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Flat knot 6.501

Min(phi) over symmetries of the knot is: [-4,-1,-1,1,2,3,0,2,3,2,4,1,1,1,1,2,1,3,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.501']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 8*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 :['6.143', '6.158', '6.264', '6.282', '6.501']
Outer characteristic polynomial of the knot is: t^7+94t^5+164t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.501']
2-strand cable arrow polynomial of the knot is: -320*K1**4*K2**2 + 576*K1**4*K2 - 1152*K1**4 + 224*K1**3*K2*K3 - 128*K1**3*K3 - 640*K1**2*K2**4 + 1920*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 96*K1**2*K2**2*K4 - 5776*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 480*K1**2*K2*K4 + 5288*K1**2*K2 - 352*K1**2*K3**2 - 3028*K1**2 + 1184*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 1440*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 256*K1*K2**2*K5 - 352*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5160*K1*K2*K3 + 664*K1*K3*K4 + 40*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 224*K2**4*K4 - 1872*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 1040*K2**2*K3**2 - 344*K2**2*K4**2 + 1592*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 1730*K2**2 + 624*K2*K3*K5 + 64*K2*K4*K6 + 8*K2*K5*K7 - 1248*K3**2 - 374*K4**2 - 108*K5**2 - 14*K6**2 + 2556
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.501']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19949', 'vk6.20045', 'vk6.21196', 'vk6.21329', 'vk6.26922', 'vk6.27110', 'vk6.28678', 'vk6.28801', 'vk6.38346', 'vk6.38495', 'vk6.40488', 'vk6.40698', 'vk6.45211', 'vk6.45395', 'vk6.47036', 'vk6.47145', 'vk6.56737', 'vk6.56853', 'vk6.57840', 'vk6.57994', 'vk6.61170', 'vk6.61380', 'vk6.62412', 'vk6.62545', 'vk6.66437', 'vk6.66566', 'vk6.67209', 'vk6.67359', 'vk6.69089', 'vk6.69218', 'vk6.69872', 'vk6.69961']
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 O1O2O3O4O5U1U6U2O6U4U5U3
R3 orbit {'O1O2O3O4O5U1U6U2O6U4U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U1U2O6U4U6U5
Gauss code of K* O1O2O3U2O4O5O6U1U3U6U4U5
Gauss code of -K* O1O2O3U4O5O4O6U2U3U1U5U6
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 -1 2 1 3 -1],[ 4 0 1 4 2 3 3],[ 1 -1 0 2 0 1 1],[-2 -4 -2 0 -1 1 -2],[-1 -2 0 1 0 1 -1],[-3 -3 -1 -1 -1 0 -3],[ 1 -3 -1 2 1 3 0]]
Primitive based matrix [[ 0 3 2 1 -1 -1 -4],[-3 0 -1 -1 -1 -3 -3],[-2 1 0 -1 -2 -2 -4],[-1 1 1 0 0 -1 -2],[ 1 1 2 0 0 1 -1],[ 1 3 2 1 -1 0 -3],[ 4 3 4 2 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,1,1,4,1,1,1,3,3,1,2,2,4,0,1,2,-1,1,3]
Phi over symmetry [-4,-1,-1,1,2,3,0,2,3,2,4,1,1,1,1,2,1,3,0,1,0]
Phi of -K [-4,-1,-1,1,2,3,0,2,3,2,4,1,1,1,1,2,1,3,0,1,0]
Phi of K* [-3,-2,-1,1,1,4,0,1,1,3,4,0,1,1,2,1,2,3,-1,0,2]
Phi of -K* [-4,-1,-1,1,2,3,1,3,2,4,3,1,0,2,1,1,2,3,1,1,1]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial 5w^3z^2+22w^2z+25w
Inner characteristic polynomial t^6+62t^4+83t^2
Outer characteristic polynomial t^7+94t^5+164t^3+4t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial -320*K1**4*K2**2 + 576*K1**4*K2 - 1152*K1**4 + 224*K1**3*K2*K3 - 128*K1**3*K3 - 640*K1**2*K2**4 + 1920*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 96*K1**2*K2**2*K4 - 5776*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 480*K1**2*K2*K4 + 5288*K1**2*K2 - 352*K1**2*K3**2 - 3028*K1**2 + 1184*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 1440*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 256*K1*K2**2*K5 - 352*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5160*K1*K2*K3 + 664*K1*K3*K4 + 40*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 224*K2**4*K4 - 1872*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 1040*K2**2*K3**2 - 344*K2**2*K4**2 + 1592*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 1730*K2**2 + 624*K2*K3*K5 + 64*K2*K4*K6 + 8*K2*K5*K7 - 1248*K3**2 - 374*K4**2 - 108*K5**2 - 14*K6**2 + 2556
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}]]
If K is slice False
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