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Flat knot 6.1530

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,1,2,0,0,1,0,1,1,1,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1530']
Arrow polynomial of the knot is: 4*K1**3 - 14*K1**2 - 4*K1*K2 - K1 + 7*K2 + K3 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1023', '6.1530', '6.1662', '6.1668', '6.1801']
Outer characteristic polynomial of the knot is: t^7+20t^5+32t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1530']
2-strand cable arrow polynomial of the knot is: -512*K1**4*K2**2 + 1888*K1**4*K2 - 5024*K1**4 + 640*K1**3*K2*K3 + 32*K1**3*K3*K4 - 928*K1**3*K3 - 256*K1**2*K2**4 + 1312*K1**2*K2**3 - 7376*K1**2*K2**2 - 512*K1**2*K2*K4 + 10864*K1**2*K2 - 480*K1**2*K3**2 - 32*K1**2*K3*K5 - 32*K1**2*K4**2 - 4640*K1**2 + 480*K1*K2**3*K3 - 864*K1*K2**2*K3 - 64*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 6448*K1*K2*K3 + 696*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 968*K2**4 - 240*K2**2*K3**2 - 48*K2**2*K4**2 + 904*K2**2*K4 - 3758*K2**2 + 144*K2*K3*K5 + 16*K2*K4*K6 - 1436*K3**2 - 302*K4**2 - 28*K5**2 - 2*K6**2 + 4124
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1530']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4855', 'vk6.4857', 'vk6.5198', 'vk6.5201', 'vk6.6430', 'vk6.6434', 'vk6.6856', 'vk6.8387', 'vk6.8399', 'vk6.8810', 'vk6.8819', 'vk6.9750', 'vk6.9759', 'vk6.10047', 'vk6.20783', 'vk6.20789', 'vk6.22186', 'vk6.29748', 'vk6.39817', 'vk6.39823', 'vk6.46377', 'vk6.46386', 'vk6.47954', 'vk6.47964', 'vk6.49083', 'vk6.49092', 'vk6.49920', 'vk6.51339', 'vk6.51342', 'vk6.51555', 'vk6.58806', 'vk6.63272']
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 O1O2O3U2O4U5U1O6O5U6U4U3
R3 orbit {'O1O2O3U2O4U5U1O6O5U6U4U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4U5O6O5U3U6O4U2
Gauss code of K* O1O2O3U4U5U3O5U2O6O4U1U6
Gauss code of -K* O1O2O3U4U3O5O4U2O6U1U6U5
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 -1 2 1 0 -1],[ 1 0 0 2 0 1 -1],[ 1 0 0 1 0 1 0],[-2 -2 -1 0 0 -1 -1],[-1 0 0 0 0 0 -1],[ 0 -1 -1 1 0 0 -1],[ 1 1 0 1 1 1 0]]
Primitive based matrix [[ 0 2 1 0 -1 -1 -1],[-2 0 0 -1 -1 -1 -2],[-1 0 0 0 0 -1 0],[ 0 1 0 0 -1 -1 -1],[ 1 1 0 1 0 0 0],[ 1 1 1 1 0 0 1],[ 1 2 0 1 0 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,1,1,1,0,1,1,1,2,0,0,1,0,1,1,1,0,0,-1]
Phi over symmetry [-2,-1,0,1,1,1,0,1,1,1,2,0,0,1,0,1,1,1,0,0,-1]
Phi of -K [-1,-1,-1,0,1,2,-1,0,0,1,2,0,0,2,1,0,2,2,1,1,1]
Phi of K* [-2,-1,0,1,1,1,1,1,1,2,2,1,2,1,2,0,0,0,-1,0,0]
Phi of -K* [-1,-1,-1,0,1,2,-1,0,1,0,2,0,1,1,1,1,0,1,0,1,0]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 2z^2+21z+35
Enhanced Jones-Krushkal polynomial 2w^3z^2+21w^2z+35w
Inner characteristic polynomial t^6+12t^4+11t^2+1
Outer characteristic polynomial t^7+20t^5+32t^3+6t
Flat arrow polynomial 4*K1**3 - 14*K1**2 - 4*K1*K2 - K1 + 7*K2 + K3 + 8
2-strand cable arrow polynomial -512*K1**4*K2**2 + 1888*K1**4*K2 - 5024*K1**4 + 640*K1**3*K2*K3 + 32*K1**3*K3*K4 - 928*K1**3*K3 - 256*K1**2*K2**4 + 1312*K1**2*K2**3 - 7376*K1**2*K2**2 - 512*K1**2*K2*K4 + 10864*K1**2*K2 - 480*K1**2*K3**2 - 32*K1**2*K3*K5 - 32*K1**2*K4**2 - 4640*K1**2 + 480*K1*K2**3*K3 - 864*K1*K2**2*K3 - 64*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 6448*K1*K2*K3 + 696*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 968*K2**4 - 240*K2**2*K3**2 - 48*K2**2*K4**2 + 904*K2**2*K4 - 3758*K2**2 + 144*K2*K3*K5 + 16*K2*K4*K6 - 1436*K3**2 - 302*K4**2 - 28*K5**2 - 2*K6**2 + 4124
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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