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Flat knot 6.1152

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,1,2,2,1,1,1,1,1,0,1,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1152']
Arrow polynomial of the knot is: -2*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 3*K2 + K3 + 2*K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1152']
Outer characteristic polynomial of the knot is: t^7+56t^5+53t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1152']
2-strand cable arrow polynomial of the knot is: -16*K1**4 + 32*K1**3*K2*K3 - 800*K1**3*K3 - 288*K1**2*K2**2 + 352*K1**2*K2*K3**2 + 3600*K1**2*K2 - 1552*K1**2*K3**2 - 288*K1**2*K3*K5 - 48*K1**2*K6**2 - 4896*K1**2 + 96*K1*K2**3*K3 - 1504*K1*K2**2*K3 + 32*K1*K2*K3**3 - 192*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 5872*K1*K2*K3 - 64*K1*K2*K5*K6 + 2336*K1*K3*K4 + 216*K1*K4*K5 + 152*K1*K5*K6 + 72*K1*K6*K7 - 72*K2**4 - 32*K2**3*K6 - 960*K2**2*K3**2 - 8*K2**2*K4**2 + 584*K2**2*K4 - 48*K2**2*K6**2 - 3414*K2**2 - 64*K2*K3**2*K4 + 872*K2*K3*K5 + 200*K2*K4*K6 + 16*K2*K5*K7 + 32*K2*K6*K8 - 32*K3**4 + 120*K3**2*K6 - 2680*K3**2 + 8*K3*K4*K7 - 710*K4**2 - 276*K5**2 - 170*K6**2 - 20*K7**2 - 4*K8**2 + 3640
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1152']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11017', 'vk6.11096', 'vk6.12183', 'vk6.12290', 'vk6.18196', 'vk6.18532', 'vk6.24655', 'vk6.25082', 'vk6.30590', 'vk6.30685', 'vk6.31856', 'vk6.31902', 'vk6.36784', 'vk6.37235', 'vk6.44028', 'vk6.44369', 'vk6.51828', 'vk6.51895', 'vk6.52696', 'vk6.52790', 'vk6.55991', 'vk6.56264', 'vk6.60526', 'vk6.60870', 'vk6.63511', 'vk6.63555', 'vk6.63989', 'vk6.64033', 'vk6.65650', 'vk6.65930', 'vk6.68698', 'vk6.68908']
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 O1O2O3O4U1U3U5O6O5U2U4U6
R3 orbit {'O1O2O3O4U1U3U5O6O5U2U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U1U3O6O5U6U2U4
Gauss code of K* O1O2O3U4U3O5O6O4U1U5U2U6
Gauss code of -K* O1O2O3U4U1O4O5O6U2U5U3U6
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 -3 -1 0 2 1 1],[ 3 0 2 1 3 3 2],[ 1 -2 0 0 2 1 1],[ 0 -1 0 0 1 0 1],[-2 -3 -2 -1 0 -2 0],[-1 -3 -1 0 2 0 1],[-1 -2 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 -2 -1 -2 -3],[-1 0 0 -1 -1 -1 -2],[-1 2 1 0 0 -1 -3],[ 0 1 1 0 0 0 -1],[ 1 2 1 1 0 0 -2],[ 3 3 2 3 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,2,1,2,3,1,1,1,2,0,1,3,0,1,2]
Phi over symmetry [-3,-1,0,1,1,2,0,2,1,2,2,1,1,1,1,1,0,1,-1,-1,1]
Phi of -K [-3,-1,0,1,1,2,0,2,1,2,2,1,1,1,1,1,0,1,-1,-1,1]
Phi of K* [-2,-1,-1,0,1,3,-1,1,1,1,2,1,1,1,1,0,1,2,1,2,0]
Phi of -K* [-3,-1,0,1,1,2,2,1,2,3,3,0,1,1,2,1,0,1,-1,0,2]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 3z^2+20z+29
Enhanced Jones-Krushkal polynomial 3w^3z^2+20w^2z+29w
Inner characteristic polynomial t^6+40t^4+16t^2
Outer characteristic polynomial t^7+56t^5+53t^3+4t
Flat arrow polynomial -2*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 3*K2 + K3 + 2*K4 + 2
2-strand cable arrow polynomial -16*K1**4 + 32*K1**3*K2*K3 - 800*K1**3*K3 - 288*K1**2*K2**2 + 352*K1**2*K2*K3**2 + 3600*K1**2*K2 - 1552*K1**2*K3**2 - 288*K1**2*K3*K5 - 48*K1**2*K6**2 - 4896*K1**2 + 96*K1*K2**3*K3 - 1504*K1*K2**2*K3 + 32*K1*K2*K3**3 - 192*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 5872*K1*K2*K3 - 64*K1*K2*K5*K6 + 2336*K1*K3*K4 + 216*K1*K4*K5 + 152*K1*K5*K6 + 72*K1*K6*K7 - 72*K2**4 - 32*K2**3*K6 - 960*K2**2*K3**2 - 8*K2**2*K4**2 + 584*K2**2*K4 - 48*K2**2*K6**2 - 3414*K2**2 - 64*K2*K3**2*K4 + 872*K2*K3*K5 + 200*K2*K4*K6 + 16*K2*K5*K7 + 32*K2*K6*K8 - 32*K3**4 + 120*K3**2*K6 - 2680*K3**2 + 8*K3*K4*K7 - 710*K4**2 - 276*K5**2 - 170*K6**2 - 20*K7**2 - 4*K8**2 + 3640
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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