| Min(phi) over symmetries of the knot is: [-4,-2,1,1,1,3,0,1,3,4,4,0,1,2,2,0,0,0,0,1,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.227'] |
| Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.80', '6.163', '6.227', '6.447'] |
| Outer characteristic polynomial of the knot is: t^7+88t^5+67t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.227'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**2*K2**4 + 832*K1**2*K2**3 - 2448*K1**2*K2**2 - 224*K1**2*K2*K4 + 2368*K1**2*K2 - 1816*K1**2 + 384*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 896*K1*K2**2*K3 - 96*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 2536*K1*K2*K3 + 384*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1200*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 496*K2**2*K3**2 - 184*K2**2*K4**2 + 1144*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 1000*K2**2 - 32*K2*K3**2*K4 + 288*K2*K3*K5 + 72*K2*K4*K6 + 8*K3**2*K6 - 712*K3**2 - 310*K4**2 - 56*K5**2 - 16*K6**2 + 1404 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.227'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16791', 'vk6.16798', 'vk6.16822', 'vk6.16831', 'vk6.18159', 'vk6.18170', 'vk6.18493', 'vk6.18505', 'vk6.23203', 'vk6.23210', 'vk6.23412', 'vk6.23721', 'vk6.24617', 'vk6.25029', 'vk6.25044', 'vk6.35220', 'vk6.35934', 'vk6.36751', 'vk6.37169', 'vk6.37191', 'vk6.39382', 'vk6.41569', 'vk6.42699', 'vk6.42709', 'vk6.44331', 'vk6.44342', 'vk6.45955', 'vk6.47634', 'vk6.54978', 'vk6.55011', 'vk6.55968', 'vk6.57390', 'vk6.59364', 'vk6.59375', 'vk6.59561', 'vk6.62048', 'vk6.65190', 'vk6.65628', 'vk6.68165', 'vk6.68172'] |
| 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 | O1O2O3O4O5U3O6U1U6U4U5U2 |
| R3 orbit | {'O1O2O3O4O5U3O6U1U6U4U5U2', 'O1O2O3O4U2O5O6U1U6U4U3U5'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U4U1U2U6U5O6U3 |
| Gauss code of K* | O1O2O3O4O5U1U5U6U3U4O6U2 |
| Gauss code of -K* | O1O2O3O4O5U4O6U2U3U6U1U5 |
| 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 1 3 1],[ 4 0 4 0 3 4 1],[-1 -4 0 -2 0 2 0],[ 2 0 2 0 1 2 0],[-1 -3 0 -1 0 1 0],[-3 -4 -2 -2 -1 0 0],[-1 -1 0 0 0 0 0]] |
| Primitive based matrix | [[ 0 3 1 1 1 -2 -4],[-3 0 0 -1 -2 -2 -4],[-1 0 0 0 0 0 -1],[-1 1 0 0 0 -1 -3],[-1 2 0 0 0 -2 -4],[ 2 2 0 1 2 0 0],[ 4 4 1 3 4 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,-1,-1,2,4,0,1,2,2,4,0,0,0,1,0,1,3,2,4,0] |
| Phi over symmetry | [-4,-2,1,1,1,3,0,1,3,4,4,0,1,2,2,0,0,0,0,1,2] |
| Phi of -K | [-4,-2,1,1,1,3,2,1,2,4,3,1,2,3,3,0,0,0,0,1,2] |
| Phi of K* | [-3,-1,-1,-1,2,4,0,1,2,3,3,0,0,1,1,0,2,2,3,4,2] |
| Phi of -K* | [-4,-2,1,1,1,3,0,1,3,4,4,0,1,2,2,0,0,0,0,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-t^3+t^2-3t |
| Normalized Jones-Krushkal polynomial | 6z^2+19z+15 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+19w^2z+15w |
| Inner characteristic polynomial | t^6+56t^4+26t^2 |
| Outer characteristic polynomial | t^7+88t^5+67t^3+3t |
| Flat arrow polynomial | 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -192*K1**2*K2**4 + 832*K1**2*K2**3 - 2448*K1**2*K2**2 - 224*K1**2*K2*K4 + 2368*K1**2*K2 - 1816*K1**2 + 384*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 896*K1*K2**2*K3 - 96*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 2536*K1*K2*K3 + 384*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1200*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 496*K2**2*K3**2 - 184*K2**2*K4**2 + 1144*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 1000*K2**2 - 32*K2*K3**2*K4 + 288*K2*K3*K5 + 72*K2*K4*K6 + 8*K3**2*K6 - 712*K3**2 - 310*K4**2 - 56*K5**2 - 16*K6**2 + 1404 |
| 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}], [{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}, {3, 5}, {4}, {2}, {1}], [{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}, {3, 4}, {1, 2}]] |
| If K is slice | False |