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Flat knot 6.1911

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,1,2,1,0,1,2,-1,0,1,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1911']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 - 2*K2**2 + 4*K2 + K3 + K4 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1911']
Outer characteristic polynomial of the knot is: t^7+25t^5+57t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1911']
2-strand cable arrow polynomial of the knot is: -512*K1**4*K2**2 + 736*K1**4*K2 - 1088*K1**4 + 448*K1**3*K2*K3 - 160*K1**3*K3 - 256*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 2400*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 256*K1**2*K2**2*K4**2 + 288*K1**2*K2**2*K4 - 7792*K1**2*K2**2 + 64*K1**2*K2*K4**2 - 1280*K1**2*K2*K4 + 7784*K1**2*K2 - 256*K1**2*K3**2 - 240*K1**2*K4**2 - 5784*K1**2 - 128*K1*K2**3*K3*K4 + 1568*K1*K2**3*K3 + 864*K1*K2**2*K3*K4 - 1504*K1*K2**2*K3 + 224*K1*K2**2*K4*K5 - 352*K1*K2**2*K5 + 64*K1*K2*K3**3 + 160*K1*K2*K3*K4**2 - 800*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 7096*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1616*K1*K3*K4 + 432*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 416*K2**4*K4 - 2152*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 1168*K2**2*K3**2 + 32*K2**2*K4**3 - 752*K2**2*K4**2 + 2232*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 3294*K2**2 - 96*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 696*K2*K3*K5 - 32*K2*K4**2*K6 + 272*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 32*K3**4 - 32*K3**2*K4**2 + 40*K3**2*K6 - 2008*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 988*K4**2 - 176*K5**2 - 50*K6**2 - 2*K8**2 + 4380
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1911']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4756', 'vk6.5083', 'vk6.6306', 'vk6.6745', 'vk6.8271', 'vk6.8720', 'vk6.9649', 'vk6.9964', 'vk6.20398', 'vk6.21747', 'vk6.27736', 'vk6.29274', 'vk6.39172', 'vk6.41400', 'vk6.45900', 'vk6.47541', 'vk6.48788', 'vk6.48999', 'vk6.49608', 'vk6.49811', 'vk6.50812', 'vk6.51027', 'vk6.51291', 'vk6.51486', 'vk6.57259', 'vk6.58476', 'vk6.61903', 'vk6.63012', 'vk6.66872', 'vk6.67742', 'vk6.69496', 'vk6.70218']
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 O1O2O3U1U2U4O5O6O4U6U3U5
R3 orbit {'O1O2O3U1U2U4O5O6O4U6U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4U5O6O4O5U3U6U2
Gauss code of K* O1O2O3U4U5U2O4O5O6U3U1U6
Gauss code of -K* O1O2O3U4U3U1O4O5O6U2U5U6
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 -2 0 1 2 0 -1],[ 2 0 1 2 2 1 0],[ 0 -1 0 1 0 1 0],[-1 -2 -1 0 0 0 -1],[-2 -2 0 0 0 -1 -1],[ 0 -1 -1 0 1 0 0],[ 1 0 0 1 1 0 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 0 0 -1 -1 -2],[-1 0 0 -1 0 -1 -2],[ 0 0 1 0 1 0 -1],[ 0 1 0 -1 0 0 -1],[ 1 1 1 0 0 0 0],[ 2 2 2 1 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,0,0,1,1,2,1,0,1,2,-1,0,1,0,1,0]
Phi over symmetry [-2,-1,0,0,1,2,0,0,1,1,2,1,0,1,2,-1,0,1,0,1,0]
Phi of -K [-2,-1,0,0,1,2,1,1,1,1,2,1,1,1,2,-1,0,2,1,1,1]
Phi of K* [-2,-1,0,0,1,2,1,1,2,2,2,1,0,1,1,-1,1,1,1,1,1]
Phi of -K* [-2,-1,0,0,1,2,0,1,1,2,2,0,0,1,1,-1,0,1,1,0,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 4z^2+23z+31
Enhanced Jones-Krushkal polynomial 4w^3z^2-2w^3z+25w^2z+31w
Inner characteristic polynomial t^6+15t^4+27t^2+4
Outer characteristic polynomial t^7+25t^5+57t^3+13t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 - 2*K2**2 + 4*K2 + K3 + K4 + 6
2-strand cable arrow polynomial -512*K1**4*K2**2 + 736*K1**4*K2 - 1088*K1**4 + 448*K1**3*K2*K3 - 160*K1**3*K3 - 256*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 2400*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 256*K1**2*K2**2*K4**2 + 288*K1**2*K2**2*K4 - 7792*K1**2*K2**2 + 64*K1**2*K2*K4**2 - 1280*K1**2*K2*K4 + 7784*K1**2*K2 - 256*K1**2*K3**2 - 240*K1**2*K4**2 - 5784*K1**2 - 128*K1*K2**3*K3*K4 + 1568*K1*K2**3*K3 + 864*K1*K2**2*K3*K4 - 1504*K1*K2**2*K3 + 224*K1*K2**2*K4*K5 - 352*K1*K2**2*K5 + 64*K1*K2*K3**3 + 160*K1*K2*K3*K4**2 - 800*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 7096*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1616*K1*K3*K4 + 432*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 416*K2**4*K4 - 2152*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 1168*K2**2*K3**2 + 32*K2**2*K4**3 - 752*K2**2*K4**2 + 2232*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 3294*K2**2 - 96*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 696*K2*K3*K5 - 32*K2*K4**2*K6 + 272*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 32*K3**4 - 32*K3**2*K4**2 + 40*K3**2*K6 - 2008*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 988*K4**2 - 176*K5**2 - 50*K6**2 - 2*K8**2 + 4380
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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