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Flat knot 6.410

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,1,0,3,2,1,1,1,1,1,1,1,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.410', '7.19581']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 4*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.406', '6.410', '6.412', '6.1151', '6.1175', '6.1176']
Outer characteristic polynomial of the knot is: t^7+40t^5+64t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.410', '6.797', '7.19581']
2-strand cable arrow polynomial of the knot is: -512*K1**6 + 1024*K1**4*K2**3 - 2944*K1**4*K2**2 + 3072*K1**4*K2 - 2944*K1**4 - 128*K1**3*K2**2*K3 + 1088*K1**3*K2*K3 - 448*K1**3*K3 + 1152*K1**2*K2**5 - 4992*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 6592*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 9584*K1**2*K2**2 - 960*K1**2*K2*K4 + 6328*K1**2*K2 - 128*K1**2*K3**2 - 32*K1**2*K3*K5 - 1676*K1**2 + 256*K1*K2**5*K3 - 768*K1*K2**4*K3 - 128*K1*K2**3*K3*K4 + 3904*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1792*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 512*K1*K2**2*K5 - 160*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5056*K1*K2*K3 - 32*K1*K2*K4*K5 + 360*K1*K3*K4 + 88*K1*K4*K5 + 8*K1*K5*K6 - 128*K2**8 + 256*K2**6*K4 - 1472*K2**6 - 320*K2**4*K3**2 - 192*K2**4*K4**2 + 1280*K2**4*K4 - 2880*K2**4 + 256*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 912*K2**2*K3**2 - 376*K2**2*K4**2 + 1768*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 66*K2**2 + 360*K2*K3*K5 + 112*K2*K4*K6 + 8*K2*K5*K7 - 632*K3**2 - 238*K4**2 - 60*K5**2 - 14*K6**2 + 1924
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.410']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.485', 'vk6.553', 'vk6.617', 'vk6.955', 'vk6.1050', 'vk6.1119', 'vk6.1642', 'vk6.1753', 'vk6.1834', 'vk6.2139', 'vk6.2234', 'vk6.2303', 'vk6.2578', 'vk6.2849', 'vk6.3055', 'vk6.3177', 'vk6.12056', 'vk6.13047', 'vk6.20484', 'vk6.20999', 'vk6.21837', 'vk6.22423', 'vk6.27877', 'vk6.28451', 'vk6.29385', 'vk6.32702', 'vk6.39315', 'vk6.40220', 'vk6.41493', 'vk6.46719', 'vk6.46857', 'vk6.57352']
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 O1O2O3O4O5U4U5O6U2U3U1U6
R3 orbit {'O1O2O3O4O5U4U5O6U2U3U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U5U3U4O6U1U2
Gauss code of K* O1O2O3O4U3U1U2U5U6O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U5U6U3U4U2
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 -1 -2 0 -1 1 3],[ 1 0 -1 1 -1 1 3],[ 2 1 0 1 -1 1 2],[ 0 -1 -1 0 -1 1 1],[ 1 1 1 1 0 1 0],[-1 -1 -1 -1 -1 0 0],[-3 -3 -2 -1 0 0 0]]
Primitive based matrix [[ 0 3 1 0 -1 -1 -2],[-3 0 0 -1 0 -3 -2],[-1 0 0 -1 -1 -1 -1],[ 0 1 1 0 -1 -1 -1],[ 1 0 1 1 0 1 1],[ 1 3 1 1 -1 0 -1],[ 2 2 1 1 -1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,1,1,2,0,1,0,3,2,1,1,1,1,1,1,1,-1,-1,1]
Phi over symmetry [-3,-1,0,1,1,2,0,1,0,3,2,1,1,1,1,1,1,1,-1,-1,1]
Phi of -K [-2,-1,-1,0,1,3,0,2,1,2,3,1,0,1,1,0,1,4,0,2,2]
Phi of K* [-3,-1,0,1,1,2,2,2,1,4,3,0,1,1,2,0,0,1,-1,0,2]
Phi of -K* [-2,-1,-1,0,1,3,-1,1,1,1,2,1,1,1,0,1,1,3,1,1,0]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 3z^2+16z+21
Enhanced Jones-Krushkal polynomial -2w^4z^2+5w^3z^2-6w^3z+22w^2z+21w
Inner characteristic polynomial t^6+24t^4+31t^2
Outer characteristic polynomial t^7+40t^5+64t^3+7t
Flat arrow polynomial -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 4*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial -512*K1**6 + 1024*K1**4*K2**3 - 2944*K1**4*K2**2 + 3072*K1**4*K2 - 2944*K1**4 - 128*K1**3*K2**2*K3 + 1088*K1**3*K2*K3 - 448*K1**3*K3 + 1152*K1**2*K2**5 - 4992*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 6592*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 9584*K1**2*K2**2 - 960*K1**2*K2*K4 + 6328*K1**2*K2 - 128*K1**2*K3**2 - 32*K1**2*K3*K5 - 1676*K1**2 + 256*K1*K2**5*K3 - 768*K1*K2**4*K3 - 128*K1*K2**3*K3*K4 + 3904*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1792*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 512*K1*K2**2*K5 - 160*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5056*K1*K2*K3 - 32*K1*K2*K4*K5 + 360*K1*K3*K4 + 88*K1*K4*K5 + 8*K1*K5*K6 - 128*K2**8 + 256*K2**6*K4 - 1472*K2**6 - 320*K2**4*K3**2 - 192*K2**4*K4**2 + 1280*K2**4*K4 - 2880*K2**4 + 256*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 912*K2**2*K3**2 - 376*K2**2*K4**2 + 1768*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 66*K2**2 + 360*K2*K3*K5 + 112*K2*K4*K6 + 8*K2*K5*K7 - 632*K3**2 - 238*K4**2 - 60*K5**2 - 14*K6**2 + 1924
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}]]
If K is slice False
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