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Flat knot 6.157

Min(phi) over symmetries of the knot is: [-4,-2,-1,1,3,3,1,1,4,3,4,0,2,2,3,1,1,2,2,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.157']
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+100t^5+49t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.157']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 352*K1**4*K2 - 496*K1**4 + 384*K1**3*K2*K3 - 672*K1**3*K3 + 160*K1**2*K2**3 - 1184*K1**2*K2**2 - 448*K1**2*K2*K4 + 2368*K1**2*K2 - 208*K1**2*K3**2 - 64*K1**2*K4**2 - 1936*K1**2 + 64*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 224*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 64*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 2192*K1*K2*K3 - 64*K1*K2*K4*K5 + 680*K1*K3*K4 + 184*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**4 - 128*K2**2*K3**2 - 176*K2**2*K4**2 + 520*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 1456*K2**2 + 320*K2*K3*K5 + 136*K2*K4*K6 + 48*K2*K5*K7 + 8*K2*K6*K8 + 8*K3**2*K6 - 828*K3**2 - 446*K4**2 - 168*K5**2 - 40*K6**2 - 12*K7**2 - 2*K8**2 + 1526
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.157']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16922', 'vk6.17165', 'vk6.19965', 'vk6.20230', 'vk6.21129', 'vk6.21525', 'vk6.23310', 'vk6.26823', 'vk6.26970', 'vk6.27438', 'vk6.28605', 'vk6.29048', 'vk6.35341', 'vk6.38259', 'vk6.38378', 'vk6.38855', 'vk6.40379', 'vk6.41045', 'vk6.42830', 'vk6.45122', 'vk6.45254', 'vk6.45608', 'vk6.46982', 'vk6.47366', 'vk6.55082', 'vk6.56690', 'vk6.56763', 'vk6.57769', 'vk6.58189', 'vk6.59475', 'vk6.61085', 'vk6.61238', 'vk6.62346', 'vk6.62761', 'vk6.64919', 'vk6.66469', 'vk6.67525', 'vk6.68218', 'vk6.69120', 'vk6.69830']
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 O1O2O3O4O5U1O6U3U6U2U5U4
R3 orbit {'O1O2O3O4O5U1U2O6U3U6U5U4', 'O1O2O3O4O5U1O6U3U6U2U5U4'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5U2U1U4U6U3O6U5
Gauss code of K* O1O2O3O4O5U6U3U1U5U4O6U2
Gauss code of -K* O1O2O3O4O5U4O6U2U1U5U3U6
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 3 3 1],[ 4 0 2 1 4 3 1],[ 1 -2 0 -1 3 2 1],[ 2 -1 1 0 3 2 1],[-3 -4 -3 -3 0 0 0],[-3 -3 -2 -2 0 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 3 3 1 -1 -2 -4],[-3 0 0 0 -2 -2 -3],[-3 0 0 0 -3 -3 -4],[-1 0 0 0 -1 -1 -1],[ 1 2 3 1 0 -1 -2],[ 2 2 3 1 1 0 -1],[ 4 3 4 1 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,-1,1,2,4,0,0,2,2,3,0,3,3,4,1,1,1,1,2,1]
Phi over symmetry [-4,-2,-1,1,3,3,1,1,4,3,4,0,2,2,3,1,1,2,2,2,0]
Phi of -K [-4,-2,-1,1,3,3,1,1,4,3,4,0,2,2,3,1,1,2,2,2,0]
Phi of K* [-3,-3,-1,1,2,4,0,2,1,2,3,2,2,3,4,1,2,4,0,1,1]
Phi of -K* [-4,-2,-1,1,3,3,1,2,1,3,4,1,1,2,3,1,2,3,0,0,0]
Symmetry type of based matrix c
u-polynomial t^4-2t^3+t^2
Normalized Jones-Krushkal polynomial 4z^2+17z+19
Enhanced Jones-Krushkal polynomial 4w^3z^2+17w^2z+19w
Inner characteristic polynomial t^6+60t^4+8t^2
Outer characteristic polynomial t^7+100t^5+49t^3+3t
Flat arrow polynomial -4*K1*K2 - 2*K1*K3 + 2*K1 + K2 + 2*K3 + K4 + 1
2-strand cable arrow polynomial -256*K1**4*K2**2 + 352*K1**4*K2 - 496*K1**4 + 384*K1**3*K2*K3 - 672*K1**3*K3 + 160*K1**2*K2**3 - 1184*K1**2*K2**2 - 448*K1**2*K2*K4 + 2368*K1**2*K2 - 208*K1**2*K3**2 - 64*K1**2*K4**2 - 1936*K1**2 + 64*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 224*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 64*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 2192*K1*K2*K3 - 64*K1*K2*K4*K5 + 680*K1*K3*K4 + 184*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**4 - 128*K2**2*K3**2 - 176*K2**2*K4**2 + 520*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 1456*K2**2 + 320*K2*K3*K5 + 136*K2*K4*K6 + 48*K2*K5*K7 + 8*K2*K6*K8 + 8*K3**2*K6 - 828*K3**2 - 446*K4**2 - 168*K5**2 - 40*K6**2 - 12*K7**2 - 2*K8**2 + 1526
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}]]
If K is slice False
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