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

Flat knot 6.321

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,0,1,3,4,3,0,1,1,1,0,1,1,0,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.321']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 16*K1**2 - 6*K1*K2 - 2*K2**2 + 6*K2 + 2*K3 + 9
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.159', '6.321']
Outer characteristic polynomial of the knot is: t^7+76t^5+54t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.321']
2-strand cable arrow polynomial of the knot is: -192*K1**6 - 448*K1**4*K2**2 + 2560*K1**4*K2 - 5856*K1**4 - 256*K1**3*K2**2*K3 + 1184*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1280*K1**3*K3 - 128*K1**2*K2**4 + 1504*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 9552*K1**2*K2**2 + 320*K1**2*K2*K3**2 + 128*K1**2*K2*K4**2 - 1024*K1**2*K2*K4 + 12952*K1**2*K2 - 1600*K1**2*K3**2 - 336*K1**2*K4**2 - 6564*K1**2 + 608*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1824*K1*K2**2*K3 - 224*K1*K2**2*K5 + 64*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 608*K1*K2*K3*K4 + 10800*K1*K2*K3 - 96*K1*K2*K4*K5 + 2648*K1*K3*K4 + 448*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1648*K2**4 + 32*K2**3*K3*K5 + 64*K2**2*K3**2*K4 - 752*K2**2*K3**2 + 32*K2**2*K4**3 - 344*K2**2*K4**2 + 2336*K2**2*K4 - 5860*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 736*K2*K3*K5 + 200*K2*K4*K6 - 64*K3**4 - 48*K3**2*K4**2 + 56*K3**2*K6 - 3204*K3**2 + 16*K3*K4*K7 - 8*K4**4 - 1288*K4**2 - 288*K5**2 - 44*K6**2 + 6526
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.321']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13891', 'vk6.13988', 'vk6.14147', 'vk6.14370', 'vk6.14966', 'vk6.15089', 'vk6.15599', 'vk6.16069', 'vk6.16290', 'vk6.16313', 'vk6.17412', 'vk6.22601', 'vk6.22632', 'vk6.23920', 'vk6.33702', 'vk6.33779', 'vk6.34149', 'vk6.34262', 'vk6.34585', 'vk6.36199', 'vk6.36224', 'vk6.42277', 'vk6.53865', 'vk6.53908', 'vk6.54104', 'vk6.54410', 'vk6.54575', 'vk6.55571', 'vk6.59024', 'vk6.59043', 'vk6.60062', 'vk6.64560']
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 O1O2O3O4O5U2U4O6U3U6U1U5
R3 orbit {'O1O2O3O4O5U2U4O6U3U6U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U5U6U3O6U2U4
Gauss code of K* O1O2O3O4U3U5U1U6U4O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U1U5U4U6U2
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 -1 -3 -1 0 4 1],[ 1 0 -2 0 1 4 1],[ 3 2 0 2 1 3 1],[ 1 0 -2 0 0 3 1],[ 0 -1 -1 0 0 1 0],[-4 -4 -3 -3 -1 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 4 1 0 -1 -1 -3],[-4 0 0 -1 -3 -4 -3],[-1 0 0 0 -1 -1 -1],[ 0 1 0 0 0 -1 -1],[ 1 3 1 0 0 0 -2],[ 1 4 1 1 0 0 -2],[ 3 3 1 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,0,1,1,3,0,1,3,4,3,0,1,1,1,0,1,1,0,2,2]
Phi over symmetry [-4,-1,0,1,1,3,0,1,3,4,3,0,1,1,1,0,1,1,0,2,2]
Phi of -K [-3,-1,-1,0,1,4,0,0,2,3,4,0,0,1,1,1,1,2,1,3,3]
Phi of K* [-4,-1,0,1,1,3,3,3,1,2,4,1,1,1,3,0,1,2,0,0,0]
Phi of -K* [-3,-1,-1,0,1,4,2,2,1,1,3,0,0,1,3,1,1,4,0,1,0]
Symmetry type of based matrix c
u-polynomial -t^4+t^3+t
Normalized Jones-Krushkal polynomial 4z^2+25z+35
Enhanced Jones-Krushkal polynomial 4w^3z^2+25w^2z+35w
Inner characteristic polynomial t^6+48t^4+26t^2+1
Outer characteristic polynomial t^7+76t^5+54t^3+5t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 16*K1**2 - 6*K1*K2 - 2*K2**2 + 6*K2 + 2*K3 + 9
2-strand cable arrow polynomial -192*K1**6 - 448*K1**4*K2**2 + 2560*K1**4*K2 - 5856*K1**4 - 256*K1**3*K2**2*K3 + 1184*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1280*K1**3*K3 - 128*K1**2*K2**4 + 1504*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 9552*K1**2*K2**2 + 320*K1**2*K2*K3**2 + 128*K1**2*K2*K4**2 - 1024*K1**2*K2*K4 + 12952*K1**2*K2 - 1600*K1**2*K3**2 - 336*K1**2*K4**2 - 6564*K1**2 + 608*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1824*K1*K2**2*K3 - 224*K1*K2**2*K5 + 64*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 608*K1*K2*K3*K4 + 10800*K1*K2*K3 - 96*K1*K2*K4*K5 + 2648*K1*K3*K4 + 448*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1648*K2**4 + 32*K2**3*K3*K5 + 64*K2**2*K3**2*K4 - 752*K2**2*K3**2 + 32*K2**2*K4**3 - 344*K2**2*K4**2 + 2336*K2**2*K4 - 5860*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 736*K2*K3*K5 + 200*K2*K4*K6 - 64*K3**4 - 48*K3**2*K4**2 + 56*K3**2*K6 - 3204*K3**2 + 16*K3*K4*K7 - 8*K4**4 - 1288*K4**2 - 288*K5**2 - 44*K6**2 + 6526
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{6}, {4, 5}, {1, 3}, {2}]]
If K is slice False
Contact