| Min(phi) over symmetries of the knot is: [-3,-3,1,1,2,2,0,1,2,2,3,1,3,2,3,0,0,0,0,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.824'] |
| Arrow polynomial of the knot is: -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.546', '6.591', '6.598', '6.666', '6.680', '6.742', '6.778', '6.805', '6.822', '6.824', '6.1129', '6.1512', '6.1647', '6.1678', '6.1705', '6.1847', '6.1857'] |
| Outer characteristic polynomial of the knot is: t^7+82t^5+91t^3+11t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.824'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 - 192*K1**4*K2**2 + 1344*K1**4*K2 - 4816*K1**4 + 928*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1152*K1**3*K3 + 128*K1**2*K2**2*K4 - 5424*K1**2*K2**2 - 704*K1**2*K2*K4 + 11464*K1**2*K2 - 1104*K1**2*K3**2 - 32*K1**2*K3*K5 - 128*K1**2*K4**2 - 6924*K1**2 + 128*K1*K2**3*K3 - 864*K1*K2**2*K3 - 192*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 8704*K1*K2*K3 - 32*K1*K2*K4*K5 + 1832*K1*K3*K4 + 272*K1*K4*K5 + 16*K1*K5*K6 - 528*K2**4 - 432*K2**2*K3**2 - 48*K2**2*K4**2 + 1384*K2**2*K4 - 6124*K2**2 - 32*K2*K3**2*K4 + 568*K2*K3*K5 + 80*K2*K4*K6 + 16*K3**2*K6 - 2960*K3**2 - 940*K4**2 - 220*K5**2 - 28*K6**2 + 6202 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.824'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20027', 'vk6.20067', 'vk6.21297', 'vk6.21347', 'vk6.27078', 'vk6.27128', 'vk6.28781', 'vk6.28815', 'vk6.38467', 'vk6.38525', 'vk6.40654', 'vk6.40720', 'vk6.45351', 'vk6.45421', 'vk6.47118', 'vk6.47161', 'vk6.56826', 'vk6.56888', 'vk6.57958', 'vk6.58024', 'vk6.61344', 'vk6.61414', 'vk6.62518', 'vk6.62569', 'vk6.66538', 'vk6.66588', 'vk6.67325', 'vk6.67377', 'vk6.69184', 'vk6.69236', 'vk6.69933', 'vk6.69975'] |
| 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 | O1O2O3O4U2O5U3U1U6U4O6U5 |
| R3 orbit | {'O1O2O3O4U2O5U3U1U6U4O6U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5O6U1U6U4U2O5U3 |
| Gauss code of K* | O1O2O3O4U3O5U2U6U1U4O6U5 |
| Gauss code of -K* | O1O2O3O4U5O6U1U4U6U3O5U2 |
| 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 -2 -1 3 3 -1],[ 2 0 -1 1 3 3 1],[ 2 1 0 1 2 2 1],[ 1 -1 -1 0 1 2 0],[-3 -3 -2 -1 0 0 -3],[-3 -3 -2 -2 0 0 -3],[ 1 -1 -1 0 3 3 0]] |
| Primitive based matrix | [[ 0 3 3 -1 -1 -2 -2],[-3 0 0 -1 -3 -2 -3],[-3 0 0 -2 -3 -2 -3],[ 1 1 2 0 0 -1 -1],[ 1 3 3 0 0 -1 -1],[ 2 2 2 1 1 0 1],[ 2 3 3 1 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-3,1,1,2,2,0,1,3,2,3,2,3,2,3,0,1,1,1,1,-1] |
| Phi over symmetry | [-3,-3,1,1,2,2,0,1,2,2,3,1,3,2,3,0,0,0,0,0,-1] |
| Phi of -K | [-2,-2,-1,-1,3,3,-1,0,0,3,3,0,0,2,2,0,1,1,2,3,0] |
| Phi of K* | [-3,-3,1,1,2,2,0,1,2,2,3,1,3,2,3,0,0,0,0,0,-1] |
| Phi of -K* | [-2,-2,-1,-1,3,3,-1,1,1,3,3,1,1,2,2,0,1,2,3,3,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -2t^3+2t^2+2t |
| Normalized Jones-Krushkal polynomial | z^2+22z+41 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+22w^2z+41w |
| Inner characteristic polynomial | t^6+54t^4+41t^2+4 |
| Outer characteristic polynomial | t^7+82t^5+91t^3+11t |
| Flat arrow polynomial | -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7 |
| 2-strand cable arrow polynomial | -64*K1**6 - 192*K1**4*K2**2 + 1344*K1**4*K2 - 4816*K1**4 + 928*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1152*K1**3*K3 + 128*K1**2*K2**2*K4 - 5424*K1**2*K2**2 - 704*K1**2*K2*K4 + 11464*K1**2*K2 - 1104*K1**2*K3**2 - 32*K1**2*K3*K5 - 128*K1**2*K4**2 - 6924*K1**2 + 128*K1*K2**3*K3 - 864*K1*K2**2*K3 - 192*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 8704*K1*K2*K3 - 32*K1*K2*K4*K5 + 1832*K1*K3*K4 + 272*K1*K4*K5 + 16*K1*K5*K6 - 528*K2**4 - 432*K2**2*K3**2 - 48*K2**2*K4**2 + 1384*K2**2*K4 - 6124*K2**2 - 32*K2*K3**2*K4 + 568*K2*K3*K5 + 80*K2*K4*K6 + 16*K3**2*K6 - 2960*K3**2 - 940*K4**2 - 220*K5**2 - 28*K6**2 + 6202 |
| 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}], [{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}, {4}, {2, 3}, {1}]] |
| If K is slice | False |