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Flat knot 6.399

Min(phi) over symmetries of the knot is: [-4,-1,1,1,1,2,0,1,2,4,4,0,0,1,1,0,0,0,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.399']
Arrow polynomial of the knot is: -4*K1*K2 - 2*K1*K3 + 2*K1 + K2 + 2*K3 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.108', '6.157', '6.283', '6.399', '6.445', '6.510']
Outer characteristic polynomial of the knot is: t^7+64t^5+44t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.399']
2-strand cable arrow polynomial of the knot is: -976*K1**4 + 352*K1**3*K2*K3 + 32*K1**3*K3*K4 - 224*K1**3*K3 + 96*K1**2*K2**2*K4 - 1312*K1**2*K2**2 - 448*K1**2*K2*K4 + 2624*K1**2*K2 - 240*K1**2*K3**2 - 64*K1**2*K4**2 - 1664*K1**2 - 512*K1*K2**2*K3 - 256*K1*K2**2*K5 + 2392*K1*K2*K3 + 832*K1*K3*K4 + 208*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**4 - 32*K2**3*K6 - 64*K2**2*K3**2 - 48*K2**2*K4**2 + 632*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 1768*K2**2 + 432*K2*K3*K5 + 96*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 + 24*K3**2*K6 - 988*K3**2 - 510*K4**2 - 240*K5**2 - 48*K6**2 - 4*K7**2 - 2*K8**2 + 1710
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.399']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11498', 'vk6.11812', 'vk6.12828', 'vk6.13155', 'vk6.13396', 'vk6.13491', 'vk6.13682', 'vk6.13778', 'vk6.14211', 'vk6.14466', 'vk6.15687', 'vk6.16135', 'vk6.16761', 'vk6.16774', 'vk6.16895', 'vk6.19042', 'vk6.19301', 'vk6.19594', 'vk6.22480', 'vk6.23179', 'vk6.23276', 'vk6.23801', 'vk6.26487', 'vk6.28370', 'vk6.33147', 'vk6.33208', 'vk6.33303', 'vk6.35166', 'vk6.36046', 'vk6.40012', 'vk6.40293', 'vk6.42665', 'vk6.42680', 'vk6.44721', 'vk6.46760', 'vk6.48015', 'vk6.52253', 'vk6.53410', 'vk6.53572', 'vk6.53702']
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 O1O2O3O4O5U4U3O6U2U6U1U5
R3 orbit {'O1O2O3O4O5U4U3U1O6U2U6U5', 'O1O2O3O4O5U4U3O6U2U6U1U5'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5U1U5U6U4O6U3U2
Gauss code of K* O1O2O3O4U3U1U5U6U4O6O5U2
Gauss code of -K* O1O2O3O4U3O5O6U1U6U5U4U2
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 -1 -1 4 1],[ 1 0 -1 0 0 4 1],[ 2 1 0 0 0 4 1],[ 1 0 0 0 0 2 0],[ 1 0 0 0 0 1 0],[-4 -4 -4 -2 -1 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 4 1 -1 -1 -1 -2],[-4 0 0 -1 -2 -4 -4],[-1 0 0 0 0 -1 -1],[ 1 1 0 0 0 0 0],[ 1 2 0 0 0 0 0],[ 1 4 1 0 0 0 -1],[ 2 4 1 0 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,1,1,1,2,0,1,2,4,4,0,0,1,1,0,0,0,0,0,1]
Phi over symmetry [-4,-1,1,1,1,2,0,1,2,4,4,0,0,1,1,0,0,0,0,0,1]
Phi of -K [-2,-1,-1,-1,1,4,0,1,1,2,2,0,0,1,1,0,2,3,2,4,3]
Phi of K* [-4,-1,1,1,1,2,3,1,3,4,2,1,2,2,2,0,0,0,0,1,1]
Phi of -K* [-2,-1,-1,-1,1,4,0,0,1,1,4,0,0,0,1,0,0,2,1,4,0]
Symmetry type of based matrix c
u-polynomial -t^4+t^2+2t
Normalized Jones-Krushkal polynomial 4z^2+17z+19
Enhanced Jones-Krushkal polynomial 4w^3z^2+17w^2z+19w
Inner characteristic polynomial t^6+40t^4+15t^2
Outer characteristic polynomial t^7+64t^5+44t^3+3t
Flat arrow polynomial -4*K1*K2 - 2*K1*K3 + 2*K1 + K2 + 2*K3 + K4 + 1
2-strand cable arrow polynomial -976*K1**4 + 352*K1**3*K2*K3 + 32*K1**3*K3*K4 - 224*K1**3*K3 + 96*K1**2*K2**2*K4 - 1312*K1**2*K2**2 - 448*K1**2*K2*K4 + 2624*K1**2*K2 - 240*K1**2*K3**2 - 64*K1**2*K4**2 - 1664*K1**2 - 512*K1*K2**2*K3 - 256*K1*K2**2*K5 + 2392*K1*K2*K3 + 832*K1*K3*K4 + 208*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**4 - 32*K2**3*K6 - 64*K2**2*K3**2 - 48*K2**2*K4**2 + 632*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 1768*K2**2 + 432*K2*K3*K5 + 96*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 + 24*K3**2*K6 - 988*K3**2 - 510*K4**2 - 240*K5**2 - 48*K6**2 - 4*K7**2 - 2*K8**2 + 1710
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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