| Min(phi) over symmetries of the knot is: [-4,-3,0,1,3,3,0,2,4,3,4,1,3,2,3,1,1,2,2,2,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.95'] |
| Arrow polynomial of the knot is: -2*K1*K2 + K1 - 2*K2**2 + K3 + K4 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.95', '6.107', '6.276', '6.292', '6.394', '6.429', '6.463'] |
| Outer characteristic polynomial of the knot is: t^7+110t^5+87t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.95'] |
| 2-strand cable arrow polynomial of the knot is: -400*K1**4 + 384*K1**3*K2*K3 + 32*K1**3*K3*K4 - 576*K1**3*K3 - 544*K1**2*K2**2 - 160*K1**2*K2*K4 + 1928*K1**2*K2 - 848*K1**2*K3**2 - 96*K1**2*K4**2 - 1748*K1**2 + 64*K1*K2**3*K3 - 224*K1*K2**2*K3 + 128*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 320*K1*K2*K3*K4 + 2248*K1*K2*K3 - 64*K1*K3**2*K5 + 928*K1*K3*K4 + 184*K1*K4*K5 - 32*K2**4 - 256*K2**2*K3**2 - 40*K2**2*K4**2 + 280*K2**2*K4 - 1246*K2**2 + 320*K2*K3*K5 + 8*K2*K4*K6 - 192*K3**4 - 80*K3**2*K4**2 + 112*K3**2*K6 - 808*K3**2 + 32*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 312*K4**2 - 104*K5**2 - 10*K6**2 - 4*K7**2 - 2*K8**2 + 1352 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.95'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16989', 'vk6.17230', 'vk6.19927', 'vk6.20219', 'vk6.21145', 'vk6.21513', 'vk6.23391', 'vk6.26843', 'vk6.26866', 'vk6.27415', 'vk6.28616', 'vk6.29027', 'vk6.35450', 'vk6.38275', 'vk6.38298', 'vk6.38826', 'vk6.40407', 'vk6.41017', 'vk6.42884', 'vk6.45150', 'vk6.45161', 'vk6.45589', 'vk6.46996', 'vk6.47354', 'vk6.55146', 'vk6.56706', 'vk6.56709', 'vk6.57795', 'vk6.58170', 'vk6.59518', 'vk6.61117', 'vk6.61563', 'vk6.62368', 'vk6.62737', 'vk6.64962', 'vk6.66396', 'vk6.67159', 'vk6.68252', 'vk6.69051', 'vk6.69840'] |
| 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 | O1O2O3O4O5O6U2U6U1U3U5U4 |
| R3 orbit | {'O1O2O3O4O5O6U2U6U1U3U5U4', 'O1O2O3O4O5U1O6U2U6U3U5U4'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5O6U3U2U4U6U1U5 |
| Gauss code of K* | O1O2O3O4O5O6U3U1U4U6U5U2 |
| Gauss code of -K* | O1O2O3O4O5O6U5U2U1U3U6U4 |
| 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 -3 -4 0 3 3 1],[ 3 0 -1 2 4 3 1],[ 4 1 0 2 4 3 1],[ 0 -2 -2 0 2 1 0],[-3 -4 -4 -2 0 0 0],[-3 -3 -3 -1 0 0 0],[-1 -1 -1 0 0 0 0]] |
| Primitive based matrix | [[ 0 3 3 1 0 -3 -4],[-3 0 0 0 -1 -3 -3],[-3 0 0 0 -2 -4 -4],[-1 0 0 0 0 -1 -1],[ 0 1 2 0 0 -2 -2],[ 3 3 4 1 2 0 -1],[ 4 3 4 1 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-3,-1,0,3,4,0,0,1,3,3,0,2,4,4,0,1,1,2,2,1] |
| Phi over symmetry | [-4,-3,0,1,3,3,0,2,4,3,4,1,3,2,3,1,1,2,2,2,0] |
| Phi of -K | [-4,-3,0,1,3,3,0,2,4,3,4,1,3,2,3,1,1,2,2,2,0] |
| Phi of K* | [-3,-3,-1,0,3,4,0,2,1,2,3,2,2,3,4,1,3,4,1,2,0] |
| Phi of -K* | [-4,-3,0,1,3,3,1,2,1,3,4,2,1,3,4,0,1,2,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-t^3-t |
| Normalized Jones-Krushkal polynomial | 4z^2+17z+19 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+17w^2z+19w |
| Inner characteristic polynomial | t^6+66t^4+23t^2 |
| Outer characteristic polynomial | t^7+110t^5+87t^3+3t |
| Flat arrow polynomial | -2*K1*K2 + K1 - 2*K2**2 + K3 + K4 + 2 |
| 2-strand cable arrow polynomial | -400*K1**4 + 384*K1**3*K2*K3 + 32*K1**3*K3*K4 - 576*K1**3*K3 - 544*K1**2*K2**2 - 160*K1**2*K2*K4 + 1928*K1**2*K2 - 848*K1**2*K3**2 - 96*K1**2*K4**2 - 1748*K1**2 + 64*K1*K2**3*K3 - 224*K1*K2**2*K3 + 128*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 320*K1*K2*K3*K4 + 2248*K1*K2*K3 - 64*K1*K3**2*K5 + 928*K1*K3*K4 + 184*K1*K4*K5 - 32*K2**4 - 256*K2**2*K3**2 - 40*K2**2*K4**2 + 280*K2**2*K4 - 1246*K2**2 + 320*K2*K3*K5 + 8*K2*K4*K6 - 192*K3**4 - 80*K3**2*K4**2 + 112*K3**2*K6 - 808*K3**2 + 32*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 312*K4**2 - 104*K5**2 - 10*K6**2 - 4*K7**2 - 2*K8**2 + 1352 |
| Genus of based matrix | 2 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{2, 6}, {5}, {4}, {3}, {1}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {3, 5}, {4}, {2}, {1}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]] |
| If K is slice | False |