Invariant Table Check a Knot Higher Crossing Crossref Virtual Knots Please cite FlatKnotInfo

Flat knot 6.1259

Min(phi) over symmetries of the knot is: [-2,-1,1,2,0,1,2,1,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.1259']
Arrow polynomial of the knot is: 4K12K2 - 4K1K3 + K4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.111', '6.139', '6.519', '6.566', '6.1228', '6.1254', '6.1259', '6.1912', '6.1936']
Outer characteristic polynomial of the knot is: t^5+21t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1259']
2-strand cable arrow polynomial of the knot is: -1152K14K2*2 + 3072K1*4K2 - 3200K14 + 512K1*3K2K3 - 384K1*3K3 + 1920K12K2*3 - 5312K1*2K2*2 - 576K1*2K2K4 + 3888K1*2K2 - 288K12K3*2 - 192K1*2K3K5 - 384K1*2K4*2 - 128K1*2K4K6 - 160K1*2K5*2 - 1504K1*2 + 192K1K22K3K4 - 768K1K22K3 + 192K1K2*2K4K5 - 448K1K22K5 - 192K1K2K3K4 - 320K1K2K3K6 + 3616K1K2K3 - 64K1K2K4K5 - 128K1K2K4K7 - 64K1K2K5K6 + 1152K1K3K4 + 1152K1K4K5 + 432K1K5K6 + 48K1K6K7 - 32K2*4K4*2 + 128K2*4K4 - 976K24 + 64K2*3K3K5 + 64K2*3K4K6 - 64K2*3K6 - 192K22K3*2 - 352K2*2K4*2 + 1120K2*2K4 - 192K22K5*2 - 48K2*2K6*2 - 1424K2*2 + 640K2K3K5 + 416K2K4K6 + 144K2K5K7 + 16K2K6K8 - 800K3*2 - 784K4*2 - 544K5*2 - 192K6*2 - 32K7*2 - 2K8**2 + 2128
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1259']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4812', 'vk6.5154', 'vk6.6373', 'vk6.6803', 'vk6.8337', 'vk6.8773', 'vk6.9710', 'vk6.10014', 'vk6.21099', 'vk6.22531', 'vk6.28546', 'vk6.42196', 'vk6.46821', 'vk6.48060', 'vk6.48823', 'vk6.49869', 'vk6.50838', 'vk6.51521', 'vk6.58985', 'vk6.69819']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is r.
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

Min(phi) over symmetries of the knot is: [-2,-1,1,2,0,1,2,1,2,1]

Flat knots (up to 7 crossings) with same phi are :['6.1259']

Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1*K3 + K4

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.111', '6.139', '6.519', '6.566', '6.1228', '6.1254', '6.1259', '6.1912', '6.1936']

Outer characteristic polynomial of the knot is: t^5+21t^3+10t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1259']

2-strand cable arrow polynomial of the knot is: -1152*K1**4*K2**2 + 3072*K1**4*K2 - 3200*K1**4 + 512*K1**3*K2*K3 - 384*K1**3*K3 + 1920*K1**2*K2**3 - 5312*K1**2*K2**2 - 576*K1**2*K2*K4 + 3888*K1**2*K2 - 288*K1**2*K3**2 - 192*K1**2*K3*K5 - 384*K1**2*K4**2 - 128*K1**2*K4*K6 - 160*K1**2*K5**2 - 1504*K1**2 + 192*K1*K2**2*K3*K4 - 768*K1*K2**2*K3 + 192*K1*K2**2*K4*K5 - 448*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 320*K1*K2*K3*K6 + 3616*K1*K2*K3 - 64*K1*K2*K4*K5 - 128*K1*K2*K4*K7 - 64*K1*K2*K5*K6 + 1152*K1*K3*K4 + 1152*K1*K4*K5 + 432*K1*K5*K6 + 48*K1*K6*K7 - 32*K2**4*K4**2 + 128*K2**4*K4 - 976*K2**4 + 64*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 192*K2**2*K3**2 - 352*K2**2*K4**2 + 1120*K2**2*K4 - 192*K2**2*K5**2 - 48*K2**2*K6**2 - 1424*K2**2 + 640*K2*K3*K5 + 416*K2*K4*K6 + 144*K2*K5*K7 + 16*K2*K6*K8 - 800*K3**2 - 784*K4**2 - 544*K5**2 - 192*K6**2 - 32*K7**2 - 2*K8**2 + 2128

