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Flat knot 6.566

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,0,0,0,1,2,1,2,2,2,0,-1,-1,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.566']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1*K3 + K4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.111', '6.139', '6.519', '6.566', '6.1228', '6.1254', '6.1259', '6.1912', '6.1936']
Outer characteristic polynomial of the knot is: t^7+34t^5+71t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.549', '6.566']
2-strand cable arrow polynomial of the knot is: 1024*K1**4*K2 - 1344*K1**4 - 384*K1**3*K2**2*K3 + 512*K1**3*K2*K3 - 576*K1**3*K3 + 960*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3552*K1**2*K2**2 + 768*K1**2*K2*K3**2 - 800*K1**2*K2*K4 + 4368*K1**2*K2 - 1152*K1**2*K3**2 - 384*K1**2*K3*K5 - 96*K1**2*K4**2 - 2880*K1**2 + 416*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1952*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 320*K1*K2**2*K5 - 800*K1*K2*K3*K4 + 5048*K1*K2*K3 - 192*K1*K2*K4*K5 - 32*K1*K2*K4*K7 - 64*K1*K2*K5*K6 + 2128*K1*K3*K4 + 544*K1*K4*K5 + 48*K1*K5*K6 + 24*K1*K6*K7 - 32*K2**4*K4**2 + 128*K2**4*K4 - 976*K2**4 + 32*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 928*K2**2*K3**2 - 32*K2**2*K3*K7 - 288*K2**2*K4**2 - 32*K2**2*K4*K8 + 1696*K2**2*K4 - 64*K2**2*K5**2 - 16*K2**2*K6**2 - 2328*K2**2 - 32*K2*K3**2*K4 + 992*K2*K3*K5 + 256*K2*K4*K6 + 72*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1512*K3**2 - 824*K4**2 - 280*K5**2 - 56*K6**2 - 16*K7**2 - 2*K8**2 + 2456
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.566']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4804', 'vk6.4814', 'vk6.5142', 'vk6.5158', 'vk6.6363', 'vk6.6799', 'vk6.6805', 'vk6.8329', 'vk6.8335', 'vk6.8771', 'vk6.9704', 'vk6.9706', 'vk6.10010', 'vk6.10012', 'vk6.21100', 'vk6.21107', 'vk6.22530', 'vk6.22539', 'vk6.28547', 'vk6.42195', 'vk6.46822', 'vk6.46829', 'vk6.48059', 'vk6.48068', 'vk6.48820', 'vk6.49865', 'vk6.49871', 'vk6.50834', 'vk6.50836', 'vk6.51518', 'vk6.58984', 'vk6.69820']
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 O1O2O3O4O5U4U3U5O6U2U1U6
R3 orbit {'O1O2O3O4O5U4U3U5O6U2U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U5U4O6U1U3U2
Gauss code of K* O1O2O3U4O5O6O4U6U5U2U1U3
Gauss code of -K* O1O2O3U1O4O5O6U4U6U5U3U2
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 -1 -1 2 2],[ 1 0 0 -1 -1 2 2],[ 1 0 0 -1 -1 2 1],[ 1 1 1 0 0 2 0],[ 1 1 1 0 0 1 0],[-2 -2 -2 -2 -1 0 0],[-2 -2 -1 0 0 0 0]]
Primitive based matrix [[ 0 2 2 -1 -1 -1 -1],[-2 0 0 0 0 -1 -2],[-2 0 0 -1 -2 -2 -2],[ 1 0 1 0 0 1 1],[ 1 0 2 0 0 1 1],[ 1 1 2 -1 -1 0 0],[ 1 2 2 -1 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,1,1,1,1,0,0,0,1,2,1,2,2,2,0,-1,-1,-1,-1,0]
Phi over symmetry [-2,-2,1,1,1,1,0,0,0,1,2,1,2,2,2,0,-1,-1,-1,-1,0]
Phi of -K [-1,-1,-1,-1,2,2,-1,-1,0,1,3,0,1,1,1,1,1,2,2,3,0]
Phi of K* [-2,-2,1,1,1,1,0,1,1,1,2,1,2,3,3,0,-1,-1,-1,-1,0]
Phi of -K* [-1,-1,-1,-1,2,2,-1,-1,0,1,2,0,1,0,1,1,0,2,2,2,0]
Symmetry type of based matrix c
u-polynomial -2t^2+4t
Normalized Jones-Krushkal polynomial 9z^2+30z+25
Enhanced Jones-Krushkal polynomial 9w^3z^2+30w^2z+25w
Inner characteristic polynomial t^6+22t^4+33t^2+1
Outer characteristic polynomial t^7+34t^5+71t^3+7t
Flat arrow polynomial 4*K1**2*K2 - 4*K1*K3 + K4
2-strand cable arrow polynomial 1024*K1**4*K2 - 1344*K1**4 - 384*K1**3*K2**2*K3 + 512*K1**3*K2*K3 - 576*K1**3*K3 + 960*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3552*K1**2*K2**2 + 768*K1**2*K2*K3**2 - 800*K1**2*K2*K4 + 4368*K1**2*K2 - 1152*K1**2*K3**2 - 384*K1**2*K3*K5 - 96*K1**2*K4**2 - 2880*K1**2 + 416*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1952*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 320*K1*K2**2*K5 - 800*K1*K2*K3*K4 + 5048*K1*K2*K3 - 192*K1*K2*K4*K5 - 32*K1*K2*K4*K7 - 64*K1*K2*K5*K6 + 2128*K1*K3*K4 + 544*K1*K4*K5 + 48*K1*K5*K6 + 24*K1*K6*K7 - 32*K2**4*K4**2 + 128*K2**4*K4 - 976*K2**4 + 32*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 928*K2**2*K3**2 - 32*K2**2*K3*K7 - 288*K2**2*K4**2 - 32*K2**2*K4*K8 + 1696*K2**2*K4 - 64*K2**2*K5**2 - 16*K2**2*K6**2 - 2328*K2**2 - 32*K2*K3**2*K4 + 992*K2*K3*K5 + 256*K2*K4*K6 + 72*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1512*K3**2 - 824*K4**2 - 280*K5**2 - 56*K6**2 - 16*K7**2 - 2*K8**2 + 2456
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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