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Flat knot 6.1030

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,1,1,2,1,1,2,2,0,0,0,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1030']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 8*K1*K2 + K1 + K2 + 3*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1030', '6.1062', '6.1226', '6.1508', '6.1525', '6.1596', '6.1724', '6.1729', '6.1735', '6.1738', '6.1789', '6.1809', '6.1921']
Outer characteristic polynomial of the knot is: t^7+30t^5+65t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1030']
2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 1152*K1**4*K2 - 1920*K1**4 + 256*K1**3*K2*K3 - 1056*K1**3*K3 + 480*K1**2*K2**3 - 3344*K1**2*K2**2 - 1120*K1**2*K2*K4 + 6296*K1**2*K2 - 288*K1**2*K3**2 - 32*K1**2*K4**2 - 5012*K1**2 + 384*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 32*K1*K2**2*K5 - 352*K1*K2*K3*K4 + 6120*K1*K2*K3 + 2008*K1*K3*K4 + 128*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 760*K2**4 - 32*K2**3*K6 - 384*K2**2*K3**2 - 128*K2**2*K4**2 + 1712*K2**2*K4 - 3890*K2**2 + 352*K2*K3*K5 + 104*K2*K4*K6 - 2332*K3**2 - 1130*K4**2 - 88*K5**2 - 22*K6**2 + 4080
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1030']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10141', 'vk6.10202', 'vk6.10345', 'vk6.10428', 'vk6.16694', 'vk6.19081', 'vk6.19126', 'vk6.19253', 'vk6.19548', 'vk6.23009', 'vk6.23126', 'vk6.25710', 'vk6.25754', 'vk6.26063', 'vk6.26442', 'vk6.29928', 'vk6.29983', 'vk6.30085', 'vk6.35002', 'vk6.35129', 'vk6.37810', 'vk6.37868', 'vk6.42572', 'vk6.44652', 'vk6.51625', 'vk6.51730', 'vk6.54907', 'vk6.56592', 'vk6.59331', 'vk6.64878', 'vk6.66190', 'vk6.66221']
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 O1O2O3O4U3U5O6U4O5U2U1U6
R3 orbit {'O1O2O3O4U3U5O6U4O5U2U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U4U3O6U1O5U6U2
Gauss code of K* O1O2O3U2U1U4U5O4O6U3O5U6
Gauss code of -K* O1O2O3U4O5U1O4O6U5U6U3U2
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 -1 1 0 2],[ 1 0 0 -1 2 0 2],[ 1 0 0 -1 2 0 1],[ 1 1 1 0 1 0 1],[-1 -2 -2 -1 0 -1 0],[ 0 0 0 0 1 0 2],[-2 -2 -1 -1 0 -2 0]]
Primitive based matrix [[ 0 2 1 0 -1 -1 -1],[-2 0 0 -2 -1 -1 -2],[-1 0 0 -1 -1 -2 -2],[ 0 2 1 0 0 0 0],[ 1 1 1 0 0 1 1],[ 1 1 2 0 -1 0 0],[ 1 2 2 0 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,1,1,1,0,2,1,1,2,1,1,2,2,0,0,0,-1,-1,0]
Phi over symmetry [-2,-1,0,1,1,1,0,2,1,1,2,1,1,2,2,0,0,0,-1,-1,0]
Phi of -K [-1,-1,-1,0,1,2,-1,-1,1,1,2,0,1,0,1,1,0,2,0,0,1]
Phi of K* [-2,-1,0,1,1,1,1,0,1,2,2,0,0,0,1,1,1,1,0,-1,-1]
Phi of -K* [-1,-1,-1,0,1,2,-1,0,0,2,1,1,0,1,1,0,2,2,1,2,0]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 7z^2+28z+29
Enhanced Jones-Krushkal polynomial 7w^3z^2+28w^2z+29w
Inner characteristic polynomial t^6+22t^4+30t^2+1
Outer characteristic polynomial t^7+30t^5+65t^3+7t
Flat arrow polynomial 4*K1**3 - 2*K1**2 - 8*K1*K2 + K1 + K2 + 3*K3 + 2
2-strand cable arrow polynomial -128*K1**4*K2**2 + 1152*K1**4*K2 - 1920*K1**4 + 256*K1**3*K2*K3 - 1056*K1**3*K3 + 480*K1**2*K2**3 - 3344*K1**2*K2**2 - 1120*K1**2*K2*K4 + 6296*K1**2*K2 - 288*K1**2*K3**2 - 32*K1**2*K4**2 - 5012*K1**2 + 384*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 32*K1*K2**2*K5 - 352*K1*K2*K3*K4 + 6120*K1*K2*K3 + 2008*K1*K3*K4 + 128*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 760*K2**4 - 32*K2**3*K6 - 384*K2**2*K3**2 - 128*K2**2*K4**2 + 1712*K2**2*K4 - 3890*K2**2 + 352*K2*K3*K5 + 104*K2*K4*K6 - 2332*K3**2 - 1130*K4**2 - 88*K5**2 - 22*K6**2 + 4080
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{6}, {4, 5}, {2, 3}, {1}]]
If K is slice False
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