| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,1,1,2,1,1,1,0,0,1,1,1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1840'] |
| 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+24t^5+47t^3+11t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1840'] |
| 2-strand cable arrow polynomial of the knot is: -16*K1**4 + 32*K1**3*K2*K3 - 192*K1**2*K2**4 + 352*K1**2*K2**3 - 4208*K1**2*K2**2 - 448*K1**2*K2*K4 + 3368*K1**2*K2 - 112*K1**2*K3**2 - 3204*K1**2 + 608*K1*K2**3*K3 - 320*K1*K2**2*K3 - 32*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 5488*K1*K2*K3 + 968*K1*K3*K4 + 72*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 256*K2**4*K4 - 1384*K2**4 - 448*K2**2*K3**2 - 528*K2**2*K4**2 + 1968*K2**2*K4 - 2638*K2**2 + 264*K2*K3*K5 + 304*K2*K4*K6 + 8*K3**2*K6 - 1936*K3**2 - 986*K4**2 - 76*K5**2 - 66*K6**2 + 3104 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1840'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11073', 'vk6.11151', 'vk6.12239', 'vk6.12346', 'vk6.18330', 'vk6.18668', 'vk6.24767', 'vk6.25226', 'vk6.30648', 'vk6.30741', 'vk6.31880', 'vk6.31949', 'vk6.36945', 'vk6.37409', 'vk6.44140', 'vk6.44462', 'vk6.51872', 'vk6.51919', 'vk6.52741', 'vk6.52852', 'vk6.56108', 'vk6.56328', 'vk6.60625', 'vk6.60959', 'vk6.63530', 'vk6.63574', 'vk6.64012', 'vk6.64056', 'vk6.65749', 'vk6.66015', 'vk6.68761', 'vk6.68970'] |
| 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 | O1O2O3U1U4O5U3O6O4U6U2U5 |
| R3 orbit | {'O1O2O3U1U4O5U3O6O4U6U2U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U2U5O6O5U1O4U6U3 |
| Gauss code of K* | O1O2O3U4U2U5O4O6U3O5U1U6 |
| Gauss code of -K* | O1O2O3U4U3O5U1O4O6U5U2U6 |
| 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 1 -1],[ 2 0 2 1 1 2 0],[ 0 -2 0 1 0 1 -1],[-1 -1 -1 0 -1 0 -1],[-1 -1 0 1 0 0 -1],[-1 -2 -1 0 0 0 0],[ 1 0 1 1 1 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 0 -1 -1],[-1 -1 0 0 -1 -1 -1],[-1 0 0 0 -1 0 -2],[ 0 0 1 1 0 -1 -2],[ 1 1 1 0 1 0 0],[ 2 1 1 2 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,0,1,1,0,1,1,1,1,0,2,1,2,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,2,1,1,2,1,1,1,0,0,1,1,1,0,0] |
| Phi of -K | [-2,-1,0,1,1,1,1,0,1,2,2,0,2,1,1,0,0,1,0,0,1] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,0,1,2,0,1,1,2,0,2,1,0,0,1] |
| Phi of -K* | [-2,-1,0,1,1,1,0,2,1,1,2,1,1,1,0,0,1,1,1,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 4z^2+17z+19 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2-8w^3z+25w^2z+19w |
| Inner characteristic polynomial | t^6+16t^4+24t^2 |
| Outer characteristic polynomial | t^7+24t^5+47t^3+11t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -16*K1**4 + 32*K1**3*K2*K3 - 192*K1**2*K2**4 + 352*K1**2*K2**3 - 4208*K1**2*K2**2 - 448*K1**2*K2*K4 + 3368*K1**2*K2 - 112*K1**2*K3**2 - 3204*K1**2 + 608*K1*K2**3*K3 - 320*K1*K2**2*K3 - 32*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 5488*K1*K2*K3 + 968*K1*K3*K4 + 72*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 256*K2**4*K4 - 1384*K2**4 - 448*K2**2*K3**2 - 528*K2**2*K4**2 + 1968*K2**2*K4 - 2638*K2**2 + 264*K2*K3*K5 + 304*K2*K4*K6 + 8*K3**2*K6 - 1936*K3**2 - 986*K4**2 - 76*K5**2 - 66*K6**2 + 3104 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {4, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}]] |
| If K is slice | False |