| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,1,3,2,1,2,1,1,0,1,-1,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1861'] |
| 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+34t^5+151t^3+18t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1861'] |
| 2-strand cable arrow polynomial of the knot is: -16*K1**4 - 1216*K1**2*K2**4 + 992*K1**2*K2**3 - 5680*K1**2*K2**2 - 352*K1**2*K2*K4 + 3672*K1**2*K2 - 16*K1**2*K3**2 - 2572*K1**2 + 1952*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 448*K1*K2**2*K3 - 416*K1*K2**2*K5 + 5448*K1*K2*K3 - 96*K1*K2*K4*K5 + 544*K1*K3*K4 + 40*K1*K4*K5 - 288*K2**6 + 544*K2**4*K4 - 2440*K2**4 - 160*K2**3*K6 - 768*K2**2*K3**2 - 336*K2**2*K4**2 + 2088*K2**2*K4 - 1334*K2**2 - 96*K2*K3**2*K4 + 264*K2*K3*K5 + 200*K2*K4*K6 + 8*K3**2*K6 - 1440*K3**2 - 618*K4**2 - 28*K5**2 - 10*K6**2 + 2352 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1861'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10506', 'vk6.10515', 'vk6.10573', 'vk6.10590', 'vk6.10762', 'vk6.10777', 'vk6.10883', 'vk6.10890', 'vk6.17681', 'vk6.17685', 'vk6.17728', 'vk6.17732', 'vk6.24291', 'vk6.24295', 'vk6.30187', 'vk6.30196', 'vk6.30252', 'vk6.30269', 'vk6.30381', 'vk6.30396', 'vk6.36515', 'vk6.36523', 'vk6.43623', 'vk6.43631', 'vk6.43725', 'vk6.43735', 'vk6.60349', 'vk6.60367', 'vk6.63451', 'vk6.63460', 'vk6.65416', 'vk6.65426'] |
| 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 | O1O2O3U4U5O4U1O6O5U6U2U3 |
| R3 orbit | {'O1O2O3U4U5O4U1O6O5U6U2U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U2U4O5O4U3O6U5U6 |
| Gauss code of K* | O1O2O3U4U2U3O5O6U5O4U1U6 |
| Gauss code of -K* | O1O2O3U4U3O5U6O4O6U1U2U5 |
| 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 0 2 -1 1 -1],[ 1 0 0 1 1 2 -1],[ 0 0 0 1 -1 2 -1],[-2 -1 -1 0 -3 0 -1],[ 1 -1 1 3 0 1 0],[-1 -2 -2 0 -1 0 -1],[ 1 1 1 1 0 1 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 0 -1 -1 -1 -3],[-1 0 0 -2 -1 -2 -1],[ 0 1 2 0 -1 0 -1],[ 1 1 1 1 0 1 0],[ 1 1 2 0 -1 0 1],[ 1 3 1 1 0 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,0,1,1,1,3,2,1,2,1,1,0,1,-1,0,-1] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,1,1,3,2,1,2,1,1,0,1,-1,0,-1] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,0,0,1,2,-1,1,0,2,0,1,0,-1,1,1] |
| Phi of K* | [-2,-1,0,1,1,1,1,1,0,2,2,-1,1,0,1,0,1,0,-1,0,-1] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,0,1,1,3,-1,0,2,1,1,1,1,2,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | z^2+6z+9 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+5w^3z^2-16w^3z+22w^2z+9w |
| Inner characteristic polynomial | t^6+26t^4+104t^2+4 |
| Outer characteristic polynomial | t^7+34t^5+151t^3+18t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -16*K1**4 - 1216*K1**2*K2**4 + 992*K1**2*K2**3 - 5680*K1**2*K2**2 - 352*K1**2*K2*K4 + 3672*K1**2*K2 - 16*K1**2*K3**2 - 2572*K1**2 + 1952*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 448*K1*K2**2*K3 - 416*K1*K2**2*K5 + 5448*K1*K2*K3 - 96*K1*K2*K4*K5 + 544*K1*K3*K4 + 40*K1*K4*K5 - 288*K2**6 + 544*K2**4*K4 - 2440*K2**4 - 160*K2**3*K6 - 768*K2**2*K3**2 - 336*K2**2*K4**2 + 2088*K2**2*K4 - 1334*K2**2 - 96*K2*K3**2*K4 + 264*K2*K3*K5 + 200*K2*K4*K6 + 8*K3**2*K6 - 1440*K3**2 - 618*K4**2 - 28*K5**2 - 10*K6**2 + 2352 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {3, 5}, {1, 2}]] |
| If K is slice | False |