| Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,3,2,3,1,1,1,1,1,1,1,2,1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.211'] |
| Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 6*K1*K2 + 4*K2 + 2*K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.211', '6.557', '6.676', '6.685', '6.750', '6.751', '6.856', '6.919', '6.1093', '6.1371'] |
| Outer characteristic polynomial of the knot is: t^7+60t^5+38t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.211'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 + 64*K1**4*K2 - 1456*K1**4 - 384*K1**3*K3 + 128*K1**2*K2**3 - 2096*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 288*K1**2*K2*K4 + 4208*K1**2*K2 - 592*K1**2*K3**2 - 48*K1**2*K4**2 - 2596*K1**2 + 96*K1*K2**3*K3 - 384*K1*K2**2*K3 - 64*K1*K2**2*K5 + 32*K1*K2*K3**3 - 224*K1*K2*K3*K4 + 4000*K1*K2*K3 + 912*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 384*K2**4 - 432*K2**2*K3**2 - 56*K2**2*K4**2 + 616*K2**2*K4 - 2148*K2**2 - 32*K2*K3**2*K4 + 376*K2*K3*K5 + 24*K2*K4*K6 - 64*K3**4 + 40*K3**2*K6 - 1244*K3**2 - 348*K4**2 - 64*K5**2 - 4*K6**2 + 2266 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.211'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4210', 'vk6.4290', 'vk6.5468', 'vk6.5579', 'vk6.7571', 'vk6.7660', 'vk6.9072', 'vk6.9152', 'vk6.11167', 'vk6.12251', 'vk6.12358', 'vk6.19387', 'vk6.19680', 'vk6.19789', 'vk6.26167', 'vk6.26222', 'vk6.26583', 'vk6.26667', 'vk6.30757', 'vk6.31304', 'vk6.31699', 'vk6.31958', 'vk6.32458', 'vk6.32873', 'vk6.38171', 'vk6.38202', 'vk6.39098', 'vk6.41354', 'vk6.44828', 'vk6.44943', 'vk6.45850', 'vk6.48520', 'vk6.49321', 'vk6.52296', 'vk6.53136', 'vk6.58446', 'vk6.62966', 'vk6.63589', 'vk6.66323', 'vk6.66342'] |
| 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 | O1O2O3O4O5U2O6U4U6U5U1U3 |
| R3 orbit | {'O1O2O3O4U1O5O6U4U6U2U5U3', 'O1O2O3O4O5U2O6U4U6U5U1U3'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U3U5U1U6U2O6U4 |
| Gauss code of K* | O1O2O3O4O5U4U6U5U1U3O6U2 |
| Gauss code of -K* | O1O2O3O4O5U4O6U3U5U1U6U2 |
| 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 -1 -3 2 -1 2 1],[ 1 0 -2 2 -1 2 1],[ 3 2 0 3 1 2 1],[-2 -2 -3 0 -2 1 1],[ 1 1 -1 2 0 2 1],[-2 -2 -2 -1 -2 0 0],[-1 -1 -1 -1 -1 0 0]] |
| Primitive based matrix | [[ 0 2 2 1 -1 -1 -3],[-2 0 1 1 -2 -2 -3],[-2 -1 0 0 -2 -2 -2],[-1 -1 0 0 -1 -1 -1],[ 1 2 2 1 0 1 -1],[ 1 2 2 1 -1 0 -2],[ 3 3 2 1 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-1,1,1,3,-1,-1,2,2,3,0,2,2,2,1,1,1,-1,1,2] |
| Phi over symmetry | [-3,-1,-1,1,2,2,0,1,3,2,3,1,1,1,1,1,1,1,2,1,-1] |
| Phi of -K | [-3,-1,-1,1,2,2,0,1,3,2,3,1,1,1,1,1,1,1,2,1,-1] |
| Phi of K* | [-2,-2,-1,1,1,3,-1,1,1,1,3,2,1,1,2,1,1,3,-1,0,1] |
| Phi of -K* | [-3,-1,-1,1,2,2,1,2,1,2,3,1,1,2,2,1,2,2,0,-1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-2t^2+t |
| Normalized Jones-Krushkal polynomial | z^2+14z+25 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+14w^2z+25w |
| Inner characteristic polynomial | t^6+40t^4+8t^2 |
| Outer characteristic polynomial | t^7+60t^5+38t^3+3t |
| Flat arrow polynomial | 4*K1**3 - 8*K1**2 - 6*K1*K2 + 4*K2 + 2*K3 + 5 |
| 2-strand cable arrow polynomial | -64*K1**6 + 64*K1**4*K2 - 1456*K1**4 - 384*K1**3*K3 + 128*K1**2*K2**3 - 2096*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 288*K1**2*K2*K4 + 4208*K1**2*K2 - 592*K1**2*K3**2 - 48*K1**2*K4**2 - 2596*K1**2 + 96*K1*K2**3*K3 - 384*K1*K2**2*K3 - 64*K1*K2**2*K5 + 32*K1*K2*K3**3 - 224*K1*K2*K3*K4 + 4000*K1*K2*K3 + 912*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 384*K2**4 - 432*K2**2*K3**2 - 56*K2**2*K4**2 + 616*K2**2*K4 - 2148*K2**2 - 32*K2*K3**2*K4 + 376*K2*K3*K5 + 24*K2*K4*K6 - 64*K3**4 + 40*K3**2*K6 - 1244*K3**2 - 348*K4**2 - 64*K5**2 - 4*K6**2 + 2266 |
| 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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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}, {2, 5}, {4}, {3}, {1}], [{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 |