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Flat knot 6.915

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,-1,2,1,1,3,1,0,1,0,0,1,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.915', '7.28360']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 6*K1**2 - 6*K1*K2 - 3*K1 - 2*K2**2 + 3*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.853', '6.915']
Outer characteristic polynomial of the knot is: t^7+36t^5+55t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.915', '7.28360']
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 576*K1**4*K2**2 + 3776*K1**4*K2 - 6576*K1**4 - 384*K1**3*K2**2*K3 + 2112*K1**3*K2*K3 - 1760*K1**3*K3 + 384*K1**2*K2**5 - 1664*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 3936*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 1664*K1**2*K2**2*K4 - 13536*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 96*K1**2*K2*K4**2 - 1568*K1**2*K2*K4 + 11064*K1**2*K2 - 1680*K1**2*K3**2 - 288*K1**2*K4**2 - 1860*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 2336*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 2112*K1*K2**2*K3 - 608*K1*K2**2*K5 + 192*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 928*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8880*K1*K2*K3 - 32*K1*K2*K4*K5 + 1312*K1*K3*K4 + 208*K1*K4*K5 + 8*K1*K5*K6 - 128*K2**8 + 256*K2**6*K4 - 1088*K2**6 - 192*K2**4*K3**2 - 192*K2**4*K4**2 + 1632*K2**4*K4 - 3864*K2**4 + 32*K2**3*K3*K5 - 288*K2**3*K6 + 192*K2**2*K3**2*K4 - 1232*K2**2*K3**2 - 32*K2**2*K3*K7 + 64*K2**2*K4**3 - 504*K2**2*K4**2 + 2520*K2**2*K4 - 1174*K2**2 + 624*K2*K3*K5 + 144*K2*K4*K6 - 64*K3**4 - 48*K3**2*K4**2 + 16*K3**2*K6 - 1200*K3**2 + 8*K3*K4*K7 - 8*K4**4 - 402*K4**2 - 52*K5**2 - 10*K6**2 + 3040
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.915']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.319', 'vk6.358', 'vk6.417', 'vk6.714', 'vk6.761', 'vk6.823', 'vk6.862', 'vk6.1502', 'vk6.1571', 'vk6.1949', 'vk6.1988', 'vk6.2044', 'vk6.2482', 'vk6.2648', 'vk6.2726', 'vk6.3111', 'vk6.10269', 'vk6.10414', 'vk6.18318', 'vk6.18657', 'vk6.19403', 'vk6.19696', 'vk6.25210', 'vk6.25854', 'vk6.26183', 'vk6.36931', 'vk6.37397', 'vk6.37965', 'vk6.38028', 'vk6.44856', 'vk6.56100', 'vk6.65741']
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 O1O2O3O4U3U1O5O6U5U6U2U4
R3 orbit {'O1O2O3O4U3U1O5O6U5U6U2U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U3U5U6O5O6U4U2
Gauss code of K* O1O2O3O4U5U3U6U4O6O5U1U2
Gauss code of -K* O1O2O3O4U3U4O5O6U1U6U2U5
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 0 -1 3 -1 1],[ 2 0 1 0 3 0 0],[ 0 -1 0 0 2 -1 1],[ 1 0 0 0 1 0 0],[-3 -3 -2 -1 0 -1 1],[ 1 0 1 0 1 0 1],[-1 0 -1 0 -1 -1 0]]
Primitive based matrix [[ 0 3 1 0 -1 -1 -2],[-3 0 1 -2 -1 -1 -3],[-1 -1 0 -1 0 -1 0],[ 0 2 1 0 0 -1 -1],[ 1 1 0 0 0 0 0],[ 1 1 1 1 0 0 0],[ 2 3 0 1 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,1,1,2,-1,2,1,1,3,1,0,1,0,0,1,1,0,0,0]
Phi over symmetry [-3,-1,0,1,1,2,-1,2,1,1,3,1,0,1,0,0,1,1,0,0,0]
Phi of -K [-2,-1,-1,0,1,3,1,1,1,3,2,0,0,1,3,1,2,3,0,1,3]
Phi of K* [-3,-1,0,1,1,2,3,1,3,3,2,0,1,2,3,0,1,1,0,1,1]
Phi of -K* [-2,-1,-1,0,1,3,0,0,1,0,3,0,0,0,1,1,1,1,1,2,-1]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 6z^2+25z+27
Enhanced Jones-Krushkal polynomial -2w^4z^2+8w^3z^2+25w^2z+27w
Inner characteristic polynomial t^6+20t^4+26t^2+1
Outer characteristic polynomial t^7+36t^5+55t^3+10t
Flat arrow polynomial -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 6*K1**2 - 6*K1*K2 - 3*K1 - 2*K2**2 + 3*K2 + K3 + 6
2-strand cable arrow polynomial -256*K1**6 - 576*K1**4*K2**2 + 3776*K1**4*K2 - 6576*K1**4 - 384*K1**3*K2**2*K3 + 2112*K1**3*K2*K3 - 1760*K1**3*K3 + 384*K1**2*K2**5 - 1664*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 3936*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 1664*K1**2*K2**2*K4 - 13536*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 96*K1**2*K2*K4**2 - 1568*K1**2*K2*K4 + 11064*K1**2*K2 - 1680*K1**2*K3**2 - 288*K1**2*K4**2 - 1860*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 2336*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 2112*K1*K2**2*K3 - 608*K1*K2**2*K5 + 192*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 928*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8880*K1*K2*K3 - 32*K1*K2*K4*K5 + 1312*K1*K3*K4 + 208*K1*K4*K5 + 8*K1*K5*K6 - 128*K2**8 + 256*K2**6*K4 - 1088*K2**6 - 192*K2**4*K3**2 - 192*K2**4*K4**2 + 1632*K2**4*K4 - 3864*K2**4 + 32*K2**3*K3*K5 - 288*K2**3*K6 + 192*K2**2*K3**2*K4 - 1232*K2**2*K3**2 - 32*K2**2*K3*K7 + 64*K2**2*K4**3 - 504*K2**2*K4**2 + 2520*K2**2*K4 - 1174*K2**2 + 624*K2*K3*K5 + 144*K2*K4*K6 - 64*K3**4 - 48*K3**2*K4**2 + 16*K3**2*K6 - 1200*K3**2 + 8*K3*K4*K7 - 8*K4**4 - 402*K4**2 - 52*K5**2 - 10*K6**2 + 3040
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {2, 3}, {1}]]
If K is slice False
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