| Min(phi) over symmetries of the knot is: [-5,0,0,0,2,3,1,2,4,5,3,0,1,1,1,1,1,2,1,2,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.47'] |
| Arrow polynomial of the knot is: 4*K1**2*K3 - 2*K1**2 - 4*K1*K2 - 2*K1*K4 + K1 + K2 + K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.47'] |
| Outer characteristic polynomial of the knot is: t^7+111t^5+157t^3+19t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.47'] |
| 2-strand cable arrow polynomial of the knot is: -1264*K1**2*K2**2 + 496*K1**2*K2 - 16*K1**2*K3**2 - 1412*K1**2 + 1824*K1*K2**3*K3 - 256*K1*K2**2*K3 + 32*K1*K2**2*K5*K6 - 512*K1*K2**2*K5 - 160*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 4432*K1*K2*K3 - 32*K1*K2*K4*K5 + 168*K1*K3*K4 + 32*K1*K4*K5 + 16*K1*K5*K6 - 768*K2**4*K3**2 - 32*K2**4*K6**2 - 1304*K2**4 + 832*K2**3*K3*K5 + 64*K2**3*K4*K6 + 32*K2**3*K6*K8 + 128*K2**2*K3**2*K4 + 64*K2**2*K3**2*K6 - 3328*K2**2*K3**2 - 96*K2**2*K3*K7 - 48*K2**2*K4**2 - 32*K2**2*K4*K8 + 648*K2**2*K4 - 320*K2**2*K5**2 - 96*K2**2*K6**2 - 8*K2**2*K8**2 - 1138*K2**2 - 64*K2*K3**2*K4 + 2488*K2*K3*K5 + 184*K2*K4*K6 + 56*K2*K5*K7 + 40*K2*K6*K8 - 16*K3**4 + 112*K3**2*K6 - 1884*K3**2 + 8*K3*K5*K8 - 126*K4**2 - 464*K5**2 - 94*K6**2 - 12*K8**2 + 1920 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.47'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70498', 'vk6.70510', 'vk6.70557', 'vk6.70573', 'vk6.70702', 'vk6.70726', 'vk6.70807', 'vk6.70825', 'vk6.70975', 'vk6.70993', 'vk6.71053', 'vk6.71077', 'vk6.71192', 'vk6.71212', 'vk6.71271', 'vk6.71283', 'vk6.71752', 'vk6.72171', 'vk6.74065', 'vk6.74150', 'vk6.74626', 'vk6.74718', 'vk6.76210', 'vk6.76220', 'vk6.77549', 'vk6.79069', 'vk6.79167', 'vk6.80645', 'vk6.81252', 'vk6.86998', 'vk6.87932', 'vk6.89143'] |
| 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 | O1O2O3O4O5O6U1U5U4U6U2U3 |
| R3 orbit | {'O1O2O3O4O5O6U1U5U4U6U2U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5O6U4U5U1U3U2U6 |
| Gauss code of K* | O1O2O3O4O5O6U1U5U6U3U2U4 |
| Gauss code of -K* | O1O2O3O4O5O6U3U5U4U1U2U6 |
| 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 -5 0 2 0 0 3],[ 5 0 4 5 2 1 3],[ 0 -4 0 1 -1 -1 2],[-2 -5 -1 0 -1 -1 2],[ 0 -2 1 1 0 0 2],[ 0 -1 1 1 0 0 1],[-3 -3 -2 -2 -2 -1 0]] |
| Primitive based matrix | [[ 0 3 2 0 0 0 -5],[-3 0 -2 -1 -2 -2 -3],[-2 2 0 -1 -1 -1 -5],[ 0 1 1 0 1 0 -1],[ 0 2 1 -1 0 -1 -4],[ 0 2 1 0 1 0 -2],[ 5 3 5 1 4 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,0,0,0,5,2,1,2,2,3,1,1,1,5,-1,0,1,1,4,2] |
| Phi over symmetry | [-5,0,0,0,2,3,1,2,4,5,3,0,1,1,1,1,1,2,1,2,2] |
| Phi of -K | [-5,0,0,0,2,3,1,3,4,2,5,1,1,1,1,0,1,1,1,2,-1] |
| Phi of K* | [-3,-2,0,0,0,5,-1,1,1,2,5,1,1,1,2,-1,-1,1,0,3,4] |
| Phi of -K* | [-5,0,0,0,2,3,1,2,4,5,3,0,1,1,1,1,1,2,1,2,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^5-t^3-t^2 |
| Normalized Jones-Krushkal polynomial | z^2+6z+9 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+5w^3z^2-16w^3z+22w^2z+9w |
| Inner characteristic polynomial | t^6+73t^4+34t^2+1 |
| Outer characteristic polynomial | t^7+111t^5+157t^3+19t |
| Flat arrow polynomial | 4*K1**2*K3 - 2*K1**2 - 4*K1*K2 - 2*K1*K4 + K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -1264*K1**2*K2**2 + 496*K1**2*K2 - 16*K1**2*K3**2 - 1412*K1**2 + 1824*K1*K2**3*K3 - 256*K1*K2**2*K3 + 32*K1*K2**2*K5*K6 - 512*K1*K2**2*K5 - 160*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 4432*K1*K2*K3 - 32*K1*K2*K4*K5 + 168*K1*K3*K4 + 32*K1*K4*K5 + 16*K1*K5*K6 - 768*K2**4*K3**2 - 32*K2**4*K6**2 - 1304*K2**4 + 832*K2**3*K3*K5 + 64*K2**3*K4*K6 + 32*K2**3*K6*K8 + 128*K2**2*K3**2*K4 + 64*K2**2*K3**2*K6 - 3328*K2**2*K3**2 - 96*K2**2*K3*K7 - 48*K2**2*K4**2 - 32*K2**2*K4*K8 + 648*K2**2*K4 - 320*K2**2*K5**2 - 96*K2**2*K6**2 - 8*K2**2*K8**2 - 1138*K2**2 - 64*K2*K3**2*K4 + 2488*K2*K3*K5 + 184*K2*K4*K6 + 56*K2*K5*K7 + 40*K2*K6*K8 - 16*K3**4 + 112*K3**2*K6 - 1884*K3**2 + 8*K3*K5*K8 - 126*K4**2 - 464*K5**2 - 94*K6**2 - 12*K8**2 + 1920 |
| 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}], [{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}], [{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}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {1, 3}, {2}]] |
| If K is slice | False |