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1259']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4812', 'vk6.5154', 'vk6.6373', 'vk6.6803', 'vk6.8337', 'vk6.8773', 'vk6.9710', 'vk6.10014', 'vk6.21099', 'vk6.22531', 'vk6.28546', 'vk6.42196', 'vk6.46821', 'vk6.48060', 'vk6.48823', 'vk6.49869', 'vk6.50838', 'vk6.51521', 'vk6.58985', 'vk6.69819']

The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is r.

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	O1O2O3O4U5U4U3O5O6U2U1U6
R3 orbit	{'O1O2O3O4U5U4U3O5O6U2U1U6'}
R3 orbit length	1
Gauss code of -K	Same
Gauss code of K*	O1O2O3U1U4O5O6O4U6U5U3U2
Gauss code of -K*	O1O2O3U1U4O5O6O4U6U5U3U2
Diagrammatic symmetry type	r
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 -1 1 1 -2 2],[ 1 0 0 1 1 -1 2],[ 1 0 0 1 1 -1 1],[-1 -1 -1 0 0 -2 0],[-1 -1 -1 0 0 -1 0],[ 2 1 1 2 1 0 2],[-2 -2 -1 0 0 -2 0]]
Primitive based matrix	[[ 0 2 1 -1 -2],[-2 0 0 -1 -2],[-1 0 0 -1 -2],[ 1 1 1 0 -1],[ 2 2 2 1 0]]
If based matrix primitive	False
Phi of primitive based matrix	[-2,-1,1,2,0,1,2,1,2,1]
Phi over symmetry	[-2,-1,1,2,0,1,2,1,2,1]
Phi of -K	[-2,-1,1,2,0,1,2,1,2,1]
Phi of K*	[-2,-1,1,2,1,2,2,1,1,0]
Phi of -K*	[-2,-1,1,2,1,2,2,1,1,0]
Symmetry type of based matrix	r
u-polynomial	0
Normalized Jones-Krushkal polynomial	9z^2+30z+25
Enhanced Jones-Krushkal polynomial	9w^3z^2+30w^2z+25w
Inner characteristic polynomial	t^4+11t^2
Outer characteristic polynomial	t^5+21t^3+10t
Flat arrow polynomial	4K12K2 - 4K1K3 + K4
2-strand cable arrow polynomial	-1152K14K2*2 + 3072K1*4K2 - 3200K14 + 512K1*3K2K3 - 384K1*3K3 + 1920K12K2*3 - 5312K1*2K2*2 - 576K1*2K2K4 + 3888K1*2K2 - 288K12K3*2 - 192K1*2K3K5 - 384K1*2K4*2 - 128K1*2K4K6 - 160K1*2K5*2 - 1504K1*2 + 192K1K22K3K4 - 768K1K22K3 + 192K1K2*2K4K5 - 448K1K22K5 - 192K1K2K3K4 - 320K1K2K3K6 + 3616K1K2K3 - 64K1K2K4K5 - 128K1K2K4K7 - 64K1K2K5K6 + 1152K1K3K4 + 1152K1K4K5 + 432K1K5K6 + 48K1K6K7 - 32K2*4K4*2 + 128K2*4K4 - 976K24 + 64K2*3K3K5 + 64K2*3K4K6 - 64K2*3K6 - 192K22K3*2 - 352K2*2K4*2 + 1120K2*2K4 - 192K22K5*2 - 48K2*2K6*2 - 1424K2*2 + 640K2K3K5 + 416K2K4K6 + 144K2K5K7 + 16K2K6K8 - 800K3*2 - 784K4*2 - 544K5*2 - 192K6*2 - 32K7*2 - 2K8**2 + 2128
Genus of based matrix	1
Fillings of based matrix	[[{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice	False

Contact