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Flat knot 6.936

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,0,1,3,0,1,0,0,0,0,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.936', '7.16071']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.121', '6.125', '6.866', '6.894', '6.936', '6.937']
Outer characteristic polynomial of the knot is: t^7+70t^5+202t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.936', '7.16071']
2-strand cable arrow polynomial of the knot is: -512*K1**6 - 1792*K1**4*K2**2 + 3520*K1**4*K2 - 4864*K1**4 - 384*K1**3*K2**2*K3 + 2304*K1**3*K2*K3 - 928*K1**3*K3 + 384*K1**2*K2**5 - 2944*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 5472*K1**2*K2**3 - 640*K1**2*K2**2*K3**2 + 1088*K1**2*K2**2*K4 - 14096*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 1632*K1**2*K2*K4 + 10416*K1**2*K2 - 704*K1**2*K3**2 - 160*K1**2*K4**2 - 2388*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 4512*K1*K2**3*K3 + 608*K1*K2**2*K3*K4 - 2816*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 992*K1*K2**2*K5 + 64*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8424*K1*K2*K3 - 32*K1*K2*K4*K5 + 952*K1*K3*K4 + 128*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1472*K2**6 - 576*K2**4*K3**2 - 192*K2**4*K4**2 + 1760*K2**4*K4 - 4688*K2**4 + 416*K2**3*K3*K5 + 64*K2**3*K4*K6 - 288*K2**3*K6 + 64*K2**2*K3**2*K4 - 1616*K2**2*K3**2 - 32*K2**2*K3*K7 - 536*K2**2*K4**2 + 3136*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 602*K2**2 + 664*K2*K3*K5 + 136*K2*K4*K6 - 1160*K3**2 - 410*K4**2 - 60*K5**2 - 6*K6**2 + 2976
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.936']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.483', 'vk6.552', 'vk6.613', 'vk6.953', 'vk6.1049', 'vk6.1117', 'vk6.1643', 'vk6.1756', 'vk6.1830', 'vk6.2140', 'vk6.2237', 'vk6.2301', 'vk6.2580', 'vk6.2848', 'vk6.3057', 'vk6.3179', 'vk6.12047', 'vk6.13040', 'vk6.20482', 'vk6.20995', 'vk6.21836', 'vk6.22419', 'vk6.27875', 'vk6.28449', 'vk6.29384', 'vk6.32691', 'vk6.39316', 'vk6.40214', 'vk6.41496', 'vk6.46717', 'vk6.46874', 'vk6.57350']
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 O1O2O3O4U5U6O5O6U1U3U4U2
R3 orbit {'O1O2O3O4U5U6O5O6U1U3U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U1U2U4O5O6U5U6
Gauss code of K* O1O2O3O4U1U4U2U3O5O6U5U6
Gauss code of -K* O1O2O3O4U5U6O5O6U2U3U1U4
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 1 0 2 -1 1],[ 3 0 3 1 2 2 4],[-1 -3 0 -1 1 -2 0],[ 0 -1 1 0 1 -1 1],[-2 -2 -1 -1 0 -3 -1],[ 1 -2 2 1 3 0 1],[-1 -4 0 -1 1 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 -1 -1 -1 -3 -2],[-1 1 0 0 -1 -1 -4],[-1 1 0 0 -1 -2 -3],[ 0 1 1 1 0 -1 -1],[ 1 3 1 2 1 0 -2],[ 3 2 4 3 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,1,1,1,3,2,0,1,1,4,1,2,3,1,1,2]
Phi over symmetry [-3,-1,0,1,1,2,0,2,0,1,3,0,1,0,0,0,0,1,0,0,0]
Phi of -K [-3,-1,0,1,1,2,0,2,0,1,3,0,1,0,0,0,0,1,0,0,0]
Phi of K* [-2,-1,-1,0,1,3,0,0,1,0,3,0,0,0,1,0,1,0,0,2,0]
Phi of -K* [-3,-1,0,1,1,2,2,1,3,4,2,1,2,1,3,1,1,1,0,1,1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 6z^2+25z+27
Enhanced Jones-Krushkal polynomial -2w^4z^2+8w^3z^2+25w^2z+27w
Inner characteristic polynomial t^6+54t^4+151t^2+4
Outer characteristic polynomial t^7+70t^5+202t^3+13t
Flat arrow polynomial -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial -512*K1**6 - 1792*K1**4*K2**2 + 3520*K1**4*K2 - 4864*K1**4 - 384*K1**3*K2**2*K3 + 2304*K1**3*K2*K3 - 928*K1**3*K3 + 384*K1**2*K2**5 - 2944*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 5472*K1**2*K2**3 - 640*K1**2*K2**2*K3**2 + 1088*K1**2*K2**2*K4 - 14096*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 1632*K1**2*K2*K4 + 10416*K1**2*K2 - 704*K1**2*K3**2 - 160*K1**2*K4**2 - 2388*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 4512*K1*K2**3*K3 + 608*K1*K2**2*K3*K4 - 2816*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 992*K1*K2**2*K5 + 64*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8424*K1*K2*K3 - 32*K1*K2*K4*K5 + 952*K1*K3*K4 + 128*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1472*K2**6 - 576*K2**4*K3**2 - 192*K2**4*K4**2 + 1760*K2**4*K4 - 4688*K2**4 + 416*K2**3*K3*K5 + 64*K2**3*K4*K6 - 288*K2**3*K6 + 64*K2**2*K3**2*K4 - 1616*K2**2*K3**2 - 32*K2**2*K3*K7 - 536*K2**2*K4**2 + 3136*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 602*K2**2 + 664*K2*K3*K5 + 136*K2*K4*K6 - 1160*K3**2 - 410*K4**2 - 60*K5**2 - 6*K6**2 + 2976
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {2, 3}, {1}]]
If K is slice False
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