Table of flat knot invariants
Invariant Table Check a Knot Higher Crossing Crossref Virtual Knots Please cite FlatKnotInfo
Glossary Reference List

Flat knot 6.122

Min(phi) over symmetries of the knot is: [-3,-3,1,1,1,3,-1,1,2,3,3,1,2,3,3,0,-1,0,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.122']
Arrow polynomial of the knot is: 8*K1**3 + 8*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.122', '6.327', '6.371', '6.1185']
Outer characteristic polynomial of the knot is: t^7+83t^5+43t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.122']
2-strand cable arrow polynomial of the knot is: 512*K1**4*K2**3 - 1792*K1**4*K2**2 + 2944*K1**4*K2 - 4608*K1**4 - 512*K1**3*K2**2*K3 + 704*K1**3*K2*K3 + 64*K1**3*K3*K4 - 576*K1**3*K3 - 768*K1**2*K2**4 + 2816*K1**2*K2**3 - 10240*K1**2*K2**2 - 960*K1**2*K2*K4 + 11248*K1**2*K2 - 256*K1**2*K3**2 - 224*K1**2*K4**2 - 6312*K1**2 + 1792*K1*K2**3*K3 + 512*K1*K2**2*K3*K4 - 1472*K1*K2**2*K3 + 384*K1*K2**2*K4*K5 + 128*K1*K2**2*K5*K6 - 576*K1*K2**2*K5 - 576*K1*K2*K3*K4 - 320*K1*K2*K3*K6 + 9456*K1*K2*K3 - 192*K1*K2*K4*K5 + 1952*K1*K3*K4 + 384*K1*K4*K5 + 64*K1*K5*K6 - 64*K2**6 - 128*K2**4*K3**2 - 64*K2**4*K4**2 + 320*K2**4*K4 - 2448*K2**4 + 320*K2**3*K3*K5 + 128*K2**3*K4*K6 - 64*K2**3*K6 + 128*K2**2*K3**2*K6 - 1568*K2**2*K3**2 - 856*K2**2*K4**2 + 2856*K2**2*K4 - 416*K2**2*K5**2 - 176*K2**2*K6**2 - 4782*K2**2 - 128*K2*K3**2*K4 + 1264*K2*K3*K5 + 520*K2*K4*K6 + 16*K2*K5*K7 + 64*K3**2*K6 - 2856*K3**2 - 1388*K4**2 - 272*K5**2 - 90*K6**2 + 6106
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.122']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11601', 'vk6.11952', 'vk6.12943', 'vk6.13254', 'vk6.20435', 'vk6.21795', 'vk6.27803', 'vk6.29313', 'vk6.31396', 'vk6.32570', 'vk6.32954', 'vk6.39227', 'vk6.41440', 'vk6.47560', 'vk6.53188', 'vk6.53501', 'vk6.57296', 'vk6.61974', 'vk6.64281', 'vk6.64491']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3O4O5O6U3U6U1U4U5U2
R3 orbit {'O1O2O3O4O5O6U3U6U1U4U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U2U3U6U1U4
Gauss code of K* Same
Gauss code of -K* O1O2O3O4O5O6U5U2U3U6U1U4
Diagrammatic symmetry type +
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 -3 1 3 1],[ 3 0 3 -1 2 3 1],[-1 -3 0 -3 0 2 1],[ 3 1 3 0 2 3 1],[-1 -2 0 -2 0 1 0],[-3 -3 -2 -3 -1 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 3 1 1 1 -3 -3],[-3 0 0 -1 -2 -3 -3],[-1 0 0 0 -1 -1 -1],[-1 1 0 0 0 -2 -2],[-1 2 1 0 0 -3 -3],[ 3 3 1 2 3 0 1],[ 3 3 1 2 3 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,-1,3,3,0,1,2,3,3,0,1,1,1,0,2,2,3,3,-1]
Phi over symmetry [-3,-3,1,1,1,3,-1,1,2,3,3,1,2,3,3,0,-1,0,0,1,2]
Phi of -K [-3,-3,1,1,1,3,-1,1,2,3,3,1,2,3,3,0,-1,0,0,1,2]
Phi of K* [-3,-1,-1,-1,3,3,0,1,2,3,3,0,1,1,1,0,2,2,3,3,-1]
Phi of -K* [-3,-3,1,1,1,3,-1,1,2,3,3,1,2,3,3,0,-1,0,0,1,2]
Symmetry type of based matrix +
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+53t^4+21t^2+1
Outer characteristic polynomial t^7+83t^5+43t^3+5t
Flat arrow polynomial 8*K1**3 + 8*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial 512*K1**4*K2**3 - 1792*K1**4*K2**2 + 2944*K1**4*K2 - 4608*K1**4 - 512*K1**3*K2**2*K3 + 704*K1**3*K2*K3 + 64*K1**3*K3*K4 - 576*K1**3*K3 - 768*K1**2*K2**4 + 2816*K1**2*K2**3 - 10240*K1**2*K2**2 - 960*K1**2*K2*K4 + 11248*K1**2*K2 - 256*K1**2*K3**2 - 224*K1**2*K4**2 - 6312*K1**2 + 1792*K1*K2**3*K3 + 512*K1*K2**2*K3*K4 - 1472*K1*K2**2*K3 + 384*K1*K2**2*K4*K5 + 128*K1*K2**2*K5*K6 - 576*K1*K2**2*K5 - 576*K1*K2*K3*K4 - 320*K1*K2*K3*K6 + 9456*K1*K2*K3 - 192*K1*K2*K4*K5 + 1952*K1*K3*K4 + 384*K1*K4*K5 + 64*K1*K5*K6 - 64*K2**6 - 128*K2**4*K3**2 - 64*K2**4*K4**2 + 320*K2**4*K4 - 2448*K2**4 + 320*K2**3*K3*K5 + 128*K2**3*K4*K6 - 64*K2**3*K6 + 128*K2**2*K3**2*K6 - 1568*K2**2*K3**2 - 856*K2**2*K4**2 + 2856*K2**2*K4 - 416*K2**2*K5**2 - 176*K2**2*K6**2 - 4782*K2**2 - 128*K2*K3**2*K4 + 1264*K2*K3*K5 + 520*K2*K4*K6 + 16*K2*K5*K7 + 64*K3**2*K6 - 2856*K3**2 - 1388*K4**2 - 272*K5**2 - 90*K6**2 + 6106
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {4}, {1, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {5}, {1, 3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {1, 3}]]
If K is slice False
Contact