| Min(phi) over symmetries of the knot is: [-4,0,0,0,1,3,1,2,3,1,4,1,1,1,1,0,1,1,1,2,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.181'] |
| Arrow polynomial of the knot is: -2*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 + 2*K2 + K3 + K4 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.62', '6.176', '6.181', '6.194', '6.228', '6.267', '6.268', '6.449'] |
| Outer characteristic polynomial of the knot is: t^7+69t^5+125t^3+9t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.181'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4 + 32*K1**3*K2*K3 - 80*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 32*K1**2*K2*K4 + 752*K1**2*K2 - 3440*K1**2*K3**2 - 48*K1**2*K6**2 - 2676*K1**2 - 576*K1*K2**2*K3 + 32*K1*K2*K3**3 - 352*K1*K2*K3*K4 - 224*K1*K2*K3*K6 + 5192*K1*K2*K3 + 3488*K1*K3*K4 + 48*K1*K4*K5 + 128*K1*K5*K6 + 40*K1*K6*K7 - 8*K2**4 - 32*K2**3*K6 - 352*K2**2*K3**2 - 32*K2**2*K3*K7 - 8*K2**2*K4**2 + 600*K2**2*K4 - 8*K2**2*K6**2 - 2326*K2**2 + 512*K2*K3*K5 + 152*K2*K4*K6 + 24*K2*K5*K7 + 8*K2*K6*K8 - 16*K3**4 + 104*K3**2*K6 - 2468*K3**2 + 16*K3*K4*K7 - 996*K4**2 - 116*K5**2 - 122*K6**2 - 20*K7**2 - 2*K8**2 + 2740 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.181'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4642', 'vk6.4917', 'vk6.6076', 'vk6.6577', 'vk6.8099', 'vk6.8487', 'vk6.9475', 'vk6.9846', 'vk6.20635', 'vk6.22064', 'vk6.28117', 'vk6.29560', 'vk6.39545', 'vk6.41770', 'vk6.46152', 'vk6.47796', 'vk6.48682', 'vk6.48877', 'vk6.49426', 'vk6.49655', 'vk6.50692', 'vk6.50881', 'vk6.51169', 'vk6.51382', 'vk6.57527', 'vk6.58717', 'vk6.62219', 'vk6.63167', 'vk6.67029', 'vk6.67904', 'vk6.69654', 'vk6.70337'] |
| 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 | O1O2O3O4O5U1O6U5U6U3U2U4 |
| R3 orbit | {'O1O2O3O4O5U1O6U5U6U3U2U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U2U4U3U6U1O6U5 |
| Gauss code of K* | O1O2O3O4O5U6U4U3U5U1O6U2 |
| Gauss code of -K* | O1O2O3O4O5U4O6U5U1U3U2U6 |
| 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 0 0 3 0 1],[ 4 0 3 2 4 1 1],[ 0 -3 0 0 2 -1 1],[ 0 -2 0 0 1 -1 1],[-3 -4 -2 -1 0 -1 1],[ 0 -1 1 1 1 0 1],[-1 -1 -1 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 1 0 0 0 -4],[-3 0 1 -1 -1 -2 -4],[-1 -1 0 -1 -1 -1 -1],[ 0 1 1 0 1 1 -1],[ 0 1 1 -1 0 0 -2],[ 0 2 1 -1 0 0 -3],[ 4 4 1 1 2 3 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,0,0,4,-1,1,1,2,4,1,1,1,1,-1,-1,1,0,2,3] |
| Phi over symmetry | [-4,0,0,0,1,3,1,2,3,1,4,1,1,1,1,0,1,1,1,2,-1] |
| Phi of -K | [-4,0,0,0,1,3,1,2,3,4,3,0,1,0,1,1,0,2,0,2,3] |
| Phi of K* | [-3,-1,0,0,0,4,3,1,2,2,3,0,0,0,4,-1,0,1,1,3,2] |
| Phi of -K* | [-4,0,0,0,1,3,1,2,3,1,4,1,1,1,1,0,1,1,1,2,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-t^3-t |
| Normalized Jones-Krushkal polynomial | 4z^2+21z+27 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2-4w^3z+25w^2z+27w |
| Inner characteristic polynomial | t^6+43t^4+35t^2+1 |
| Outer characteristic polynomial | t^7+69t^5+125t^3+9t |
| Flat arrow polynomial | -2*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 + 2*K2 + K3 + K4 + 2 |
| 2-strand cable arrow polynomial | -64*K1**4 + 32*K1**3*K2*K3 - 80*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 32*K1**2*K2*K4 + 752*K1**2*K2 - 3440*K1**2*K3**2 - 48*K1**2*K6**2 - 2676*K1**2 - 576*K1*K2**2*K3 + 32*K1*K2*K3**3 - 352*K1*K2*K3*K4 - 224*K1*K2*K3*K6 + 5192*K1*K2*K3 + 3488*K1*K3*K4 + 48*K1*K4*K5 + 128*K1*K5*K6 + 40*K1*K6*K7 - 8*K2**4 - 32*K2**3*K6 - 352*K2**2*K3**2 - 32*K2**2*K3*K7 - 8*K2**2*K4**2 + 600*K2**2*K4 - 8*K2**2*K6**2 - 2326*K2**2 + 512*K2*K3*K5 + 152*K2*K4*K6 + 24*K2*K5*K7 + 8*K2*K6*K8 - 16*K3**4 + 104*K3**2*K6 - 2468*K3**2 + 16*K3*K4*K7 - 996*K4**2 - 116*K5**2 - 122*K6**2 - 20*K7**2 - 2*K8**2 + 2740 |
| 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}], [{1, 6}, {5}, {4}, {3}, {2}], [{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}], [{4, 6}, {5}, {3}, {2}, {1}], [{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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {3}, {1, 2}]] |
| If K is slice | False |