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Flat knot 6.83

Min(phi) over symmetries of the knot is: [-4,-1,-1,2,2,2,0,2,2,3,4,1,1,1,1,1,1,2,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.83']
Arrow polynomial of the knot is: -2*K1**2 - 4*K1*K2 + 2*K1 - 2*K2**2 + K2 + 2*K3 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.83', '6.151', '6.160', '6.190', '6.247', '6.262', '6.491', '6.514']
Outer characteristic polynomial of the knot is: t^7+93t^5+66t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.83']
2-strand cable arrow polynomial of the knot is: -768*K1**4 + 448*K1**3*K2*K3 + 32*K1**3*K3*K4 - 480*K1**3*K3 - 1376*K1**2*K2**2 - 480*K1**2*K2*K4 + 3120*K1**2*K2 - 1344*K1**2*K3**2 - 96*K1**2*K3*K5 - 96*K1**2*K4**2 - 3240*K1**2 + 64*K1*K2**3*K3 - 320*K1*K2**2*K3 + 160*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 288*K1*K2*K3*K4 + 4560*K1*K2*K3 - 96*K1*K2*K4*K5 - 96*K1*K3**2*K5 - 64*K1*K3*K4*K6 + 2176*K1*K3*K4 + 312*K1*K4*K5 + 48*K1*K5*K6 - 104*K2**4 - 432*K2**2*K3**2 - 112*K2**2*K4**2 + 624*K2**2*K4 - 2564*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 616*K2*K3*K5 + 224*K2*K4*K6 + 16*K2*K5*K7 - 320*K3**4 - 144*K3**2*K4**2 + 264*K3**2*K6 - 1968*K3**2 + 128*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 914*K4**2 - 252*K5**2 - 100*K6**2 - 20*K7**2 - 2*K8**2 + 2946
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.83']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4646', 'vk6.4925', 'vk6.6084', 'vk6.6581', 'vk6.8103', 'vk6.8495', 'vk6.9483', 'vk6.9850', 'vk6.20636', 'vk6.22063', 'vk6.28118', 'vk6.29559', 'vk6.39546', 'vk6.41769', 'vk6.46153', 'vk6.47795', 'vk6.48678', 'vk6.48869', 'vk6.49418', 'vk6.49651', 'vk6.50688', 'vk6.50873', 'vk6.51161', 'vk6.51378', 'vk6.57528', 'vk6.58716', 'vk6.62220', 'vk6.63166', 'vk6.67030', 'vk6.67903', 'vk6.69655', 'vk6.70336']
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 O1O2O3O4O5O6U2U4U6U5U1U3
R3 orbit {'O1O2O3O4O5O6U2U4U6U5U1U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U4U6U2U1U3U5
Gauss code of K* O1O2O3O4O5O6U5U1U6U2U4U3
Gauss code of -K* O1O2O3O4O5O6U4U3U5U1U6U2
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 -4 2 -1 2 2],[ 1 0 -3 2 -1 2 2],[ 4 3 0 4 1 3 2],[-2 -2 -4 0 -2 1 1],[ 1 1 -1 2 0 2 1],[-2 -2 -3 -1 -2 0 0],[-2 -2 -2 -1 -1 0 0]]
Primitive based matrix [[ 0 2 2 2 -1 -1 -4],[-2 0 1 1 -2 -2 -4],[-2 -1 0 0 -1 -2 -2],[-2 -1 0 0 -2 -2 -3],[ 1 2 1 2 0 1 -1],[ 1 2 2 2 -1 0 -3],[ 4 4 2 3 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-2,1,1,4,-1,-1,2,2,4,0,1,2,2,2,2,3,-1,1,3]
Phi over symmetry [-4,-1,-1,2,2,2,0,2,2,3,4,1,1,1,1,1,1,2,-1,-1,0]
Phi of -K [-4,-1,-1,2,2,2,0,2,2,3,4,1,1,1,1,1,1,2,-1,-1,0]
Phi of K* [-2,-2,-2,1,1,4,-1,0,1,1,3,1,1,1,2,1,2,4,-1,0,2]
Phi of -K* [-4,-1,-1,2,2,2,1,3,2,3,4,1,1,2,2,2,2,2,0,-1,-1]
Symmetry type of based matrix c
u-polynomial t^4-3t^2+2t
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial 5w^3z^2+22w^2z+25w
Inner characteristic polynomial t^6+63t^4+19t^2
Outer characteristic polynomial t^7+93t^5+66t^3+4t
Flat arrow polynomial -2*K1**2 - 4*K1*K2 + 2*K1 - 2*K2**2 + K2 + 2*K3 + K4 + 3
2-strand cable arrow polynomial -768*K1**4 + 448*K1**3*K2*K3 + 32*K1**3*K3*K4 - 480*K1**3*K3 - 1376*K1**2*K2**2 - 480*K1**2*K2*K4 + 3120*K1**2*K2 - 1344*K1**2*K3**2 - 96*K1**2*K3*K5 - 96*K1**2*K4**2 - 3240*K1**2 + 64*K1*K2**3*K3 - 320*K1*K2**2*K3 + 160*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 288*K1*K2*K3*K4 + 4560*K1*K2*K3 - 96*K1*K2*K4*K5 - 96*K1*K3**2*K5 - 64*K1*K3*K4*K6 + 2176*K1*K3*K4 + 312*K1*K4*K5 + 48*K1*K5*K6 - 104*K2**4 - 432*K2**2*K3**2 - 112*K2**2*K4**2 + 624*K2**2*K4 - 2564*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 616*K2*K3*K5 + 224*K2*K4*K6 + 16*K2*K5*K7 - 320*K3**4 - 144*K3**2*K4**2 + 264*K3**2*K6 - 1968*K3**2 + 128*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 914*K4**2 - 252*K5**2 - 100*K6**2 - 20*K7**2 - 2*K8**2 + 2946
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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