| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,-1,1,2,3,-1,1,2,2,0,1,0,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1114'] |
| Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.209', '6.231', '6.391', '6.419', '6.600', '6.661', '6.744', '6.812', '6.826', '6.1114', '6.1125', '6.1202', '6.1275', '6.1292', '6.1305', '6.1322', '6.1365', '6.1481', '6.1483', '6.1497', '6.1543', '6.1549', '6.1572', '6.1577', '6.1580', '6.1594', '6.1641', '6.1658', '6.1683', '6.1753', '6.1830', '6.1907', '6.1928'] |
| Outer characteristic polynomial of the knot is: t^7+40t^5+49t^3+11t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1114'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**6 + 928*K1**4*K2 - 3696*K1**4 + 384*K1**3*K2*K3 - 704*K1**3*K3 - 192*K1**2*K2**4 + 608*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 6240*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 288*K1**2*K2*K4 + 9776*K1**2*K2 - 944*K1**2*K3**2 - 48*K1**2*K4**2 - 5212*K1**2 + 576*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 416*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 7760*K1*K2*K3 + 920*K1*K3*K4 + 96*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1152*K2**4 - 32*K2**3*K6 - 288*K2**2*K3**2 - 16*K2**2*K4**2 + 1280*K2**2*K4 - 4318*K2**2 + 224*K2*K3*K5 + 16*K2*K4*K6 - 2084*K3**2 - 360*K4**2 - 40*K5**2 - 2*K6**2 + 4550 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1114'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11462', 'vk6.11766', 'vk6.12783', 'vk6.13119', 'vk6.17040', 'vk6.17283', 'vk6.20869', 'vk6.20949', 'vk6.22278', 'vk6.22361', 'vk6.23765', 'vk6.28343', 'vk6.31221', 'vk6.31572', 'vk6.32800', 'vk6.35551', 'vk6.36002', 'vk6.39963', 'vk6.40119', 'vk6.42038', 'vk6.42958', 'vk6.43255', 'vk6.46504', 'vk6.46637', 'vk6.52227', 'vk6.53065', 'vk6.53382', 'vk6.55449', 'vk6.58860', 'vk6.59930', 'vk6.64404', 'vk6.69726'] |
| 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 | O1O2O3O4U3U1O5U2U4O6U5U6 |
| R3 orbit | {'O1O2O3O4U3U1O5U2U4O6U5U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U6O5U1U3O6U4U2 |
| Gauss code of K* | O1O2U3O4O5U6O3O6U2U4U1U5 |
| Gauss code of -K* | O1O2U1O3O4U2O5O6U3U6U4U5 |
| 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 -1 -1 2 1 1],[ 2 0 1 0 3 2 0],[ 1 -1 0 0 2 2 1],[ 1 0 0 0 1 1 0],[-2 -3 -2 -1 0 1 1],[-1 -2 -2 -1 -1 0 1],[-1 0 -1 0 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 1 1 -1 -2 -3],[-1 -1 0 1 -1 -2 -2],[-1 -1 -1 0 0 -1 0],[ 1 1 1 0 0 0 0],[ 1 2 2 1 0 0 -1],[ 2 3 2 0 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,-1,-1,1,2,3,-1,1,2,2,0,1,0,0,0,1] |
| Phi over symmetry | [-2,-1,-1,1,1,2,-1,-1,1,2,3,-1,1,2,2,0,1,0,0,0,1] |
| Phi of -K | [-2,-1,-1,1,1,2,0,1,1,3,1,0,0,1,1,1,2,2,-1,2,2] |
| Phi of K* | [-2,-1,-1,1,1,2,2,2,1,2,1,-1,1,2,3,0,1,1,0,0,1] |
| Phi of -K* | [-2,-1,-1,1,1,2,0,1,0,2,3,0,0,1,1,1,2,2,-1,-1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 4z^2+25z+35 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+25w^2z+35w |
| Inner characteristic polynomial | t^6+28t^4+13t^2+1 |
| Outer characteristic polynomial | t^7+40t^5+49t^3+11t |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | -192*K1**6 + 928*K1**4*K2 - 3696*K1**4 + 384*K1**3*K2*K3 - 704*K1**3*K3 - 192*K1**2*K2**4 + 608*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 6240*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 288*K1**2*K2*K4 + 9776*K1**2*K2 - 944*K1**2*K3**2 - 48*K1**2*K4**2 - 5212*K1**2 + 576*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 416*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 7760*K1*K2*K3 + 920*K1*K3*K4 + 96*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1152*K2**4 - 32*K2**3*K6 - 288*K2**2*K3**2 - 16*K2**2*K4**2 + 1280*K2**2*K4 - 4318*K2**2 + 224*K2*K3*K5 + 16*K2*K4*K6 - 2084*K3**2 - 360*K4**2 - 40*K5**2 - 2*K6**2 + 4550 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}]] |
| If K is slice | False |