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Flat knot 6.114

Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,-1,1,1,3,3,1,1,2,2,1,2,2,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.114']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 4*K1*K2 - 4*K1*K3 - K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.114', '6.224', '6.363']
Outer characteristic polynomial of the knot is: t^7+86t^5+59t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.114']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 480*K1**4*K2 - 816*K1**4 + 128*K1**3*K2**3*K3 - 384*K1**3*K2**2*K3 + 1280*K1**3*K2*K3 - 480*K1**3*K3 - 320*K1**2*K2**4 + 2080*K1**2*K2**3 - 448*K1**2*K2**2*K3**2 - 8720*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 864*K1**2*K2*K4 + 7656*K1**2*K2 - 688*K1**2*K3**2 - 5708*K1**2 + 2272*K1*K2**3*K3 + 416*K1*K2**2*K3*K4 - 1760*K1*K2**2*K3 + 192*K1*K2**2*K4*K5 + 128*K1*K2**2*K5*K6 - 288*K1*K2**2*K5 + 128*K1*K2*K3**3 - 576*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 8920*K1*K2*K3 - 160*K1*K2*K4*K5 - 32*K1*K2*K5*K6 + 1368*K1*K3*K4 + 240*K1*K4*K5 + 120*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 224*K2**4*K4 - 2720*K2**4 + 96*K2**3*K3*K5 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 2208*K2**2*K3**2 - 496*K2**2*K4**2 + 2520*K2**2*K4 - 272*K2**2*K5**2 - 176*K2**2*K6**2 - 3530*K2**2 + 1576*K2*K3*K5 + 536*K2*K4*K6 + 56*K2*K5*K7 + 48*K2*K6*K8 + 24*K3**2*K6 - 2740*K3**2 - 1016*K4**2 - 404*K5**2 - 190*K6**2 - 4*K7**2 - 2*K8**2 + 4896
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.114']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4718', 'vk6.5033', 'vk6.6241', 'vk6.6693', 'vk6.8215', 'vk6.8651', 'vk6.9593', 'vk6.9922', 'vk6.20307', 'vk6.21640', 'vk6.27599', 'vk6.29151', 'vk6.39025', 'vk6.41273', 'vk6.45789', 'vk6.47466', 'vk6.48750', 'vk6.48947', 'vk6.49545', 'vk6.49763', 'vk6.50760', 'vk6.50960', 'vk6.51233', 'vk6.51444', 'vk6.57162', 'vk6.58350', 'vk6.61784', 'vk6.62903', 'vk6.66779', 'vk6.67655', 'vk6.69423', 'vk6.70145']
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 O1O2O3O4O5O6U3U4U6U1U5U2
R3 orbit {'O1O2O3O4O5O6U3U4U6U1U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U2U6U1U3U4
Gauss code of K* O1O2O3O4O5O6U4U6U1U2U5U3
Gauss code of -K* O1O2O3O4O5O6U4U2U5U6U1U3
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 1 -3 -1 3 2],[ 2 0 2 -2 0 3 2],[-1 -2 0 -3 -1 2 2],[ 3 2 3 0 1 3 2],[ 1 0 1 -1 0 2 1],[-3 -3 -2 -3 -2 0 0],[-2 -2 -2 -2 -1 0 0]]
Primitive based matrix [[ 0 3 2 1 -1 -2 -3],[-3 0 0 -2 -2 -3 -3],[-2 0 0 -2 -1 -2 -2],[-1 2 2 0 -1 -2 -3],[ 1 2 1 1 0 0 -1],[ 2 3 2 2 0 0 -2],[ 3 3 2 3 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,1,2,3,0,2,2,3,3,2,1,2,2,1,2,3,0,1,2]
Phi over symmetry [-3,-2,-1,1,2,3,-1,1,1,3,3,1,1,2,2,1,2,2,-1,0,1]
Phi of -K [-3,-2,-1,1,2,3,-1,1,1,3,3,1,1,2,2,1,2,2,-1,0,1]
Phi of K* [-3,-2,-1,1,2,3,1,0,2,2,3,-1,2,2,3,1,1,1,1,1,-1]
Phi of -K* [-3,-2,-1,1,2,3,2,1,3,2,3,0,2,2,3,1,1,2,2,2,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+58t^4+23t^2+1
Outer characteristic polynomial t^7+86t^5+59t^3+8t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 4*K1*K2 - 4*K1*K3 - K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial -256*K1**4*K2**2 + 480*K1**4*K2 - 816*K1**4 + 128*K1**3*K2**3*K3 - 384*K1**3*K2**2*K3 + 1280*K1**3*K2*K3 - 480*K1**3*K3 - 320*K1**2*K2**4 + 2080*K1**2*K2**3 - 448*K1**2*K2**2*K3**2 - 8720*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 864*K1**2*K2*K4 + 7656*K1**2*K2 - 688*K1**2*K3**2 - 5708*K1**2 + 2272*K1*K2**3*K3 + 416*K1*K2**2*K3*K4 - 1760*K1*K2**2*K3 + 192*K1*K2**2*K4*K5 + 128*K1*K2**2*K5*K6 - 288*K1*K2**2*K5 + 128*K1*K2*K3**3 - 576*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 8920*K1*K2*K3 - 160*K1*K2*K4*K5 - 32*K1*K2*K5*K6 + 1368*K1*K3*K4 + 240*K1*K4*K5 + 120*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 224*K2**4*K4 - 2720*K2**4 + 96*K2**3*K3*K5 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 2208*K2**2*K3**2 - 496*K2**2*K4**2 + 2520*K2**2*K4 - 272*K2**2*K5**2 - 176*K2**2*K6**2 - 3530*K2**2 + 1576*K2*K3*K5 + 536*K2*K4*K6 + 56*K2*K5*K7 + 48*K2*K6*K8 + 24*K3**2*K6 - 2740*K3**2 - 1016*K4**2 - 404*K5**2 - 190*K6**2 - 4*K7**2 - 2*K8**2 + 4896
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}]]
If K is slice False
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