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Flat knot 6.467

Min(phi) over symmetries of the knot is: [-4,-1,0,1,2,2,1,4,1,2,3,2,0,1,2,1,1,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.467']
Arrow polynomial of the knot is: -4*K1*K2 + 2*K1 - 2*K2**2 + 2*K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.87', '6.88', '6.184', '6.302', '6.459', '6.467', '6.506']
Outer characteristic polynomial of the knot is: t^7+85t^5+195t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.467']
2-strand cable arrow polynomial of the knot is: -496*K1**4 + 416*K1**3*K2*K3 + 32*K1**3*K3*K4 - 640*K1**3*K3 - 768*K1**2*K2**2 - 160*K1**2*K2*K4 + 2736*K1**2*K2 - 2384*K1**2*K3**2 - 96*K1**2*K4**2 - 4332*K1**2 + 128*K1*K2**3*K3 - 288*K1*K2**2*K3 + 384*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 256*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6344*K1*K2*K3 + 128*K1*K3**3*K4 - 32*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2848*K1*K3*K4 + 208*K1*K4*K5 - 64*K2**4 - 864*K2**2*K3**2 - 48*K2**2*K4**2 + 368*K2**2*K4 - 3172*K2**2 - 288*K2*K3**2*K4 + 648*K2*K3*K5 + 128*K2*K4*K6 - 448*K3**4 - 176*K3**2*K4**2 + 336*K3**2*K6 - 2816*K3**2 + 80*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 956*K4**2 - 168*K5**2 - 68*K6**2 - 12*K7**2 - 2*K8**2 + 3812
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.467']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4647', 'vk6.4928', 'vk6.6089', 'vk6.6584', 'vk6.8108', 'vk6.8506', 'vk6.9496', 'vk6.9857', 'vk6.20621', 'vk6.22050', 'vk6.28107', 'vk6.29550', 'vk6.39523', 'vk6.41748', 'vk6.46134', 'vk6.47778', 'vk6.48687', 'vk6.48888', 'vk6.49439', 'vk6.49662', 'vk6.50701', 'vk6.50900', 'vk6.51190', 'vk6.51393', 'vk6.57513', 'vk6.58703', 'vk6.62209', 'vk6.63157', 'vk6.67023', 'vk6.67898', 'vk6.69652', 'vk6.70335']
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 O1O2O3O4O5U6U4O6U2U1U3U5
R3 orbit {'O1O2O3O4O5U6U4O6U2U1U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U3U5U4O6U2U6
Gauss code of K* O1O2O3O4U2U1U3U5U4O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U1U6U2U4U3
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 -2 1 0 4 -1],[ 2 0 0 2 1 4 1],[ 2 0 0 1 1 3 1],[-1 -2 -1 0 1 2 -2],[ 0 -1 -1 -1 0 0 0],[-4 -4 -3 -2 0 0 -4],[ 1 -1 -1 2 0 4 0]]
Primitive based matrix [[ 0 4 1 0 -1 -2 -2],[-4 0 -2 0 -4 -3 -4],[-1 2 0 1 -2 -1 -2],[ 0 0 -1 0 0 -1 -1],[ 1 4 2 0 0 -1 -1],[ 2 3 1 1 1 0 0],[ 2 4 2 1 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,0,1,2,2,2,0,4,3,4,-1,2,1,2,0,1,1,1,1,0]
Phi over symmetry [-4,-1,0,1,2,2,1,4,1,2,3,2,0,1,2,1,1,1,0,0,0]
Phi of -K [-2,-2,-1,0,1,4,0,0,1,1,2,0,1,2,3,1,0,1,2,4,1]
Phi of K* [-4,-1,0,1,2,2,1,4,1,2,3,2,0,1,2,1,1,1,0,0,0]
Phi of -K* [-2,-2,-1,0,1,4,0,1,1,1,3,1,1,2,4,0,2,4,-1,0,2]
Symmetry type of based matrix c
u-polynomial -t^4+2t^2
Normalized Jones-Krushkal polynomial 5z^2+24z+29
Enhanced Jones-Krushkal polynomial 5w^3z^2-2w^3z+26w^2z+29w
Inner characteristic polynomial t^6+59t^4+99t^2+1
Outer characteristic polynomial t^7+85t^5+195t^3+10t
Flat arrow polynomial -4*K1*K2 + 2*K1 - 2*K2**2 + 2*K3 + K4 + 2
2-strand cable arrow polynomial -496*K1**4 + 416*K1**3*K2*K3 + 32*K1**3*K3*K4 - 640*K1**3*K3 - 768*K1**2*K2**2 - 160*K1**2*K2*K4 + 2736*K1**2*K2 - 2384*K1**2*K3**2 - 96*K1**2*K4**2 - 4332*K1**2 + 128*K1*K2**3*K3 - 288*K1*K2**2*K3 + 384*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 256*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6344*K1*K2*K3 + 128*K1*K3**3*K4 - 32*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2848*K1*K3*K4 + 208*K1*K4*K5 - 64*K2**4 - 864*K2**2*K3**2 - 48*K2**2*K4**2 + 368*K2**2*K4 - 3172*K2**2 - 288*K2*K3**2*K4 + 648*K2*K3*K5 + 128*K2*K4*K6 - 448*K3**4 - 176*K3**2*K4**2 + 336*K3**2*K6 - 2816*K3**2 + 80*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 956*K4**2 - 168*K5**2 - 68*K6**2 - 12*K7**2 - 2*K8**2 + 3812
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}]]
If K is slice False
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