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Flat knot 6.488

Min(phi) over symmetries of the knot is: [-4,-1,0,1,2,2,1,3,4,2,3,1,1,1,1,0,1,1,2,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.488']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 14*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 6*K2 + K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.488']
Outer characteristic polynomial of the knot is: t^7+71t^5+59t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.488']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 960*K1**4*K2 - 3872*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 800*K1**3*K2*K3 - 896*K1**3*K3 - 384*K1**2*K2**4 + 736*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 6880*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 12136*K1**2*K2 - 832*K1**2*K3**2 - 160*K1**2*K4**2 - 8076*K1**2 + 1024*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1792*K1*K2**2*K3 - 288*K1*K2**2*K5 + 32*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 10320*K1*K2*K3 - 96*K1*K2*K4*K5 + 1944*K1*K3*K4 + 304*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1400*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1056*K2**2*K3**2 - 32*K2**2*K3*K7 - 176*K2**2*K4**2 + 2248*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 6658*K2**2 - 32*K2*K3**2*K4 + 824*K2*K3*K5 + 224*K2*K4*K6 + 8*K2*K5*K7 + 8*K3**2*K6 - 3428*K3**2 - 1136*K4**2 - 200*K5**2 - 54*K6**2 + 6894
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.488']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16492', 'vk6.16583', 'vk6.18100', 'vk6.18438', 'vk6.22923', 'vk6.23018', 'vk6.24551', 'vk6.24970', 'vk6.34902', 'vk6.35007', 'vk6.36690', 'vk6.37114', 'vk6.42469', 'vk6.42580', 'vk6.43970', 'vk6.44287', 'vk6.54735', 'vk6.54830', 'vk6.55912', 'vk6.56202', 'vk6.59199', 'vk6.59262', 'vk6.60442', 'vk6.60801', 'vk6.64753', 'vk6.64812', 'vk6.65558', 'vk6.65870', 'vk6.68051', 'vk6.68114', 'vk6.68640', 'vk6.68855']
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 O1O2O3O4O5U1U4U2O6U5U6U3
R3 orbit {'O1O2O3O4O5U1U4U2O6U5U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U6U1O6U4U2U5
Gauss code of K* O1O2O3U4O5O4O6U1U3U6U2U5
Gauss code of -K* O1O2O3U2O4O5O6U3U5U1U4U6
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 -1 2 0 2 1],[ 4 0 2 4 1 3 1],[ 1 -2 0 2 0 2 1],[-2 -4 -2 0 -1 0 1],[ 0 -1 0 1 0 1 1],[-2 -3 -2 0 -1 0 1],[-1 -1 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 2 1 0 -1 -4],[-2 0 0 1 -1 -2 -3],[-2 0 0 1 -1 -2 -4],[-1 -1 -1 0 -1 -1 -1],[ 0 1 1 1 0 0 -1],[ 1 2 2 1 0 0 -2],[ 4 3 4 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,0,1,4,0,-1,1,2,3,-1,1,2,4,1,1,1,0,1,2]
Phi over symmetry [-4,-1,0,1,2,2,1,3,4,2,3,1,1,1,1,0,1,1,2,2,0]
Phi of -K [-4,-1,0,1,2,2,1,3,4,2,3,1,1,1,1,0,1,1,2,2,0]
Phi of K* [-2,-2,-1,0,1,4,0,2,1,1,2,2,1,1,3,0,1,4,1,3,1]
Phi of -K* [-4,-1,0,1,2,2,2,1,1,3,4,0,1,2,2,1,1,1,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^4-2t^2
Normalized Jones-Krushkal polynomial 3z^2+24z+37
Enhanced Jones-Krushkal polynomial 3w^3z^2+24w^2z+37w
Inner characteristic polynomial t^6+45t^4+15t^2+1
Outer characteristic polynomial t^7+71t^5+59t^3+5t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 14*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 6*K2 + K3 + 7
2-strand cable arrow polynomial -192*K1**4*K2**2 + 960*K1**4*K2 - 3872*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 800*K1**3*K2*K3 - 896*K1**3*K3 - 384*K1**2*K2**4 + 736*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 6880*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 12136*K1**2*K2 - 832*K1**2*K3**2 - 160*K1**2*K4**2 - 8076*K1**2 + 1024*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1792*K1*K2**2*K3 - 288*K1*K2**2*K5 + 32*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 10320*K1*K2*K3 - 96*K1*K2*K4*K5 + 1944*K1*K3*K4 + 304*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1400*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1056*K2**2*K3**2 - 32*K2**2*K3*K7 - 176*K2**2*K4**2 + 2248*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 6658*K2**2 - 32*K2*K3**2*K4 + 824*K2*K3*K5 + 224*K2*K4*K6 + 8*K2*K5*K7 + 8*K3**2*K6 - 3428*K3**2 - 1136*K4**2 - 200*K5**2 - 54*K6**2 + 6894
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}]]
If K is slice False
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