| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,0,1,2,2,1,1,0,1,1,0,2,0,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1813'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.568', '6.806', '6.1000', '6.1049', '6.1081', '6.1101', '6.1112', '6.1122', '6.1193', '6.1195', '6.1208', '6.1235', '6.1263', '6.1517', '6.1528', '6.1537', '6.1542', '6.1545', '6.1558', '6.1569', '6.1575', '6.1644', '6.1650', '6.1681', '6.1692', '6.1702', '6.1706', '6.1728', '6.1734', '6.1739', '6.1799', '6.1813', '6.1820', '6.1834', '6.1840', '6.1851', '6.1861', '6.1878'] |
| Outer characteristic polynomial of the knot is: t^7+28t^5+121t^3+11t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1813'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**2*K2**4 + 1088*K1**2*K2**3 - 4944*K1**2*K2**2 - 576*K1**2*K2*K4 + 4520*K1**2*K2 - 3392*K1**2 + 448*K1*K2**3*K3 - 864*K1*K2**2*K3 - 416*K1*K2**2*K5 + 5128*K1*K2*K3 + 376*K1*K3*K4 + 128*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1752*K2**4 - 32*K2**3*K6 - 464*K2**2*K3**2 - 16*K2**2*K4**2 + 1792*K2**2*K4 - 2222*K2**2 + 640*K2*K3*K5 + 16*K2*K4*K6 - 1272*K3**2 - 410*K4**2 - 208*K5**2 - 2*K6**2 + 2592 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1813'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17003', 'vk6.17246', 'vk6.20524', 'vk6.21918', 'vk6.23410', 'vk6.23719', 'vk6.27975', 'vk6.29446', 'vk6.35479', 'vk6.35930', 'vk6.39381', 'vk6.41568', 'vk6.42912', 'vk6.43213', 'vk6.45956', 'vk6.47635', 'vk6.55184', 'vk6.55428', 'vk6.57389', 'vk6.58558', 'vk6.59563', 'vk6.59899', 'vk6.62049', 'vk6.63041', 'vk6.64982', 'vk6.65194', 'vk6.66938', 'vk6.67793', 'vk6.68270', 'vk6.68426', 'vk6.69547', 'vk6.70247'] |
| 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 | O1O2O3U4U5O6O5U2U1O4U6U3 |
| R3 orbit | {'O1O2O3U4U5O6O5U2U1O4U6U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4O5U3U2O6O4U6U5 |
| Gauss code of K* | O1O2U3O4O5U2U1U5O3O6U4U6 |
| Gauss code of -K* | O1O2U3O4O5U6U2O6O3U1U5U4 |
| 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 -1 -1 2 -1 1 0],[ 1 0 0 2 0 1 0],[ 1 0 0 1 1 1 -1],[-2 -2 -1 0 -1 -1 -2],[ 1 0 -1 1 0 2 1],[-1 -1 -1 1 -2 0 0],[ 0 0 1 2 -1 0 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 -1 -2 -1 -1 -2],[-1 1 0 0 -1 -2 -1],[ 0 2 0 0 1 -1 0],[ 1 1 1 -1 0 1 0],[ 1 1 2 1 -1 0 0],[ 1 2 1 0 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,1,2,1,1,2,0,1,2,1,-1,1,0,-1,0,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,0,1,2,2,1,1,0,1,1,0,2,0,0,-1] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,0,2,1,2,0,0,0,2,1,1,1,1,0,0] |
| Phi of K* | [-2,-1,0,1,1,1,0,0,1,2,2,1,1,0,1,1,0,2,0,0,-1] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,0,1,2,1,0,-1,1,1,0,1,2,0,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 6z^2+19z+15 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+8w^3z^2-6w^3z+25w^2z+15w |
| Inner characteristic polynomial | t^6+20t^4+76t^2+4 |
| Outer characteristic polynomial | t^7+28t^5+121t^3+11t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -192*K1**2*K2**4 + 1088*K1**2*K2**3 - 4944*K1**2*K2**2 - 576*K1**2*K2*K4 + 4520*K1**2*K2 - 3392*K1**2 + 448*K1*K2**3*K3 - 864*K1*K2**2*K3 - 416*K1*K2**2*K5 + 5128*K1*K2*K3 + 376*K1*K3*K4 + 128*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1752*K2**4 - 32*K2**3*K6 - 464*K2**2*K3**2 - 16*K2**2*K4**2 + 1792*K2**2*K4 - 2222*K2**2 + 640*K2*K3*K5 + 16*K2*K4*K6 - 1272*K3**2 - 410*K4**2 - 208*K5**2 - 2*K6**2 + 2592 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}]] |
| If K is slice | False |