| Min(phi) over symmetries of the knot is: [-4,-2,1,1,2,2,1,1,4,2,3,1,3,2,3,-1,0,0,1,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.160'] |
| Arrow polynomial of the knot is: -2*K1**2 - 4*K1*K2 + 2*K1 - 2*K2**2 + K2 + 2*K3 + K4 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.83', '6.151', '6.160', '6.190', '6.247', '6.262', '6.491', '6.514'] |
| Outer characteristic polynomial of the knot is: t^7+87t^5+70t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.160'] |
| 2-strand cable arrow polynomial of the knot is: -2080*K1**4 + 960*K1**3*K2*K3 + 96*K1**3*K3*K4 - 736*K1**3*K3 + 192*K1**2*K2**2*K4 - 1888*K1**2*K2**2 - 1248*K1**2*K2*K4 + 4712*K1**2*K2 - 1856*K1**2*K3**2 - 544*K1**2*K4**2 - 4328*K1**2 - 384*K1*K2**2*K3 - 32*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6224*K1*K2*K3 - 32*K1*K2*K4*K7 + 3616*K1*K3*K4 + 560*K1*K4*K5 + 48*K1*K5*K6 + 16*K1*K6*K7 - 72*K2**4 - 48*K2**2*K3**2 - 80*K2**2*K4**2 + 960*K2**2*K4 - 3596*K2**2 - 32*K2*K3**2*K4 + 216*K2*K3*K5 + 176*K2*K4*K6 + 16*K2*K5*K7 - 16*K3**2*K4**2 + 48*K3**2*K6 - 2936*K3**2 + 32*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1554*K4**2 - 204*K5**2 - 84*K6**2 - 20*K7**2 - 2*K8**2 + 4250 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.160'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16971', 'vk6.16983', 'vk6.17212', 'vk6.17224', 'vk6.20879', 'vk6.20886', 'vk6.22286', 'vk6.22295', 'vk6.23370', 'vk6.23670', 'vk6.23694', 'vk6.28350', 'vk6.35425', 'vk6.35856', 'vk6.35880', 'vk6.39982', 'vk6.39997', 'vk6.42052', 'vk6.43167', 'vk6.43179', 'vk6.46518', 'vk6.46533', 'vk6.55132', 'vk6.55136', 'vk6.55389', 'vk6.57678', 'vk6.57693', 'vk6.58874', 'vk6.59848', 'vk6.59856', 'vk6.68403', 'vk6.69738'] |
| 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 | O1O2O3O4O5U1O6U3U6U5U4U2 |
| R3 orbit | {'O1O2O3O4O5U1O6U3U6U5U4U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U4U2U1U6U3O6U5 |
| Gauss code of K* | O1O2O3O4O5U6U5U1U4U3O6U2 |
| Gauss code of -K* | O1O2O3O4O5U4O6U3U2U5U1U6 |
| 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 -4 1 -2 2 2 1],[ 4 0 4 1 3 2 1],[-1 -4 0 -3 1 1 1],[ 2 -1 3 0 3 2 1],[-2 -3 -1 -3 0 0 0],[-2 -2 -1 -2 0 0 0],[-1 -1 -1 -1 0 0 0]] |
| Primitive based matrix | [[ 0 2 2 1 1 -2 -4],[-2 0 0 0 -1 -2 -2],[-2 0 0 0 -1 -3 -3],[-1 0 0 0 -1 -1 -1],[-1 1 1 1 0 -3 -4],[ 2 2 3 1 3 0 -1],[ 4 2 3 1 4 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-1,-1,2,4,0,0,1,2,2,0,1,3,3,1,1,1,3,4,1] |
| Phi over symmetry | [-4,-2,1,1,2,2,1,1,4,2,3,1,3,2,3,-1,0,0,1,1,0] |
| Phi of -K | [-4,-2,1,1,2,2,1,1,4,3,4,0,2,1,2,-1,0,0,1,1,0] |
| Phi of K* | [-2,-2,-1,-1,2,4,0,0,1,1,3,0,1,2,4,1,0,1,2,4,1] |
| Phi of -K* | [-4,-2,1,1,2,2,1,1,4,2,3,1,3,2,3,-1,0,0,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-t^2-2t |
| Normalized Jones-Krushkal polynomial | 6z^2+27z+31 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+27w^2z+31w |
| Inner characteristic polynomial | t^6+57t^4+17t^2 |
| Outer characteristic polynomial | t^7+87t^5+70t^3+7t |
| Flat arrow polynomial | -2*K1**2 - 4*K1*K2 + 2*K1 - 2*K2**2 + K2 + 2*K3 + K4 + 3 |
| 2-strand cable arrow polynomial | -2080*K1**4 + 960*K1**3*K2*K3 + 96*K1**3*K3*K4 - 736*K1**3*K3 + 192*K1**2*K2**2*K4 - 1888*K1**2*K2**2 - 1248*K1**2*K2*K4 + 4712*K1**2*K2 - 1856*K1**2*K3**2 - 544*K1**2*K4**2 - 4328*K1**2 - 384*K1*K2**2*K3 - 32*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6224*K1*K2*K3 - 32*K1*K2*K4*K7 + 3616*K1*K3*K4 + 560*K1*K4*K5 + 48*K1*K5*K6 + 16*K1*K6*K7 - 72*K2**4 - 48*K2**2*K3**2 - 80*K2**2*K4**2 + 960*K2**2*K4 - 3596*K2**2 - 32*K2*K3**2*K4 + 216*K2*K3*K5 + 176*K2*K4*K6 + 16*K2*K5*K7 - 16*K3**2*K4**2 + 48*K3**2*K6 - 2936*K3**2 + 32*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1554*K4**2 - 204*K5**2 - 84*K6**2 - 20*K7**2 - 2*K8**2 + 4250 |
| 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}, {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}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]] |
| If K is slice | False |