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Flat knot 6.20

Min(phi) over symmetries of the knot is: [-5,-2,-1,1,3,4,1,3,2,5,4,1,1,3,2,1,3,3,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.20']
Arrow polynomial of the knot is: -4*K1**2 - 2*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2*K3 + 3*K2 + K3 + K4 + K5 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.20']
Outer characteristic polynomial of the knot is: t^7+148t^5+89t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.20']
2-strand cable arrow polynomial of the knot is: -1072*K1**4 + 64*K1**3*K2*K3 + 32*K1**3*K3*K4 - 640*K1**3*K3 - 1248*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 5704*K1**2*K2 - 2416*K1**2*K3**2 - 256*K1**2*K3*K5 - 208*K1**2*K4**2 - 32*K1**2*K6**2 - 7260*K1**2 + 96*K1*K2**3*K3 - 896*K1*K2**2*K3 - 64*K1*K2**2*K5 + 64*K1*K2*K3**3 - 224*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 7888*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K5*K6 + 32*K1*K3**3*K4 + 4520*K1*K3*K4 + 560*K1*K4*K5 + 216*K1*K5*K6 + 64*K1*K6*K7 + 16*K1*K7*K8 - 2*K10**2 + 8*K10*K4*K6 - 208*K2**4 - 32*K2**3*K6 - 688*K2**2*K3**2 - 8*K2**2*K4**2 + 1144*K2**2*K4 - 8*K2**2*K6**2 - 5348*K2**2 - 160*K2*K3**2*K4 + 1128*K2*K3*K5 + 280*K2*K4*K6 + 40*K2*K5*K7 + 8*K2*K6*K8 - 608*K3**4 - 160*K3**2*K4**2 - 32*K3**2*K6**2 + 608*K3**2*K6 - 4108*K3**2 + 176*K3*K4*K7 + 16*K3*K6*K9 - 8*K4**2*K6**2 - 1938*K4**2 - 576*K5**2 - 306*K6**2 - 96*K7**2 - 10*K8**2 + 6426
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.20']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81570', 'vk6.81573', 'vk6.81651', 'vk6.81653', 'vk6.81729', 'vk6.81731', 'vk6.81862', 'vk6.81868', 'vk6.82235', 'vk6.82239', 'vk6.82384', 'vk6.82388', 'vk6.82498', 'vk6.82502', 'vk6.82577', 'vk6.82582', 'vk6.83164', 'vk6.83166', 'vk6.83589', 'vk6.83591', 'vk6.84130', 'vk6.84136', 'vk6.84333', 'vk6.84337', 'vk6.84564', 'vk6.84572', 'vk6.86471', 'vk6.86475', 'vk6.88733', 'vk6.88737', 'vk6.88907', 'vk6.88919']
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 O1O2O3O4O5O6U1U3U5U2U6U4
R3 orbit {'O1O2O3O4O5O6U1U3U5U2U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U3U1U5U2U4U6
Gauss code of K* O1O2O3O4O5O6U1U4U2U6U3U5
Gauss code of -K* O1O2O3O4O5O6U2U4U1U5U3U6
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 -5 -1 -2 3 1 4],[ 5 0 3 1 5 2 4],[ 1 -3 0 -1 3 1 3],[ 2 -1 1 0 3 1 2],[-3 -5 -3 -3 0 -1 1],[-1 -2 -1 -1 1 0 1],[-4 -4 -3 -2 -1 -1 0]]
Primitive based matrix [[ 0 4 3 1 -1 -2 -5],[-4 0 -1 -1 -3 -2 -4],[-3 1 0 -1 -3 -3 -5],[-1 1 1 0 -1 -1 -2],[ 1 3 3 1 0 -1 -3],[ 2 2 3 1 1 0 -1],[ 5 4 5 2 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-3,-1,1,2,5,1,1,3,2,4,1,3,3,5,1,1,2,1,3,1]
Phi over symmetry [-5,-2,-1,1,3,4,1,3,2,5,4,1,1,3,2,1,3,3,1,1,1]
Phi of -K [-5,-2,-1,1,3,4,2,1,4,3,5,0,2,2,4,1,1,2,1,2,0]
Phi of K* [-4,-3,-1,1,2,5,0,2,2,4,5,1,1,2,3,1,2,4,0,1,2]
Phi of -K* [-5,-2,-1,1,3,4,1,3,2,5,4,1,1,3,2,1,3,3,1,1,1]
Symmetry type of based matrix c
u-polynomial t^5-t^4-t^3+t^2
Normalized Jones-Krushkal polynomial 4z^2+25z+35
Enhanced Jones-Krushkal polynomial 4w^3z^2+25w^2z+35w
Inner characteristic polynomial t^6+92t^4+24t^2+1
Outer characteristic polynomial t^7+148t^5+89t^3+6t
Flat arrow polynomial -4*K1**2 - 2*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2*K3 + 3*K2 + K3 + K4 + K5 + 3
2-strand cable arrow polynomial -1072*K1**4 + 64*K1**3*K2*K3 + 32*K1**3*K3*K4 - 640*K1**3*K3 - 1248*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 5704*K1**2*K2 - 2416*K1**2*K3**2 - 256*K1**2*K3*K5 - 208*K1**2*K4**2 - 32*K1**2*K6**2 - 7260*K1**2 + 96*K1*K2**3*K3 - 896*K1*K2**2*K3 - 64*K1*K2**2*K5 + 64*K1*K2*K3**3 - 224*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 7888*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K5*K6 + 32*K1*K3**3*K4 + 4520*K1*K3*K4 + 560*K1*K4*K5 + 216*K1*K5*K6 + 64*K1*K6*K7 + 16*K1*K7*K8 - 2*K10**2 + 8*K10*K4*K6 - 208*K2**4 - 32*K2**3*K6 - 688*K2**2*K3**2 - 8*K2**2*K4**2 + 1144*K2**2*K4 - 8*K2**2*K6**2 - 5348*K2**2 - 160*K2*K3**2*K4 + 1128*K2*K3*K5 + 280*K2*K4*K6 + 40*K2*K5*K7 + 8*K2*K6*K8 - 608*K3**4 - 160*K3**2*K4**2 - 32*K3**2*K6**2 + 608*K3**2*K6 - 4108*K3**2 + 176*K3*K4*K7 + 16*K3*K6*K9 - 8*K4**2*K6**2 - 1938*K4**2 - 576*K5**2 - 306*K6**2 - 96*K7**2 - 10*K8**2 + 6426
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}]]
If K is slice False
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