| Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,0,1,3,2,4,1,2,1,2,1,1,2,2,2,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.209'] |
| 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+74t^5+47t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.209'] |
| 2-strand cable arrow polynomial of the knot is: -16*K1**4 + 32*K1**3*K2*K3 - 64*K1**3*K3 - 192*K1**2*K2**4 + 512*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5040*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 6712*K1**2*K2 - 80*K1**2*K3**2 - 144*K1**2*K4**2 - 6088*K1**2 + 288*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 896*K1*K2**2*K3 - 160*K1*K2**2*K5 - 384*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6696*K1*K2*K3 + 1248*K1*K3*K4 + 232*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 880*K2**4 - 464*K2**2*K3**2 - 304*K2**2*K4**2 + 2248*K2**2*K4 - 4742*K2**2 + 456*K2*K3*K5 + 192*K2*K4*K6 + 8*K3**2*K6 - 2172*K3**2 - 1124*K4**2 - 116*K5**2 - 42*K6**2 + 4650 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.209'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11063', 'vk6.11141', 'vk6.12225', 'vk6.12332', 'vk6.18343', 'vk6.18682', 'vk6.24781', 'vk6.25240', 'vk6.30634', 'vk6.30729', 'vk6.31866', 'vk6.31936', 'vk6.36963', 'vk6.37422', 'vk6.44150', 'vk6.44472', 'vk6.51864', 'vk6.51909', 'vk6.52727', 'vk6.52834', 'vk6.56119', 'vk6.56342', 'vk6.60636', 'vk6.60973', 'vk6.63521', 'vk6.63565', 'vk6.63999', 'vk6.64043', 'vk6.65765', 'vk6.66028', 'vk6.68770', 'vk6.68980'] |
| 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 | O1O2O3O4O5U2O6U4U6U1U5U3 |
| R3 orbit | {'O1O2O3O4O5U2O6U4U6U1U5U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U3U1U5U6U2O6U4 |
| Gauss code of K* | O1O2O3O4O5U3U6U5U1U4O6U2 |
| Gauss code of -K* | O1O2O3O4O5U4O6U2U5U1U6U3 |
| 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 -3 2 -1 3 1],[ 2 0 -1 3 0 3 1],[ 3 1 0 3 1 2 1],[-2 -3 -3 0 -2 1 1],[ 1 0 -1 2 0 2 1],[-3 -3 -2 -1 -2 0 0],[-1 -1 -1 -1 -1 0 0]] |
| Primitive based matrix | [[ 0 3 2 1 -1 -2 -3],[-3 0 -1 0 -2 -3 -2],[-2 1 0 1 -2 -3 -3],[-1 0 -1 0 -1 -1 -1],[ 1 2 2 1 0 0 -1],[ 2 3 3 1 0 0 -1],[ 3 2 3 1 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,1,2,3,1,0,2,3,2,-1,2,3,3,1,1,1,0,1,1] |
| Phi over symmetry | [-3,-2,-1,1,2,3,0,1,3,2,4,1,2,1,2,1,1,2,2,2,0] |
| Phi of -K | [-3,-2,-1,1,2,3,0,1,3,2,4,1,2,1,2,1,1,2,2,2,0] |
| Phi of K* | [-3,-2,-1,1,2,3,0,2,2,2,4,2,1,1,2,1,2,3,1,1,0] |
| Phi of -K* | [-3,-2,-1,1,2,3,1,1,1,3,2,0,1,3,3,1,2,2,-1,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 8z^2+28z+25 |
| Enhanced Jones-Krushkal polynomial | 8w^3z^2+28w^2z+25w |
| Inner characteristic polynomial | t^6+46t^4+17t^2 |
| Outer characteristic polynomial | t^7+74t^5+47t^3+6t |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | -16*K1**4 + 32*K1**3*K2*K3 - 64*K1**3*K3 - 192*K1**2*K2**4 + 512*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5040*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 6712*K1**2*K2 - 80*K1**2*K3**2 - 144*K1**2*K4**2 - 6088*K1**2 + 288*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 896*K1*K2**2*K3 - 160*K1*K2**2*K5 - 384*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6696*K1*K2*K3 + 1248*K1*K3*K4 + 232*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 880*K2**4 - 464*K2**2*K3**2 - 304*K2**2*K4**2 + 2248*K2**2*K4 - 4742*K2**2 + 456*K2*K3*K5 + 192*K2*K4*K6 + 8*K3**2*K6 - 2172*K3**2 - 1124*K4**2 - 116*K5**2 - 42*K6**2 + 4650 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{4, 6}, {2, 5}, {1, 3}]] |
| If K is slice | True |