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Flat knot 6.118

Min(phi) over symmetries of the knot is: [-3,-2,0,0,2,3,-1,1,2,3,3,1,2,2,2,0,0,1,1,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.118']
Arrow polynomial of the knot is: 16*K1**3 + 4*K1**2*K2 - 8*K1**2 - 8*K1*K2 - 4*K1*K3 - 8*K1 + 4*K2 + K4 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.118']
Outer characteristic polynomial of the knot is: t^7+73t^5+69t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.118']
2-strand cable arrow polynomial of the knot is: -544*K1**4 + 256*K1**3*K2*K3 - 384*K1**3*K3 - 128*K1**2*K2**4 + 1088*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 5728*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 512*K1**2*K2*K4 + 5872*K1**2*K2 - 544*K1**2*K3**2 - 128*K1**2*K3*K5 - 32*K1**2*K5**2 - 4136*K1**2 + 2816*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 1600*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 384*K1*K2**2*K5 + 128*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 6208*K1*K2*K3 - 64*K1*K2*K5*K6 + 912*K1*K3*K4 + 192*K1*K4*K5 + 48*K1*K5*K6 - 256*K2**6 - 512*K2**4*K3**2 - 32*K2**4*K4**2 + 320*K2**4*K4 - 3120*K2**4 + 320*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 + 128*K2**2*K3**2*K4 - 2112*K2**2*K3**2 - 64*K2**2*K3*K7 - 256*K2**2*K4**2 + 2208*K2**2*K4 - 128*K2**2*K5**2 - 48*K2**2*K6**2 - 1568*K2**2 + 1056*K2*K3*K5 + 144*K2*K4*K6 + 64*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 + 32*K3**2*K6 - 1600*K3**2 - 440*K4**2 - 168*K5**2 - 32*K6**2 - 2*K8**2 + 3064
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.118']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70416', 'vk6.70425', 'vk6.70440', 'vk6.70453', 'vk6.70542', 'vk6.70620', 'vk6.70782', 'vk6.70863', 'vk6.70875', 'vk6.70890', 'vk6.70905', 'vk6.71025', 'vk6.71133', 'vk6.71259', 'vk6.71846', 'vk6.72283', 'vk6.76680', 'vk6.77631', 'vk6.87962', 'vk6.89211']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is r.
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 O1O2O3O4O5O6U3U5U6U1U2U4
R3 orbit {'O1O2O3O4O5O6U3U5U6U1U2U4'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* O1O2O3O4O5O6U4U5U1U6U2U3
Gauss code of -K* O1O2O3O4O5O6U4U5U1U6U2U3
Diagrammatic symmetry type r
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 -3 3 0 2],[ 2 0 1 -2 3 0 2],[ 0 -1 0 -2 2 0 2],[ 3 2 2 0 3 1 2],[-3 -3 -2 -3 0 -1 1],[ 0 0 0 -1 1 0 1],[-2 -2 -2 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 2 0 0 -2 -3],[-3 0 1 -1 -2 -3 -3],[-2 -1 0 -1 -2 -2 -2],[ 0 1 1 0 0 0 -1],[ 0 2 2 0 0 -1 -2],[ 2 3 2 0 1 0 -2],[ 3 3 2 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,0,0,2,3,-1,1,2,3,3,1,2,2,2,0,0,1,1,2,2]
Phi over symmetry [-3,-2,0,0,2,3,-1,1,2,3,3,1,2,2,2,0,0,1,1,2,2]
Phi of -K [-3,-2,0,0,2,3,-1,1,2,3,3,1,2,2,2,0,0,1,1,2,2]
Phi of K* [-3,-2,0,0,2,3,2,1,2,2,3,0,1,2,3,0,1,1,2,2,-1]
Phi of -K* [-3,-2,0,0,2,3,2,1,2,2,3,0,1,2,3,0,1,1,2,2,-1]
Symmetry type of based matrix r
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial 5w^3z^2+22w^2z+25w
Inner characteristic polynomial t^6+47t^4+15t^2
Outer characteristic polynomial t^7+73t^5+69t^3+4t
Flat arrow polynomial 16*K1**3 + 4*K1**2*K2 - 8*K1**2 - 8*K1*K2 - 4*K1*K3 - 8*K1 + 4*K2 + K4 + 4
2-strand cable arrow polynomial -544*K1**4 + 256*K1**3*K2*K3 - 384*K1**3*K3 - 128*K1**2*K2**4 + 1088*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 5728*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 512*K1**2*K2*K4 + 5872*K1**2*K2 - 544*K1**2*K3**2 - 128*K1**2*K3*K5 - 32*K1**2*K5**2 - 4136*K1**2 + 2816*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 1600*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 384*K1*K2**2*K5 + 128*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 6208*K1*K2*K3 - 64*K1*K2*K5*K6 + 912*K1*K3*K4 + 192*K1*K4*K5 + 48*K1*K5*K6 - 256*K2**6 - 512*K2**4*K3**2 - 32*K2**4*K4**2 + 320*K2**4*K4 - 3120*K2**4 + 320*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 + 128*K2**2*K3**2*K4 - 2112*K2**2*K3**2 - 64*K2**2*K3*K7 - 256*K2**2*K4**2 + 2208*K2**2*K4 - 128*K2**2*K5**2 - 48*K2**2*K6**2 - 1568*K2**2 + 1056*K2*K3*K5 + 144*K2*K4*K6 + 64*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 + 32*K3**2*K6 - 1600*K3**2 - 440*K4**2 - 168*K5**2 - 32*K6**2 - 2*K8**2 + 3064
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {5}, {3, 4}, {2}]]
If K is slice False
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