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Flat knot 6.514

Min(phi) over symmetries of the knot is: [-4,-1,1,1,1,2,0,1,2,3,5,0,0,0,1,0,0,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.514']
Arrow polynomial of the knot is: -2*K1**2 - 4*K1*K2 + 2*K1 - 2*K2**2 + K2 + 2*K3 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.83', '6.151', '6.160', '6.190', '6.247', '6.262', '6.491', '6.514']
Outer characteristic polynomial of the knot is: t^7+82t^5+76t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.514']
2-strand cable arrow polynomial of the knot is: -432*K1**4 + 96*K1**3*K3*K4 - 752*K1**2*K2**2 - 576*K1**2*K2*K4 + 1992*K1**2*K2 - 208*K1**2*K3**2 - 352*K1**2*K4**2 - 2536*K1**2 - 256*K1*K2**2*K3 - 384*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 2408*K1*K2*K3 - 160*K1*K2*K4*K5 + 1688*K1*K3*K4 + 696*K1*K4*K5 + 120*K1*K5*K6 - 72*K2**4 - 80*K2**2*K4**2 + 880*K2**2*K4 - 2132*K2**2 - 32*K2*K3*K4*K5 + 560*K2*K3*K5 + 184*K2*K4*K6 + 8*K2*K5*K7 - 16*K3**2*K4**2 + 48*K3**2*K6 - 1436*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1118*K4**2 - 464*K5**2 - 116*K6**2 - 4*K7**2 - 2*K8**2 + 2454
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.514']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4709', 'vk6.5020', 'vk6.6225', 'vk6.6684', 'vk6.8208', 'vk6.8641', 'vk6.9584', 'vk6.9917', 'vk6.17409', 'vk6.20929', 'vk6.21084', 'vk6.22339', 'vk6.22512', 'vk6.23578', 'vk6.23917', 'vk6.28410', 'vk6.36190', 'vk6.40076', 'vk6.40322', 'vk6.42125', 'vk6.43404', 'vk6.46605', 'vk6.46792', 'vk6.48047', 'vk6.48743', 'vk6.49754', 'vk6.50753', 'vk6.51441', 'vk6.57740', 'vk6.58935', 'vk6.65303', 'vk6.69777']
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 O1O2O3O4O5U1U6U5O6U4U3U2
R3 orbit {'O1O2O3O4O5U1U6U5O6U4U3U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U3U2O6U1U6U5
Gauss code of K* O1O2O3U2O4O5O6U1U6U5U4U3
Gauss code of -K* O1O2O3U4O5O4O6U5U3U2U1U6
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 1 1 2 -1],[ 4 0 4 3 2 1 3],[-1 -4 0 0 0 1 -2],[-1 -3 0 0 0 1 -2],[-1 -2 0 0 0 1 -2],[-2 -1 -1 -1 -1 0 -2],[ 1 -3 2 2 2 2 0]]
Primitive based matrix [[ 0 2 1 1 1 -1 -4],[-2 0 -1 -1 -1 -2 -1],[-1 1 0 0 0 -2 -2],[-1 1 0 0 0 -2 -3],[-1 1 0 0 0 -2 -4],[ 1 2 2 2 2 0 -3],[ 4 1 2 3 4 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,-1,1,4,1,1,1,2,1,0,0,2,2,0,2,3,2,4,3]
Phi over symmetry [-4,-1,1,1,1,2,0,1,2,3,5,0,0,0,1,0,0,0,0,0,0]
Phi of -K [-4,-1,1,1,1,2,0,1,2,3,5,0,0,0,1,0,0,0,0,0,0]
Phi of K* [-2,-1,-1,-1,1,4,0,0,0,1,5,0,0,0,1,0,0,2,0,3,0]
Phi of -K* [-4,-1,1,1,1,2,3,2,3,4,1,2,2,2,2,0,0,1,0,1,1]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 5z+11
Enhanced Jones-Krushkal polynomial -4w^4z^2+4w^3z^2-12w^3z+17w^2z+11w
Inner characteristic polynomial t^6+58t^4+41t^2
Outer characteristic polynomial t^7+82t^5+76t^3+6t
Flat arrow polynomial -2*K1**2 - 4*K1*K2 + 2*K1 - 2*K2**2 + K2 + 2*K3 + K4 + 3
2-strand cable arrow polynomial -432*K1**4 + 96*K1**3*K3*K4 - 752*K1**2*K2**2 - 576*K1**2*K2*K4 + 1992*K1**2*K2 - 208*K1**2*K3**2 - 352*K1**2*K4**2 - 2536*K1**2 - 256*K1*K2**2*K3 - 384*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 2408*K1*K2*K3 - 160*K1*K2*K4*K5 + 1688*K1*K3*K4 + 696*K1*K4*K5 + 120*K1*K5*K6 - 72*K2**4 - 80*K2**2*K4**2 + 880*K2**2*K4 - 2132*K2**2 - 32*K2*K3*K4*K5 + 560*K2*K3*K5 + 184*K2*K4*K6 + 8*K2*K5*K7 - 16*K3**2*K4**2 + 48*K3**2*K6 - 1436*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1118*K4**2 - 464*K5**2 - 116*K6**2 - 4*K7**2 - 2*K8**2 + 2454
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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