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Flat knot 6.1299

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,1,1,1,3,3,0,1,0,1,1,0,1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1299']
Arrow polynomial of the knot is: 8*K1**3 - 4*K1**2 - 10*K1*K2 - K1 + 2*K2 + 3*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.770', '6.1299', '6.1366']
Outer characteristic polynomial of the knot is: t^7+38t^5+75t^3+2t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1299']
2-strand cable arrow polynomial of the knot is: 2784*K1**4*K2 - 5952*K1**4 + 1856*K1**3*K2*K3 - 1696*K1**3*K3 - 384*K1**2*K2**4 + 1376*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 9216*K1**2*K2**2 - 1280*K1**2*K2*K4 + 10088*K1**2*K2 - 2048*K1**2*K3**2 - 96*K1**2*K4**2 - 3128*K1**2 + 1184*K1*K2**3*K3 - 2112*K1*K2**2*K3 - 480*K1*K2**2*K5 - 512*K1*K2*K3*K4 + 9280*K1*K2*K3 + 2408*K1*K3*K4 + 240*K1*K4*K5 - 192*K2**6 + 384*K2**4*K4 - 1808*K2**4 - 96*K2**3*K6 - 1152*K2**2*K3**2 - 264*K2**2*K4**2 + 2208*K2**2*K4 - 3506*K2**2 + 1000*K2*K3*K5 + 168*K2*K4*K6 - 2312*K3**2 - 920*K4**2 - 232*K5**2 - 30*K6**2 + 4054
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1299']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13909', 'vk6.14006', 'vk6.14189', 'vk6.14428', 'vk6.14980', 'vk6.15103', 'vk6.15661', 'vk6.16115', 'vk6.16701', 'vk6.16726', 'vk6.16845', 'vk6.18813', 'vk6.19270', 'vk6.19562', 'vk6.23135', 'vk6.23226', 'vk6.25407', 'vk6.26459', 'vk6.33720', 'vk6.33797', 'vk6.34276', 'vk6.35134', 'vk6.37540', 'vk6.42735', 'vk6.44683', 'vk6.54124', 'vk6.54918', 'vk6.54945', 'vk6.56399', 'vk6.56605', 'vk6.59343', 'vk6.64603']
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 O1O2O3U2O4O5U6U3O6U4U1U5
R3 orbit {'O1O2O3U2O4O5U6U3O6U4U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U3U5O6U1U6O4O5U2
Gauss code of K* O1O2O3U2U4U5O4U1U3O6O5U6
Gauss code of -K* O1O2O3U4O5O4U1U3O6U5U6U2
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 -1 0 0 3 -1],[ 1 0 -1 1 1 3 0],[ 1 1 0 1 1 1 0],[ 0 -1 -1 0 0 1 0],[ 0 -1 -1 0 0 1 0],[-3 -3 -1 -1 -1 0 -3],[ 1 0 0 0 0 3 0]]
Primitive based matrix [[ 0 3 0 0 -1 -1 -1],[-3 0 -1 -1 -1 -3 -3],[ 0 1 0 0 -1 0 -1],[ 0 1 0 0 -1 0 -1],[ 1 1 1 1 0 0 1],[ 1 3 0 0 0 0 0],[ 1 3 1 1 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,0,0,1,1,1,1,1,1,3,3,0,1,0,1,1,0,1,0,-1,0]
Phi over symmetry [-3,0,0,1,1,1,1,1,1,3,3,0,1,0,1,1,0,1,0,-1,0]
Phi of -K [-1,-1,-1,0,0,3,-1,0,0,0,3,0,0,0,1,1,1,1,0,2,2]
Phi of K* [-3,0,0,1,1,1,2,2,1,1,3,0,0,1,0,0,1,0,0,-1,0]
Phi of -K* [-1,-1,-1,0,0,3,-1,0,1,1,3,0,1,1,1,0,0,3,0,1,1]
Symmetry type of based matrix c
u-polynomial -t^3+3t
Normalized Jones-Krushkal polynomial 7z^2+28z+29
Enhanced Jones-Krushkal polynomial 7w^3z^2+28w^2z+29w
Inner characteristic polynomial t^6+26t^4+47t^2
Outer characteristic polynomial t^7+38t^5+75t^3+2t
Flat arrow polynomial 8*K1**3 - 4*K1**2 - 10*K1*K2 - K1 + 2*K2 + 3*K3 + 3
2-strand cable arrow polynomial 2784*K1**4*K2 - 5952*K1**4 + 1856*K1**3*K2*K3 - 1696*K1**3*K3 - 384*K1**2*K2**4 + 1376*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 9216*K1**2*K2**2 - 1280*K1**2*K2*K4 + 10088*K1**2*K2 - 2048*K1**2*K3**2 - 96*K1**2*K4**2 - 3128*K1**2 + 1184*K1*K2**3*K3 - 2112*K1*K2**2*K3 - 480*K1*K2**2*K5 - 512*K1*K2*K3*K4 + 9280*K1*K2*K3 + 2408*K1*K3*K4 + 240*K1*K4*K5 - 192*K2**6 + 384*K2**4*K4 - 1808*K2**4 - 96*K2**3*K6 - 1152*K2**2*K3**2 - 264*K2**2*K4**2 + 2208*K2**2*K4 - 3506*K2**2 + 1000*K2*K3*K5 + 168*K2*K4*K6 - 2312*K3**2 - 920*K4**2 - 232*K5**2 - 30*K6**2 + 4054
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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