| Min(phi) over symmetries of the knot is: [-3,-2,0,0,2,3,-1,1,2,2,4,1,2,1,2,0,0,1,1,3,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.594'] |
| Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.414', '6.594', '6.608', '6.790', '6.1233', '6.1285', '6.1293', '6.1513', '6.1752', '6.1787', '6.1810', '6.1818', '6.1867', '6.1868', '6.1923'] |
| Outer characteristic polynomial of the knot is: t^7+73t^5+67t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.594'] |
| 2-strand cable arrow polynomial of the knot is: 512*K1**4*K2 - 3056*K1**4 + 128*K1**3*K2*K3 - 832*K1**3*K3 + 736*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 5488*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 640*K1**2*K2*K4 + 10448*K1**2*K2 - 1264*K1**2*K3**2 - 64*K1**2*K3*K5 - 7148*K1**2 + 416*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1056*K1*K2**2*K3 - 96*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8624*K1*K2*K3 + 1800*K1*K3*K4 + 56*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 1296*K2**4 - 768*K2**2*K3**2 - 80*K2**2*K4**2 + 1544*K2**2*K4 - 5180*K2**2 + 648*K2*K3*K5 + 40*K2*K4*K6 - 96*K3**4 + 64*K3**2*K6 - 2676*K3**2 - 716*K4**2 - 128*K5**2 - 20*K6**2 + 5674 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.594'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71389', 'vk6.71450', 'vk6.71915', 'vk6.71976', 'vk6.72458', 'vk6.72587', 'vk6.72704', 'vk6.72816', 'vk6.72881', 'vk6.73018', 'vk6.74243', 'vk6.74361', 'vk6.74436', 'vk6.74872', 'vk6.75050', 'vk6.76621', 'vk6.76923', 'vk6.77046', 'vk6.77416', 'vk6.77759', 'vk6.77811', 'vk6.79289', 'vk6.79407', 'vk6.79763', 'vk6.79825', 'vk6.79892', 'vk6.80853', 'vk6.80916', 'vk6.81388', 'vk6.85512', 'vk6.87224', 'vk6.89257'] |
| 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 | O1O2O3O4U2O5U1O6U5U3U6U4 |
| R3 orbit | {'O1O2O3O4U2O5U1O6U5U3U6U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U5U2U6O5U4O6U3 |
| Gauss code of K* | O1O2O3O4U5U6U2U4O6U1O5U3 |
| Gauss code of -K* | O1O2O3O4U2O5U4O6U1U3U6U5 |
| 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 -2 0 3 0 2],[ 3 0 0 3 4 1 2],[ 2 0 0 1 2 0 1],[ 0 -3 -1 0 2 0 2],[-3 -4 -2 -2 0 -1 1],[ 0 -1 0 0 1 0 1],[-2 -2 -1 -2 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 2 0 0 -2 -3],[-3 0 1 -1 -2 -2 -4],[-2 -1 0 -1 -2 -1 -2],[ 0 1 1 0 0 0 -1],[ 0 2 2 0 0 -1 -3],[ 2 2 1 0 1 0 0],[ 3 4 2 1 3 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,0,0,2,3,-1,1,2,2,4,1,2,1,2,0,0,1,1,3,0] |
| Phi over symmetry | [-3,-2,0,0,2,3,-1,1,2,2,4,1,2,1,2,0,0,1,1,3,0] |
| Phi of -K | [-3,-2,0,0,2,3,1,0,2,3,2,1,2,3,3,0,0,1,1,2,2] |
| Phi of K* | [-3,-2,0,0,2,3,2,1,2,3,2,0,1,3,3,0,1,0,2,2,1] |
| Phi of -K* | [-3,-2,0,0,2,3,0,1,3,2,4,0,1,1,2,0,1,1,2,2,-1] |
| 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+47t^4+19t^2 |
| Outer characteristic polynomial | t^7+73t^5+67t^3+7t |
| Flat arrow polynomial | 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5 |
| 2-strand cable arrow polynomial | 512*K1**4*K2 - 3056*K1**4 + 128*K1**3*K2*K3 - 832*K1**3*K3 + 736*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 5488*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 640*K1**2*K2*K4 + 10448*K1**2*K2 - 1264*K1**2*K3**2 - 64*K1**2*K3*K5 - 7148*K1**2 + 416*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1056*K1*K2**2*K3 - 96*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8624*K1*K2*K3 + 1800*K1*K3*K4 + 56*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 1296*K2**4 - 768*K2**2*K3**2 - 80*K2**2*K4**2 + 1544*K2**2*K4 - 5180*K2**2 + 648*K2*K3*K5 + 40*K2*K4*K6 - 96*K3**4 + 64*K3**2*K6 - 2676*K3**2 - 716*K4**2 - 128*K5**2 - 20*K6**2 + 5674 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {3, 5}, {1, 4}]] |
| If K is slice | False |