| Min(phi) over symmetries of the knot is: [-4,-3,1,1,1,4,0,1,3,4,5,0,1,2,3,0,0,0,0,1,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.195'] |
| Arrow polynomial of the knot is: -2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + 2*K4 + 1 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.126', '6.195', '6.367', '6.438', '6.869', '6.872', '6.896', '6.1147'] |
| Outer characteristic polynomial of the knot is: t^7+114t^5+165t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.195'] |
| 2-strand cable arrow polynomial of the knot is: -96*K1**3*K3 + 256*K1**2*K2*K3**2 + 1112*K1**2*K2 - 2560*K1**2*K3**2 - 224*K1**2*K3*K5 - 48*K1**2*K6**2 - 2692*K1**2 - 704*K1*K2**2*K3 - 576*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 4584*K1*K2*K3 - 32*K1*K2*K5*K6 + 3016*K1*K3*K4 + 240*K1*K4*K5 + 112*K1*K5*K6 + 56*K1*K6*K7 + 8*K1*K7*K8 - 1008*K2**2*K3**2 - 32*K2**2*K3*K7 - 8*K2**2*K4**2 + 504*K2**2*K4 - 16*K2**2*K6**2 - 2246*K2**2 + 1320*K2*K3*K5 + 96*K2*K4*K6 + 32*K2*K5*K7 + 16*K2*K6*K8 + 32*K3**2*K6 - 2344*K3**2 + 16*K3*K4*K7 + 8*K3*K5*K8 - 888*K4**2 - 384*K5**2 - 98*K6**2 - 28*K7**2 - 12*K8**2 + 2650 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.195'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16994', 'vk6.17237', 'vk6.20210', 'vk6.21496', 'vk6.23398', 'vk6.23707', 'vk6.27398', 'vk6.29019', 'vk6.35461', 'vk6.35905', 'vk6.38814', 'vk6.40993', 'vk6.42895', 'vk6.43196', 'vk6.45565', 'vk6.47345', 'vk6.55157', 'vk6.55404', 'vk6.57052', 'vk6.58162', 'vk6.59533', 'vk6.59878', 'vk6.61558', 'vk6.62733', 'vk6.64967', 'vk6.65174', 'vk6.66668', 'vk6.67502', 'vk6.68257', 'vk6.68413', 'vk6.69316', 'vk6.70074'] |
| 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 | O1O2O3O4O5U2O6U1U6U4U3U5 |
| R3 orbit | {'O1O2O3O4O5U2O6U1U6U4U3U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U1U3U2U6U5O6U4 |
| Gauss code of K* | O1O2O3O4O5U1U6U4U3U5O6U2 |
| Gauss code of -K* | O1O2O3O4O5U4O6U1U3U2U6U5 |
| 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 -3 1 1 4 1],[ 4 0 0 4 3 5 1],[ 3 0 0 2 1 3 0],[-1 -4 -2 0 0 2 0],[-1 -3 -1 0 0 1 0],[-4 -5 -3 -2 -1 0 0],[-1 -1 0 0 0 0 0]] |
| Primitive based matrix | [[ 0 4 1 1 1 -3 -4],[-4 0 0 -1 -2 -3 -5],[-1 0 0 0 0 0 -1],[-1 1 0 0 0 -1 -3],[-1 2 0 0 0 -2 -4],[ 3 3 0 1 2 0 0],[ 4 5 1 3 4 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-4,-1,-1,-1,3,4,0,1,2,3,5,0,0,0,1,0,1,3,2,4,0] |
| Phi over symmetry | [-4,-3,1,1,1,4,0,1,3,4,5,0,1,2,3,0,0,0,0,1,2] |
| Phi of -K | [-4,-3,1,1,1,4,1,1,2,4,3,2,3,4,4,0,0,1,0,2,3] |
| Phi of K* | [-4,-1,-1,-1,3,4,1,2,3,4,3,0,0,2,1,0,3,2,4,4,1] |
| Phi of -K* | [-4,-3,1,1,1,4,0,1,3,4,5,0,1,2,3,0,0,0,0,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-3t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+5w^3z^2-6w^3z+22w^2z+21w |
| Inner characteristic polynomial | t^6+70t^4+47t^2 |
| Outer characteristic polynomial | t^7+114t^5+165t^3+7t |
| Flat arrow polynomial | -2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + 2*K4 + 1 |
| 2-strand cable arrow polynomial | -96*K1**3*K3 + 256*K1**2*K2*K3**2 + 1112*K1**2*K2 - 2560*K1**2*K3**2 - 224*K1**2*K3*K5 - 48*K1**2*K6**2 - 2692*K1**2 - 704*K1*K2**2*K3 - 576*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 4584*K1*K2*K3 - 32*K1*K2*K5*K6 + 3016*K1*K3*K4 + 240*K1*K4*K5 + 112*K1*K5*K6 + 56*K1*K6*K7 + 8*K1*K7*K8 - 1008*K2**2*K3**2 - 32*K2**2*K3*K7 - 8*K2**2*K4**2 + 504*K2**2*K4 - 16*K2**2*K6**2 - 2246*K2**2 + 1320*K2*K3*K5 + 96*K2*K4*K6 + 32*K2*K5*K7 + 16*K2*K6*K8 + 32*K3**2*K6 - 2344*K3**2 + 16*K3*K4*K7 + 8*K3*K5*K8 - 888*K4**2 - 384*K5**2 - 98*K6**2 - 28*K7**2 - 12*K8**2 + 2650 |
| 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}, {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}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]] |
| If K is slice | False |