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Flat knot 6.923

Min(phi) over symmetries of the knot is: [-3,-1,0,0,1,3,0,0,2,3,4,0,0,1,1,1,0,1,1,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.923']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 14*K1**2 - 8*K1*K2 - 2*K1*K3 - 2*K1 - 2*K2**2 + 6*K2 + 2*K3 + K4 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.923']
Outer characteristic polynomial of the knot is: t^7+62t^5+92t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.923']
2-strand cable arrow polynomial of the knot is: -192*K1**6 - 192*K1**4*K2**2 + 960*K1**4*K2 - 4288*K1**4 + 1184*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1024*K1**3*K3 - 128*K1**2*K2**4 + 512*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 8176*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 + 64*K1**2*K2*K4**2 - 992*K1**2*K2*K4 + 11672*K1**2*K2 - 2208*K1**2*K3**2 - 32*K1**2*K3*K5 - 320*K1**2*K4**2 - 32*K1**2*K5**2 - 6988*K1**2 + 1088*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1120*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 832*K1*K2**2*K5 + 192*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 608*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 11864*K1*K2*K3 - 128*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 64*K1*K3**3*K4 + 2840*K1*K3*K4 + 624*K1*K4*K5 + 64*K1*K5*K6 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1608*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 + 64*K2**2*K3**2*K4 - 1520*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 336*K2**2*K4**2 - 32*K2**2*K4*K8 + 2272*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 5988*K2**2 - 224*K2*K3**2*K4 + 1496*K2*K3*K5 + 280*K2*K4*K6 + 40*K2*K5*K7 + 8*K2*K6*K8 - 224*K3**4 - 112*K3**2*K4**2 + 176*K3**2*K6 - 3576*K3**2 + 40*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1256*K4**2 - 436*K5**2 - 68*K6**2 - 8*K7**2 - 2*K8**2 + 6640
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.923']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4072', 'vk6.4103', 'vk6.5314', 'vk6.5345', 'vk6.7442', 'vk6.7471', 'vk6.8945', 'vk6.8976', 'vk6.10124', 'vk6.10295', 'vk6.10318', 'vk6.14541', 'vk6.15273', 'vk6.15400', 'vk6.15763', 'vk6.16180', 'vk6.29866', 'vk6.29897', 'vk6.33919', 'vk6.34002', 'vk6.34216', 'vk6.34383', 'vk6.48469', 'vk6.49175', 'vk6.50222', 'vk6.50253', 'vk6.51600', 'vk6.53954', 'vk6.54017', 'vk6.54167', 'vk6.54455', 'vk6.63311']
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 O1O2O3O4U3U5O6O5U1U6U2U4
R3 orbit {'O1O2O3O4U3U5O6O5U1U6U2U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U3U5U4O6O5U6U2
Gauss code of K* O1O2O3O4U1U3U5U4O5O6U2U6
Gauss code of -K* O1O2O3O4U5U3O5O6U1U6U2U4
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 0 -1 3 1 0],[ 3 0 2 0 4 3 0],[ 0 -2 0 0 2 1 -1],[ 1 0 0 0 1 1 0],[-3 -4 -2 -1 0 -2 -1],[-1 -3 -1 -1 2 0 0],[ 0 0 1 0 1 0 0]]
Primitive based matrix [[ 0 3 1 0 0 -1 -3],[-3 0 -2 -1 -2 -1 -4],[-1 2 0 0 -1 -1 -3],[ 0 1 0 0 1 0 0],[ 0 2 1 -1 0 0 -2],[ 1 1 1 0 0 0 0],[ 3 4 3 0 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,0,1,3,2,1,2,1,4,0,1,1,3,-1,0,0,0,2,0]
Phi over symmetry [-3,-1,0,0,1,3,0,0,2,3,4,0,0,1,1,1,0,1,1,2,2]
Phi of -K [-3,-1,0,0,1,3,2,1,3,1,2,1,1,1,3,1,0,1,1,2,0]
Phi of K* [-3,-1,0,0,1,3,0,1,2,3,2,0,1,1,1,-1,1,1,1,3,2]
Phi of -K* [-3,-1,0,0,1,3,0,0,2,3,4,0,0,1,1,1,0,1,1,2,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 3z^2+24z+37
Enhanced Jones-Krushkal polynomial 3w^3z^2+24w^2z+37w
Inner characteristic polynomial t^6+42t^4+48t^2+1
Outer characteristic polynomial t^7+62t^5+92t^3+8t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 14*K1**2 - 8*K1*K2 - 2*K1*K3 - 2*K1 - 2*K2**2 + 6*K2 + 2*K3 + K4 + 8
2-strand cable arrow polynomial -192*K1**6 - 192*K1**4*K2**2 + 960*K1**4*K2 - 4288*K1**4 + 1184*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1024*K1**3*K3 - 128*K1**2*K2**4 + 512*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 8176*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 + 64*K1**2*K2*K4**2 - 992*K1**2*K2*K4 + 11672*K1**2*K2 - 2208*K1**2*K3**2 - 32*K1**2*K3*K5 - 320*K1**2*K4**2 - 32*K1**2*K5**2 - 6988*K1**2 + 1088*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1120*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 832*K1*K2**2*K5 + 192*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 608*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 11864*K1*K2*K3 - 128*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 64*K1*K3**3*K4 + 2840*K1*K3*K4 + 624*K1*K4*K5 + 64*K1*K5*K6 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1608*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 + 64*K2**2*K3**2*K4 - 1520*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 336*K2**2*K4**2 - 32*K2**2*K4*K8 + 2272*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 5988*K2**2 - 224*K2*K3**2*K4 + 1496*K2*K3*K5 + 280*K2*K4*K6 + 40*K2*K5*K7 + 8*K2*K6*K8 - 224*K3**4 - 112*K3**2*K4**2 + 176*K3**2*K6 - 3576*K3**2 + 40*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1256*K4**2 - 436*K5**2 - 68*K6**2 - 8*K7**2 - 2*K8**2 + 6640
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}]]
If K is slice False
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