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Flat knot 6.255

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,1,2,1,4,3,1,1,3,2,1,2,2,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.255']
Arrow polynomial of the knot is: 8*K1**3 - 2*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.245', '6.255', '6.478']
Outer characteristic polynomial of the knot is: t^7+91t^5+78t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.255']
2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 608*K1**4*K2 - 848*K1**4 + 128*K1**3*K2**3*K3 - 384*K1**3*K2**2*K3 + 1056*K1**3*K2*K3 - 672*K1**3*K3 - 704*K1**2*K2**4 + 3200*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 32*K1**2*K2**2*K4 - 10224*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 544*K1**2*K2*K4 + 8216*K1**2*K2 - 816*K1**2*K3**2 - 5320*K1**2 + 2656*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 2048*K1*K2**2*K3 - 224*K1*K2**2*K5 + 224*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 9192*K1*K2*K3 + 928*K1*K3*K4 + 40*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 160*K2**4*K4 - 3656*K2**4 + 64*K2**3*K3*K5 - 32*K2**3*K6 - 2256*K2**2*K3**2 - 32*K2**2*K3*K7 - 152*K2**2*K4**2 + 2480*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 2434*K2**2 - 96*K2*K3**2*K4 + 1128*K2*K3*K5 + 72*K2*K4*K6 + 24*K2*K5*K7 + 8*K2*K6*K8 - 96*K3**4 + 72*K3**2*K6 - 2260*K3**2 + 8*K3*K4*K7 - 508*K4**2 - 144*K5**2 - 22*K6**2 - 4*K7**2 - 2*K8**2 + 4156
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.255']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16907', 'vk6.17149', 'vk6.20522', 'vk6.21910', 'vk6.23295', 'vk6.23594', 'vk6.27969', 'vk6.29442', 'vk6.35313', 'vk6.35749', 'vk6.39371', 'vk6.41553', 'vk6.42814', 'vk6.43096', 'vk6.45942', 'vk6.47625', 'vk6.55062', 'vk6.55307', 'vk6.57383', 'vk6.58549', 'vk6.59454', 'vk6.59743', 'vk6.62040', 'vk6.63036', 'vk6.64903', 'vk6.65116', 'vk6.66928', 'vk6.67779', 'vk6.68208', 'vk6.68352', 'vk6.69534', 'vk6.70238']
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 O1O2O3O4O5U1U2O6U4U6U5U3
R3 orbit {'O1O2O3O4O5U1U2O6U4U6U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U1U6U2O6U4U5
Gauss code of K* O1O2O3O4U5U6U4U1U3O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U2U4U1U5U6
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 -2 2 0 3 1],[ 4 0 1 4 2 3 1],[ 2 -1 0 3 1 2 1],[-2 -4 -3 0 -2 1 1],[ 0 -2 -1 2 0 2 1],[-3 -3 -2 -1 -2 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 3 2 1 0 -2 -4],[-3 0 -1 0 -2 -2 -3],[-2 1 0 1 -2 -3 -4],[-1 0 -1 0 -1 -1 -1],[ 0 2 2 1 0 -1 -2],[ 2 2 3 1 1 0 -1],[ 4 3 4 1 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,0,2,4,1,0,2,2,3,-1,2,3,4,1,1,1,1,2,1]
Phi over symmetry [-4,-2,0,1,2,3,1,2,1,4,3,1,1,3,2,1,2,2,-1,0,1]
Phi of -K [-4,-2,0,1,2,3,1,2,4,2,4,1,2,1,3,0,0,1,2,2,0]
Phi of K* [-3,-2,-1,0,2,4,0,2,1,3,4,2,0,1,2,0,2,4,1,2,1]
Phi of -K* [-4,-2,0,1,2,3,1,2,1,4,3,1,1,3,2,1,2,2,-1,0,1]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+57t^4+14t^2
Outer characteristic polynomial t^7+91t^5+78t^3+10t
Flat arrow polynomial 8*K1**3 - 2*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial -384*K1**4*K2**2 + 608*K1**4*K2 - 848*K1**4 + 128*K1**3*K2**3*K3 - 384*K1**3*K2**2*K3 + 1056*K1**3*K2*K3 - 672*K1**3*K3 - 704*K1**2*K2**4 + 3200*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 32*K1**2*K2**2*K4 - 10224*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 544*K1**2*K2*K4 + 8216*K1**2*K2 - 816*K1**2*K3**2 - 5320*K1**2 + 2656*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 2048*K1*K2**2*K3 - 224*K1*K2**2*K5 + 224*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 9192*K1*K2*K3 + 928*K1*K3*K4 + 40*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 160*K2**4*K4 - 3656*K2**4 + 64*K2**3*K3*K5 - 32*K2**3*K6 - 2256*K2**2*K3**2 - 32*K2**2*K3*K7 - 152*K2**2*K4**2 + 2480*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 2434*K2**2 - 96*K2*K3**2*K4 + 1128*K2*K3*K5 + 72*K2*K4*K6 + 24*K2*K5*K7 + 8*K2*K6*K8 - 96*K3**4 + 72*K3**2*K6 - 2260*K3**2 + 8*K3*K4*K7 - 508*K4**2 - 144*K5**2 - 22*K6**2 - 4*K7**2 - 2*K8**2 + 4156
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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