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

Flat knot 6.404

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,-1,1,1,2,3,1,1,0,1,1,1,2,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.404', '7.19808']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 4*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.404', '6.843', '6.1141']
Outer characteristic polynomial of the knot is: t^7+46t^5+82t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.404', '7.19808']
2-strand cable arrow polynomial of the knot is: -1024*K1**4*K2**2 + 416*K1**4*K2 - 704*K1**4 + 512*K1**3*K2**3*K3 + 1632*K1**3*K2*K3 - 128*K1**3*K3 - 2560*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3168*K1**2*K2**3 - 512*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7600*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 544*K1**2*K2*K4 + 4216*K1**2*K2 - 1280*K1**2*K3**2 - 64*K1**2*K3*K5 - 1700*K1**2 + 640*K1*K2**5*K3 - 640*K1*K2**4*K3 - 128*K1*K2**4*K5 + 4864*K1*K2**3*K3 + 448*K1*K2**2*K3*K4 - 2144*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 608*K1*K2**2*K5 + 96*K1*K2*K3**3 - 288*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 6248*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K5*K6 + 1160*K1*K3*K4 + 72*K1*K4*K5 + 16*K1*K5*K6 + 8*K1*K6*K7 - 576*K2**6 - 896*K2**4*K3**2 - 32*K2**4*K4**2 + 544*K2**4*K4 - 3488*K2**4 + 448*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 + 64*K2**2*K3**2*K4 - 2528*K2**2*K3**2 - 32*K2**2*K3*K7 - 184*K2**2*K4**2 + 2240*K2**2*K4 - 80*K2**2*K5**2 - 48*K2**2*K6**2 - 22*K2**2 + 1064*K2*K3*K5 + 136*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1344*K3**2 - 384*K4**2 - 104*K5**2 - 26*K6**2 - 4*K7**2 - 2*K8**2 + 1824
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.404']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.307', 'vk6.342', 'vk6.347', 'vk6.693', 'vk6.696', 'vk6.739', 'vk6.744', 'vk6.1479', 'vk6.1488', 'vk6.1937', 'vk6.1972', 'vk6.1977', 'vk6.2454', 'vk6.2633', 'vk6.2638', 'vk6.3104', 'vk6.18262', 'vk6.18265', 'vk6.18597', 'vk6.18602', 'vk6.24746', 'vk6.24749', 'vk6.25152', 'vk6.36871', 'vk6.36876', 'vk6.37332', 'vk6.44093', 'vk6.44096', 'vk6.56060', 'vk6.56065', 'vk6.60615', 'vk6.65727']
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 O1O2O3O4O5U4U5O6U1U2U6U3
R3 orbit {'O1O2O3O4O5U4U5O6U1U2U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U6U4U5O6U1U2
Gauss code of K* O1O2O3O4U1U2U4U5U6O5O6U3
Gauss code of -K* O1O2O3O4U2O5O6U5U6U1U3U4
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 -1 2 -1 1 2],[ 3 0 1 3 -1 1 2],[ 1 -1 0 2 -1 1 1],[-2 -3 -2 0 -1 1 0],[ 1 1 1 1 0 1 0],[-1 -1 -1 -1 -1 0 0],[-2 -2 -1 0 0 0 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 0 1 -1 -2 -3],[-2 0 0 0 0 -1 -2],[-1 -1 0 0 -1 -1 -1],[ 1 1 0 1 0 1 1],[ 1 2 1 1 -1 0 -1],[ 3 3 2 1 -1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,0,-1,1,2,3,0,0,1,2,1,1,1,-1,-1,1]
Phi over symmetry [-3,-1,-1,1,2,2,-1,1,1,2,3,1,1,0,1,1,1,2,0,-1,0]
Phi of -K [-3,-1,-1,1,2,2,1,3,3,2,3,1,1,1,2,1,2,3,2,1,0]
Phi of K* [-2,-2,-1,1,1,3,0,1,2,3,3,2,1,2,2,1,1,3,-1,1,3]
Phi of -K* [-3,-1,-1,1,2,2,-1,1,1,2,3,1,1,0,1,1,1,2,0,-1,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial z^2+6z+9
Enhanced Jones-Krushkal polynomial -4w^4z^2+5w^3z^2-12w^3z+18w^2z+9w
Inner characteristic polynomial t^6+26t^4+32t^2+1
Outer characteristic polynomial t^7+46t^5+82t^3+11t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 4*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial -1024*K1**4*K2**2 + 416*K1**4*K2 - 704*K1**4 + 512*K1**3*K2**3*K3 + 1632*K1**3*K2*K3 - 128*K1**3*K3 - 2560*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3168*K1**2*K2**3 - 512*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7600*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 544*K1**2*K2*K4 + 4216*K1**2*K2 - 1280*K1**2*K3**2 - 64*K1**2*K3*K5 - 1700*K1**2 + 640*K1*K2**5*K3 - 640*K1*K2**4*K3 - 128*K1*K2**4*K5 + 4864*K1*K2**3*K3 + 448*K1*K2**2*K3*K4 - 2144*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 608*K1*K2**2*K5 + 96*K1*K2*K3**3 - 288*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 6248*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K5*K6 + 1160*K1*K3*K4 + 72*K1*K4*K5 + 16*K1*K5*K6 + 8*K1*K6*K7 - 576*K2**6 - 896*K2**4*K3**2 - 32*K2**4*K4**2 + 544*K2**4*K4 - 3488*K2**4 + 448*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 + 64*K2**2*K3**2*K4 - 2528*K2**2*K3**2 - 32*K2**2*K3*K7 - 184*K2**2*K4**2 + 2240*K2**2*K4 - 80*K2**2*K5**2 - 48*K2**2*K6**2 - 22*K2**2 + 1064*K2*K3*K5 + 136*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1344*K3**2 - 384*K4**2 - 104*K5**2 - 26*K6**2 - 4*K7**2 - 2*K8**2 + 1824
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}]]
If K is slice False
Contact