Table of flat knot invariants
Invariant Table Check a Knot Higher Crossing Crossref Virtual Knots Please cite FlatKnotInfo
Glossary Reference List

Flat knot 6.327

Min(phi) over symmetries of the knot is: [-3,-3,1,1,1,3,-1,1,1,3,4,0,1,2,3,0,0,0,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.327']
Arrow polynomial of the knot is: 8*K1**3 + 8*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.122', '6.327', '6.371', '6.1185']
Outer characteristic polynomial of the knot is: t^7+73t^5+63t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.327']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 256*K1**4*K2**3 - 1152*K1**4*K2**2 + 1920*K1**4*K2 - 3040*K1**4 - 256*K1**3*K2**2*K3 + 768*K1**3*K2*K3 - 512*K1**3*K3 - 704*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3232*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 320*K1**2*K2**2*K4 - 10464*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 992*K1**2*K2*K4 + 10608*K1**2*K2 - 192*K1**2*K3**2 - 112*K1**2*K4**2 - 5768*K1**2 - 128*K1*K2**3*K3*K4 + 1888*K1*K2**3*K3 + 544*K1*K2**2*K3*K4 - 2144*K1*K2**2*K3 + 160*K1*K2**2*K4*K5 - 544*K1*K2**2*K5 - 576*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8856*K1*K2*K3 - 32*K1*K2*K4*K5 + 976*K1*K3*K4 + 208*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 - 64*K2**4*K3**2 - 64*K2**4*K4**2 + 768*K2**4*K4 - 3824*K2**4 + 352*K2**3*K3*K5 + 128*K2**3*K4*K6 - 224*K2**3*K6 - 1520*K2**2*K3**2 - 32*K2**2*K3*K7 - 824*K2**2*K4**2 + 3368*K2**2*K4 - 208*K2**2*K5**2 - 48*K2**2*K6**2 - 3678*K2**2 + 992*K2*K3*K5 + 304*K2*K4*K6 + 16*K2*K5*K7 - 2124*K3**2 - 860*K4**2 - 188*K5**2 - 42*K6**2 + 5058
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.327']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10918', 'vk6.10929', 'vk6.10935', 'vk6.12078', 'vk6.12088', 'vk6.12095', 'vk6.12105', 'vk6.14470', 'vk6.14490', 'vk6.15691', 'vk6.15711', 'vk6.16137', 'vk6.16147', 'vk6.30516', 'vk6.30528', 'vk6.30546', 'vk6.30560', 'vk6.31797', 'vk6.34067', 'vk6.34158', 'vk6.34169', 'vk6.34503', 'vk6.51751', 'vk6.51764', 'vk6.52625', 'vk6.54140', 'vk6.54150', 'vk6.54331', 'vk6.54534', 'vk6.63465', 'vk6.63476', 'vk6.63482']
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 O1O2O3O4O5U2U5O6U1U6U3U4
R3 orbit {'O1O2O3O4O5U2U5O6U1U6U3U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U3U6U5O6U1U4
Gauss code of K* O1O2O3O4U1U5U3U4U6O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U5U1U2U6U4
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 -3 -3 1 3 1 1],[ 3 0 -1 3 4 1 1],[ 3 1 0 2 3 1 0],[-1 -3 -2 0 1 0 0],[-3 -4 -3 -1 0 0 0],[-1 -1 -1 0 0 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 3 1 1 1 -3 -3],[-3 0 0 0 -1 -3 -4],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 -1 -1],[-1 1 0 0 0 -2 -3],[ 3 3 0 1 2 0 1],[ 3 4 1 1 3 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,-1,3,3,0,0,1,3,4,0,0,0,1,0,1,1,2,3,-1]
Phi over symmetry [-3,-3,1,1,1,3,-1,1,1,3,4,0,1,2,3,0,0,0,0,0,1]
Phi of -K [-3,-3,1,1,1,3,-1,2,3,4,3,1,3,3,2,0,0,1,0,2,2]
Phi of K* [-3,-1,-1,-1,3,3,1,2,2,2,3,0,0,1,2,0,3,3,3,4,-1]
Phi of -K* [-3,-3,1,1,1,3,-1,1,1,3,4,0,1,2,3,0,0,0,0,0,1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+43t^4+23t^2+1
Outer characteristic polynomial t^7+73t^5+63t^3+11t
Flat arrow polynomial 8*K1**3 + 8*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -128*K1**6 + 256*K1**4*K2**3 - 1152*K1**4*K2**2 + 1920*K1**4*K2 - 3040*K1**4 - 256*K1**3*K2**2*K3 + 768*K1**3*K2*K3 - 512*K1**3*K3 - 704*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3232*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 320*K1**2*K2**2*K4 - 10464*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 992*K1**2*K2*K4 + 10608*K1**2*K2 - 192*K1**2*K3**2 - 112*K1**2*K4**2 - 5768*K1**2 - 128*K1*K2**3*K3*K4 + 1888*K1*K2**3*K3 + 544*K1*K2**2*K3*K4 - 2144*K1*K2**2*K3 + 160*K1*K2**2*K4*K5 - 544*K1*K2**2*K5 - 576*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8856*K1*K2*K3 - 32*K1*K2*K4*K5 + 976*K1*K3*K4 + 208*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 - 64*K2**4*K3**2 - 64*K2**4*K4**2 + 768*K2**4*K4 - 3824*K2**4 + 352*K2**3*K3*K5 + 128*K2**3*K4*K6 - 224*K2**3*K6 - 1520*K2**2*K3**2 - 32*K2**2*K3*K7 - 824*K2**2*K4**2 + 3368*K2**2*K4 - 208*K2**2*K5**2 - 48*K2**2*K6**2 - 3678*K2**2 + 992*K2*K3*K5 + 304*K2*K4*K6 + 16*K2*K5*K7 - 2124*K3**2 - 860*K4**2 - 188*K5**2 - 42*K6**2 + 5058
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
Contact