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Flat knot 6.1223

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,1,1,2,1,1,1,1,1,1,1,1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1223']
Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 8*K1*K2 + K1 + 6*K2 + 3*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.906', '6.1223', '6.1338', '6.1351', '6.1571', '6.1670', '6.1718', '6.1743', '6.1765', '6.1793', '6.1852', '6.2070']
Outer characteristic polynomial of the knot is: t^7+44t^5+40t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1223']
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 384*K1**4*K2**2 + 1280*K1**4*K2 - 3824*K1**4 + 768*K1**3*K2*K3 - 736*K1**3*K3 - 320*K1**2*K2**4 + 704*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7136*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 10728*K1**2*K2 - 1264*K1**2*K3**2 - 32*K1**2*K4**2 - 5632*K1**2 + 736*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1344*K1*K2**2*K3 - 288*K1*K2**2*K5 + 128*K1*K2*K3**3 - 224*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8400*K1*K2*K3 + 1472*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1776*K2**4 - 32*K2**3*K6 - 1056*K2**2*K3**2 - 96*K2**2*K4**2 + 1752*K2**2*K4 - 4306*K2**2 - 64*K2*K3**2*K4 + 744*K2*K3*K5 + 80*K2*K4*K6 - 32*K3**4 + 32*K3**2*K6 - 2236*K3**2 - 588*K4**2 - 116*K5**2 - 14*K6**2 + 4946
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1223']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11045', 'vk6.11123', 'vk6.12207', 'vk6.12314', 'vk6.16430', 'vk6.19228', 'vk6.19328', 'vk6.19521', 'vk6.19623', 'vk6.22739', 'vk6.22838', 'vk6.26038', 'vk6.26096', 'vk6.26422', 'vk6.26520', 'vk6.30622', 'vk6.30717', 'vk6.31924', 'vk6.34777', 'vk6.38100', 'vk6.38126', 'vk6.42394', 'vk6.44623', 'vk6.44750', 'vk6.51850', 'vk6.52712', 'vk6.52814', 'vk6.56573', 'vk6.56625', 'vk6.64722', 'vk6.66267', 'vk6.66290']
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 O1O2O3O4U3U1U5O6O5U2U4U6
R3 orbit {'O1O2O3O4U3U1U5O6O5U2U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U1U3O6O5U6U4U2
Gauss code of K* O1O2O3U4U3O5O6O4U2U5U1U6
Gauss code of -K* O1O2O3U4U1O4O5O6U2U6U3U5
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 1 1],[ 2 0 1 0 3 2 2],[ 1 -1 0 0 2 1 1],[ 1 0 0 0 1 1 1],[-2 -3 -2 -1 0 -2 0],[-1 -2 -1 -1 2 0 1],[-1 -2 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -2 -1 -2 -3],[-1 0 0 -1 -1 -1 -2],[-1 2 1 0 -1 -1 -2],[ 1 1 1 1 0 0 0],[ 1 2 1 1 0 0 -1],[ 2 3 2 2 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,0,2,1,2,3,1,1,1,2,1,1,2,0,0,1]
Phi over symmetry [-2,-1,-1,1,1,2,-1,1,1,2,1,1,1,1,1,1,1,1,0,0,1]
Phi of -K [-2,-1,-1,1,1,2,0,1,1,1,1,0,1,1,1,1,1,2,-1,-1,1]
Phi of K* [-2,-1,-1,1,1,2,-1,1,1,2,1,1,1,1,1,1,1,1,0,0,1]
Phi of -K* [-2,-1,-1,1,1,2,0,1,2,2,3,0,1,1,1,1,1,2,-1,0,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 3z^2+23z+35
Enhanced Jones-Krushkal polynomial 3w^3z^2+23w^2z+35w
Inner characteristic polynomial t^6+32t^4+12t^2+1
Outer characteristic polynomial t^7+44t^5+40t^3+7t
Flat arrow polynomial 4*K1**3 - 12*K1**2 - 8*K1*K2 + K1 + 6*K2 + 3*K3 + 7
2-strand cable arrow polynomial -256*K1**6 - 384*K1**4*K2**2 + 1280*K1**4*K2 - 3824*K1**4 + 768*K1**3*K2*K3 - 736*K1**3*K3 - 320*K1**2*K2**4 + 704*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7136*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 10728*K1**2*K2 - 1264*K1**2*K3**2 - 32*K1**2*K4**2 - 5632*K1**2 + 736*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1344*K1*K2**2*K3 - 288*K1*K2**2*K5 + 128*K1*K2*K3**3 - 224*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8400*K1*K2*K3 + 1472*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1776*K2**4 - 32*K2**3*K6 - 1056*K2**2*K3**2 - 96*K2**2*K4**2 + 1752*K2**2*K4 - 4306*K2**2 - 64*K2*K3**2*K4 + 744*K2*K3*K5 + 80*K2*K4*K6 - 32*K3**4 + 32*K3**2*K6 - 2236*K3**2 - 588*K4**2 - 116*K5**2 - 14*K6**2 + 4946
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}]]
If K is slice False
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