| Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,1,1,2,1,3,1,2,2,2,0,1,0,2,1,-2] |
| Flat knots (up to 7 crossings) with same phi are :['6.360'] |
| Arrow polynomial of the knot is: 12*K1**3 - 8*K1**2 - 6*K1*K2 - 6*K1 + 4*K2 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.360', '6.988', '6.1003'] |
| Outer characteristic polynomial of the knot is: t^7+57t^5+97t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.360'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 320*K1**4*K2 - 1120*K1**4 + 128*K1**3*K2*K3 - 320*K1**3*K3 + 512*K1**2*K2**5 - 2688*K1**2*K2**4 + 5184*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 9616*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 480*K1**2*K2*K4 + 8048*K1**2*K2 - 224*K1**2*K3**2 - 4608*K1**2 - 512*K1*K2**4*K3 + 3168*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 2304*K1*K2**2*K3 - 256*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 6648*K1*K2*K3 + 520*K1*K3*K4 + 16*K1*K4*K5 - 608*K2**6 + 480*K2**4*K4 - 3984*K2**4 - 1264*K2**2*K3**2 - 104*K2**2*K4**2 + 2416*K2**2*K4 - 1176*K2**2 - 32*K2*K3**2*K4 + 376*K2*K3*K5 + 32*K2*K4*K6 - 1292*K3**2 - 316*K4**2 - 12*K5**2 + 3258 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.360'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70414', 'vk6.70421', 'vk6.70433', 'vk6.70436', 'vk6.70448', 'vk6.70451', 'vk6.70457', 'vk6.70538', 'vk6.70541', 'vk6.70617', 'vk6.70774', 'vk6.70781', 'vk6.70855', 'vk6.70864', 'vk6.70879', 'vk6.70883', 'vk6.70894', 'vk6.70898', 'vk6.70907', 'vk6.71017', 'vk6.71024', 'vk6.71125', 'vk6.71134', 'vk6.71256', 'vk6.71842', 'vk6.72279', 'vk6.72299', 'vk6.76664', 'vk6.76679', 'vk6.77635', 'vk6.87971', 'vk6.89207'] |
| 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 | O1O2O3O4O5U3U4O6U5U1U2U6 |
| R3 orbit | {'O1O2O3O4O5U3U4O6U5U1U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U6U4U5U1O6U2U3 |
| Gauss code of K* | O1O2O3O4U2U3U5U6U1O5O6U4 |
| Gauss code of -K* | O1O2O3O4U1O5O6U4U5U6U2U3 |
| 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 -2 0 1 3],[ 2 0 1 -2 0 2 3],[ 0 -1 0 -2 0 2 2],[ 2 2 2 0 1 2 1],[ 0 0 0 -1 0 1 1],[-1 -2 -2 -2 -1 0 1],[-3 -3 -2 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 1 0 0 -2 -2],[-3 0 -1 -1 -2 -1 -3],[-1 1 0 -1 -2 -2 -2],[ 0 1 1 0 0 -1 0],[ 0 2 2 0 0 -2 -1],[ 2 1 2 1 2 0 2],[ 2 3 2 0 1 -2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,0,2,2,1,1,2,1,3,1,2,2,2,0,1,0,2,1,-2] |
| Phi over symmetry | [-3,-1,0,0,2,2,1,1,2,1,3,1,2,2,2,0,1,0,2,1,-2] |
| Phi of -K | [-2,-2,0,0,1,3,-2,0,1,1,4,1,2,1,2,0,-1,1,0,2,1] |
| Phi of K* | [-3,-1,0,0,2,2,1,1,2,2,4,-1,0,1,1,0,1,0,2,1,-2] |
| Phi of -K* | [-2,-2,0,0,1,3,-2,0,1,2,3,1,2,2,1,0,1,1,2,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+2t^2-t |
| Normalized Jones-Krushkal polynomial | 5z^2+22z+25 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+7w^3z^2-2w^3z+24w^2z+25w |
| Inner characteristic polynomial | t^6+39t^4+23t^2 |
| Outer characteristic polynomial | t^7+57t^5+97t^3+8t |
| Flat arrow polynomial | 12*K1**3 - 8*K1**2 - 6*K1*K2 - 6*K1 + 4*K2 + 5 |
| 2-strand cable arrow polynomial | -192*K1**4*K2**2 + 320*K1**4*K2 - 1120*K1**4 + 128*K1**3*K2*K3 - 320*K1**3*K3 + 512*K1**2*K2**5 - 2688*K1**2*K2**4 + 5184*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 9616*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 480*K1**2*K2*K4 + 8048*K1**2*K2 - 224*K1**2*K3**2 - 4608*K1**2 - 512*K1*K2**4*K3 + 3168*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 2304*K1*K2**2*K3 - 256*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 6648*K1*K2*K3 + 520*K1*K3*K4 + 16*K1*K4*K5 - 608*K2**6 + 480*K2**4*K4 - 3984*K2**4 - 1264*K2**2*K3**2 - 104*K2**2*K4**2 + 2416*K2**2*K4 - 1176*K2**2 - 32*K2*K3**2*K4 + 376*K2*K3*K5 + 32*K2*K4*K6 - 1292*K3**2 - 316*K4**2 - 12*K5**2 + 3258 |
| 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}], [{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}], [{5, 6}, {4}, {3}, {2}, {1}], [{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}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]] |
| If K is slice | False |