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Flat knot 6.1915

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,0,1,2,3,1,1,3,3,0,1,0,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1915']
Arrow polynomial of the knot is: -8*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + 5*K2 + 2*K3 + 2*K4 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.562', '6.1915']
Outer characteristic polynomial of the knot is: t^7+37t^5+64t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1915']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 736*K1**4*K2 - 2352*K1**4 + 608*K1**3*K2*K3 + 96*K1**3*K3*K4 - 1600*K1**3*K3 + 192*K1**2*K2**3 - 2592*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 448*K1**2*K2*K4 + 7672*K1**2*K2 - 1168*K1**2*K3**2 - 352*K1**2*K3*K5 - 128*K1**2*K4**2 - 32*K1**2*K5**2 - 6424*K1**2 - 512*K1*K2**2*K3 + 32*K1*K2*K3**3 - 192*K1*K2*K3*K4 + 6376*K1*K2*K3 - 96*K1*K3**2*K5 - 64*K1*K3*K4*K6 + 2272*K1*K3*K4 + 536*K1*K4*K5 + 152*K1*K5*K6 + 48*K1*K6*K7 - 128*K2**4 - 224*K2**2*K3**2 - 16*K2**2*K4**2 + 632*K2**2*K4 - 8*K2**2*K6**2 - 4536*K2**2 + 696*K2*K3*K5 + 120*K2*K4*K6 + 40*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 - 32*K3**2*K4**2 + 136*K3**2*K6 - 2880*K3**2 + 88*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1106*K4**2 - 488*K5**2 - 176*K6**2 - 64*K7**2 - 12*K8**2 + 5204
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1915']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3563', 'vk6.3583', 'vk6.3601', 'vk6.3810', 'vk6.3824', 'vk6.3841', 'vk6.3855', 'vk6.6980', 'vk6.6998', 'vk6.7011', 'vk6.7029', 'vk6.7202', 'vk6.7216', 'vk6.7232', 'vk6.15343', 'vk6.15355', 'vk6.15468', 'vk6.15480', 'vk6.33976', 'vk6.34020', 'vk6.34040', 'vk6.34435', 'vk6.48207', 'vk6.48225', 'vk6.48369', 'vk6.49950', 'vk6.49969', 'vk6.49987', 'vk6.53988', 'vk6.54000', 'vk6.54042', 'vk6.54492']
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 O1O2O3U1U3U4O5O4O6U5U2U6
R3 orbit {'O1O2O3U1U3U4O5O4O6U5U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U4U5O4O6O5U1U6U3
Gauss code of K* O1O2O3U4U2U5O4O5O6U1U6U3
Gauss code of -K* O1O2O3U1U4U3O4O5O6U5U2U6
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 0 1 1 -2 2],[ 2 0 2 1 2 0 1],[ 0 -2 0 0 1 -1 2],[-1 -1 0 0 -1 0 0],[-1 -2 -1 1 0 -2 1],[ 2 0 1 0 2 0 1],[-2 -1 -2 0 -1 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -2 -2],[-2 0 0 -1 -2 -1 -1],[-1 0 0 -1 0 0 -1],[-1 1 1 0 -1 -2 -2],[ 0 2 0 1 0 -1 -2],[ 2 1 0 2 1 0 0],[ 2 1 1 2 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,2,2,0,1,2,1,1,1,0,0,1,1,2,2,1,2,0]
Phi over symmetry [-2,-2,0,1,1,2,0,0,1,2,3,1,1,3,3,0,1,0,-1,0,1]
Phi of -K [-2,-2,0,1,1,2,0,0,1,2,3,1,1,3,3,0,1,0,-1,0,1]
Phi of K* [-2,-1,-1,0,2,2,0,1,0,3,3,1,0,1,1,1,2,3,0,1,0]
Phi of -K* [-2,-2,0,1,1,2,0,1,0,2,1,2,1,2,1,0,1,2,-1,0,1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 19z+39
Enhanced Jones-Krushkal polynomial 19w^2z+39w
Inner characteristic polynomial t^6+23t^4+35t^2+1
Outer characteristic polynomial t^7+37t^5+64t^3+6t
Flat arrow polynomial -8*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + 5*K2 + 2*K3 + 2*K4 + 6
2-strand cable arrow polynomial -192*K1**4*K2**2 + 736*K1**4*K2 - 2352*K1**4 + 608*K1**3*K2*K3 + 96*K1**3*K3*K4 - 1600*K1**3*K3 + 192*K1**2*K2**3 - 2592*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 448*K1**2*K2*K4 + 7672*K1**2*K2 - 1168*K1**2*K3**2 - 352*K1**2*K3*K5 - 128*K1**2*K4**2 - 32*K1**2*K5**2 - 6424*K1**2 - 512*K1*K2**2*K3 + 32*K1*K2*K3**3 - 192*K1*K2*K3*K4 + 6376*K1*K2*K3 - 96*K1*K3**2*K5 - 64*K1*K3*K4*K6 + 2272*K1*K3*K4 + 536*K1*K4*K5 + 152*K1*K5*K6 + 48*K1*K6*K7 - 128*K2**4 - 224*K2**2*K3**2 - 16*K2**2*K4**2 + 632*K2**2*K4 - 8*K2**2*K6**2 - 4536*K2**2 + 696*K2*K3*K5 + 120*K2*K4*K6 + 40*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 - 32*K3**2*K4**2 + 136*K3**2*K6 - 2880*K3**2 + 88*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1106*K4**2 - 488*K5**2 - 176*K6**2 - 64*K7**2 - 12*K8**2 + 5204
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}]]
If K is slice False
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