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Flat knot 6.1180

Min(phi) over symmetries of the knot is: [-3,0,1,2,1,3,3,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.116', '6.458', '6.1180', '7.28052']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 10*K1**2 - 10*K1*K2 - 4*K1*K3 - K1 + 5*K2 + 3*K3 + K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1180']
Outer characteristic polynomial of the knot is: t^5+25t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1180', '7.28052']
2-strand cable arrow polynomial of the knot is: -448*K1**6 + 256*K1**4*K2**3 - 1024*K1**4*K2**2 + 3072*K1**4*K2 - 4672*K1**4 - 256*K1**3*K2**2*K3 + 992*K1**3*K2*K3 + 32*K1**3*K3*K4 - 416*K1**3*K3 + 32*K1**3*K4*K5 - 384*K1**2*K2**4 + 1280*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6816*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 448*K1**2*K2*K4 + 9336*K1**2*K2 - 928*K1**2*K3**2 - 160*K1**2*K3*K5 - 240*K1**2*K4**2 - 32*K1**2*K4*K6 - 32*K1**2*K5**2 - 5420*K1**2 + 768*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1696*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 288*K1*K2**2*K5 - 640*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 7872*K1*K2*K3 - 160*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2512*K1*K3*K4 + 960*K1*K4*K5 + 112*K1*K5*K6 + 8*K1*K6*K7 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 224*K2**4*K4 - 1496*K2**4 + 160*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 880*K2**2*K3**2 - 64*K2**2*K3*K7 - 344*K2**2*K4**2 - 32*K2**2*K4*K8 + 2408*K2**2*K4 - 80*K2**2*K5**2 - 16*K2**2*K6**2 - 5230*K2**2 - 128*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1224*K2*K3*K5 + 368*K2*K4*K6 + 88*K2*K5*K7 + 16*K2*K6*K8 + 48*K3**2*K6 - 3012*K3**2 + 16*K3*K4*K7 - 1646*K4**2 - 624*K5**2 - 98*K6**2 - 16*K7**2 - 2*K8**2 + 5942
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1180']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4213', 'vk6.4292', 'vk6.5472', 'vk6.5583', 'vk6.7580', 'vk6.7672', 'vk6.9082', 'vk6.9161', 'vk6.11189', 'vk6.12277', 'vk6.12384', 'vk6.19372', 'vk6.19667', 'vk6.19770', 'vk6.26156', 'vk6.26209', 'vk6.26574', 'vk6.26652', 'vk6.30775', 'vk6.31980', 'vk6.38152', 'vk6.38185', 'vk6.44813', 'vk6.44930', 'vk6.48535', 'vk6.49230', 'vk6.49341', 'vk6.50322', 'vk6.52759', 'vk6.63595', 'vk6.66308', 'vk6.66333']
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 O1O2O3O4U1U5U3O6O5U6U4U2
R3 orbit {'O1O2O3O4U1U5U3O6O5U6U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U1U5O6O5U2U6U4
Gauss code of K* O1O2O3U4U2O4O5O6U1U6U3U5
Gauss code of -K* O1O2O3U4U3O5O4O6U2U5U1U6
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 -3 1 1 2 0 -1],[ 3 0 3 1 2 2 0],[-1 -3 0 0 1 -1 -1],[-1 -1 0 0 0 -1 -1],[-2 -2 -1 0 0 -1 -1],[ 0 -2 1 1 1 0 -1],[ 1 0 1 1 1 1 0]]
Primitive based matrix [[ 0 2 1 0 -3],[-2 0 0 -1 -2],[-1 0 0 -1 -1],[ 0 1 1 0 -2],[ 3 2 1 2 0]]
If based matrix primitive False
Phi of primitive based matrix [-2,-1,0,3,0,1,2,1,1,2]
Phi over symmetry [-3,0,1,2,1,3,3,0,1,1]
Phi of -K [-3,0,1,2,1,3,3,0,1,1]
Phi of K* [-2,-1,0,3,1,1,3,0,3,1]
Phi of -K* [-3,0,1,2,2,1,2,1,1,0]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 3z^2+24z+37
Enhanced Jones-Krushkal polynomial 3w^3z^2+24w^2z+37w
Inner characteristic polynomial t^4+11t^2+1
Outer characteristic polynomial t^5+25t^3+4t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 10*K1**2 - 10*K1*K2 - 4*K1*K3 - K1 + 5*K2 + 3*K3 + K4 + 5
2-strand cable arrow polynomial -448*K1**6 + 256*K1**4*K2**3 - 1024*K1**4*K2**2 + 3072*K1**4*K2 - 4672*K1**4 - 256*K1**3*K2**2*K3 + 992*K1**3*K2*K3 + 32*K1**3*K3*K4 - 416*K1**3*K3 + 32*K1**3*K4*K5 - 384*K1**2*K2**4 + 1280*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6816*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 448*K1**2*K2*K4 + 9336*K1**2*K2 - 928*K1**2*K3**2 - 160*K1**2*K3*K5 - 240*K1**2*K4**2 - 32*K1**2*K4*K6 - 32*K1**2*K5**2 - 5420*K1**2 + 768*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1696*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 288*K1*K2**2*K5 - 640*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 7872*K1*K2*K3 - 160*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2512*K1*K3*K4 + 960*K1*K4*K5 + 112*K1*K5*K6 + 8*K1*K6*K7 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 224*K2**4*K4 - 1496*K2**4 + 160*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 880*K2**2*K3**2 - 64*K2**2*K3*K7 - 344*K2**2*K4**2 - 32*K2**2*K4*K8 + 2408*K2**2*K4 - 80*K2**2*K5**2 - 16*K2**2*K6**2 - 5230*K2**2 - 128*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1224*K2*K3*K5 + 368*K2*K4*K6 + 88*K2*K5*K7 + 16*K2*K6*K8 + 48*K3**2*K6 - 3012*K3**2 + 16*K3*K4*K7 - 1646*K4**2 - 624*K5**2 - 98*K6**2 - 16*K7**2 - 2*K8**2 + 5942
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}]]
If K is slice False
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