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Flat knot 6.1524

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,1,1,2,3,0,0,1,1,1,1,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1524']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 8*K1*K2 + K1 + 2*K2 + 3*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.697', '6.1075', '6.1524', '6.1733']
Outer characteristic polynomial of the knot is: t^7+36t^5+39t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1524']
2-strand cable arrow polynomial of the knot is: 2688*K1**4*K2 - 6928*K1**4 + 1184*K1**3*K2*K3 + 96*K1**3*K3*K4 - 1312*K1**3*K3 - 128*K1**2*K2**4 + 928*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 8272*K1**2*K2**2 + 32*K1**2*K2*K3*K5 - 2080*K1**2*K2*K4 + 11232*K1**2*K2 - 976*K1**2*K3**2 - 368*K1**2*K4**2 - 32*K1**2*K5**2 - 4296*K1**2 + 544*K1*K2**3*K3 - 1536*K1*K2**2*K3 - 576*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 9240*K1*K2*K3 - 32*K1*K2*K4*K5 + 2880*K1*K3*K4 + 544*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 1040*K2**4 - 32*K2**3*K6 - 496*K2**2*K3**2 - 64*K2**2*K4**2 + 2216*K2**2*K4 - 4906*K2**2 + 728*K2*K3*K5 + 56*K2*K4*K6 + 8*K3**2*K6 - 2704*K3**2 - 1396*K4**2 - 312*K5**2 - 22*K6**2 + 5154
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1524']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4368', 'vk6.4399', 'vk6.5690', 'vk6.5721', 'vk6.7759', 'vk6.7790', 'vk6.9241', 'vk6.9272', 'vk6.10492', 'vk6.10541', 'vk6.10636', 'vk6.10715', 'vk6.10746', 'vk6.10827', 'vk6.14607', 'vk6.15307', 'vk6.15432', 'vk6.16226', 'vk6.17988', 'vk6.24432', 'vk6.30171', 'vk6.30220', 'vk6.30315', 'vk6.30446', 'vk6.33949', 'vk6.34354', 'vk6.34408', 'vk6.43867', 'vk6.50445', 'vk6.50476', 'vk6.54193', 'vk6.63443']
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 O1O2O3U2O4U5U3O5O6U1U6U4
R3 orbit {'O1O2O3U2O4U5U3O5O6U1U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5U3O5O6U1U6O4U2
Gauss code of K* O1O2O3U1U4U5O4U3O6O5U6U2
Gauss code of -K* O1O2O3U2U4O5O4U1O6U5U6U3
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 -1 1 2 -1 1],[ 2 0 -1 2 3 1 1],[ 1 1 0 1 1 0 0],[-1 -2 -1 0 0 -1 0],[-2 -3 -1 0 0 -2 0],[ 1 -1 0 1 2 0 1],[-1 -1 0 0 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 0 0 -1 -2 -3],[-1 0 0 0 0 -1 -1],[-1 0 0 0 -1 -1 -2],[ 1 1 0 1 0 0 1],[ 1 2 1 1 0 0 -1],[ 2 3 1 2 -1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,0,0,1,2,3,0,0,1,1,1,1,2,0,-1,1]
Phi over symmetry [-2,-1,-1,1,1,2,-1,1,1,2,3,0,0,1,1,1,1,2,0,0,0]
Phi of -K [-2,-1,-1,1,1,2,0,2,1,2,1,0,1,1,1,1,2,2,0,1,1]
Phi of K* [-2,-1,-1,1,1,2,1,1,1,2,1,0,1,1,1,1,2,2,0,0,2]
Phi of -K* [-2,-1,-1,1,1,2,-1,1,1,2,3,0,0,1,1,1,1,2,0,0,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+24t^4+23t^2
Outer characteristic polynomial t^7+36t^5+39t^3+3t
Flat arrow polynomial 4*K1**3 - 4*K1**2 - 8*K1*K2 + K1 + 2*K2 + 3*K3 + 3
2-strand cable arrow polynomial 2688*K1**4*K2 - 6928*K1**4 + 1184*K1**3*K2*K3 + 96*K1**3*K3*K4 - 1312*K1**3*K3 - 128*K1**2*K2**4 + 928*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 8272*K1**2*K2**2 + 32*K1**2*K2*K3*K5 - 2080*K1**2*K2*K4 + 11232*K1**2*K2 - 976*K1**2*K3**2 - 368*K1**2*K4**2 - 32*K1**2*K5**2 - 4296*K1**2 + 544*K1*K2**3*K3 - 1536*K1*K2**2*K3 - 576*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 9240*K1*K2*K3 - 32*K1*K2*K4*K5 + 2880*K1*K3*K4 + 544*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 1040*K2**4 - 32*K2**3*K6 - 496*K2**2*K3**2 - 64*K2**2*K4**2 + 2216*K2**2*K4 - 4906*K2**2 + 728*K2*K3*K5 + 56*K2*K4*K6 + 8*K3**2*K6 - 2704*K3**2 - 1396*K4**2 - 312*K5**2 - 22*K6**2 + 5154
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {5}, {4}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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