + 64*K2**2*K4**3 + 32*K2**2*K4**2*K8 - 632*K2**2*K4**2 - 32*K2**2*K4*K8 + 1624*K2**2*K4 - 256*K2**2*K5**2 - 8*K2**2*K8**2 - 3050*K2**2 - 96*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 992*K2*K3*K5 - 32*K2*K4**2*K6 + 408*K2*K4*K6 + 96*K2*K5*K7 + 16*K2*K6*K8 + 16*K3**2*K6 - 1748*K3**2 - 16*K4**4 + 16*K4**2*K8 - 1118*K4**2 - 440*K5**2 - 86*K6**2 - 4*K8**2 + 3112
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.55']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20179', 'vk6.20187', 'vk6.20191', 'vk6.20195', 'vk6.21461', 'vk6.21469', 'vk6.27311', 'vk6.27327', 'vk6.27335', 'vk6.27343', 'vk6.28967', 'vk6.28983', 'vk6.28989', 'vk6.28993', 'vk6.38744', 'vk6.38760', 'vk6.38770', 'vk6.38778', 'vk6.40922', 'vk6.40938', 'vk6.47302', 'vk6.47310', 'vk6.47318', 'vk6.47326', 'vk6.57016', 'vk6.57020', 'vk6.57024', 'vk6.57032', 'vk6.62693', 'vk6.62701', 'vk6.70055', 'vk6.70059']
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 O1O2O3O4O5O6U1U6U4U5U3U2
R3 orbit {'O1O2O3O4O5O6U1U6U4U5U3U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U4U2U3U1U6
Gauss code of K* O1O2O3O4O5O6U1U6U5U3U4U2
Gauss code of -K* O1O2O3O4O5O6U5U3U4U2U1U6
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 -5 1 1 0 2 1],[ 5 0 5 4 2 3 1],[-1 -5 0 0 -1 1 0],[-1 -4 0 0 -1 1 0],[ 0 -2 1 1 0 1 0],[-2 -3 -1 -1 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 1 1 1 0 -5],[-2 0 0 -1 -1 -1 -3],[-1 0 0 0 0 0 -1],[-1 1 0 0 0 -1 -4],[-1 1 0 0 0 -1 -5],[ 0 1 0 1 1 0 -2],[ 5 3 1 4 5 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,-1,0,5,0,1,1,1,3,0,0,0,1,0,1,4,1,5,2]
Phi over symmetry [-5,0,1,1,1,2,2,1,4,5,3,0,1,1,1,0,0,0,0,1,1]
Phi of -K [-5,0,1,1,1,2,3,1,2,5,4,0,0,1,1,0,0,0,0,0,1]
Phi of K* [-2,-1,-1,-1,0,5,0,0,1,1,4,0,0,0,1,0,0,2,1,5,3]
Phi of -K* [-5,0,1,1,1,2,2,1,4,5,3,0,1,1,1,0,0,0,0,1,1]
Symmetry type of based matrix c
u-polynomial t^5-t^2-3t
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial 7w^3z^2+24w^2z+21w
Inner characteristic polynomial t^6+60t^4+8t^2
Outer characteristic polynomial t^7+92t^5+69t^3+4t
Flat arrow polynomial -2*K1**2 + 4*K1*K2**2 - 6*K1*K2 - 2*K1*K4 + K1 + K2 + 3*K3 + 2
2-strand cable arrow polynomial -1440*K1**4 - 448*K1**3*K3 - 2032*K1**2*K2**2 - 1120*K1**2*K2*K4 + 4600*K1**2*K2 - 192*K1**2*K3**2 - 64*K1**2*K3*K5 - 160*K1**2*K4**2 - 3356*K1**2 + 608*K1*K2**2*K3*K4 - 736*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 416*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 4672*K1*K2*K3 - 192*K1*K2*K4*K5 + 1784*K1*K3*K4 + 560*K1*K4*K5 + 72*K1*K5*K6 - 264*K2**4 + 192*K2**3*K3*K5 - 640*K2**2*K3**2 - 32*K2**2*K3*K7 - 32*K2**2*K4**4 + 64*K2**2*K4**3 + 32*K2**2*K4**2*K8 - 632*K2**2*K4**2 - 32*K2**2*K4*K8 + 1624*K2**2*K4 - 256*K2**2*K5**2 - 8*K2**2*K8**2 - 3050*K2**2 - 96*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 992*K2*K3*K5 - 32*K2*K4**2*K6 + 408*K2*K4*K6 + 96*K2*K5*K7 + 16*K2*K6*K8 + 16*K3**2*K6 - 1748*K3**2 - 16*K4**4 + 16*K4**2*K8 - 1118*K4**2 - 440*K5**2 - 86*K6**2 - 4*K8**2 + 3112
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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