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

Flat knot 6.977

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,0,2,2,2,4,0,1,0,1,0,2,1,1,1,-2]
Flat knots (up to 7 crossings) with same phi are :['6.977']
Arrow polynomial of the knot is: 8K13 - 8K1*2 - 6K1K2 - 3K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.238', '6.431', '6.945', '6.977', '6.981', '6.997', '6.1050', '6.1070', '6.1098', '6.1376']
Outer characteristic polynomial of the knot is: t^7+55t^5+43t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.977']
2-strand cable arrow polynomial of the knot is: 160K14K2 - 1120K14 + 128K1*3K2K3 - 832K1*3K3 + 544K12K2*3 - 64K1*2K2*2K3*2 - 3328K1*2K2*2 + 192K1*2K2K32 - 224K1*2K2K4 + 7048K1*2K2 - 544K12K3*2 - 5680K1*2 + 416K1K23K3 + 64K1K2*2K3K4 - 1184K1K22K3 - 128K1K2*2K5 + 32K1K2K33 - 224K1K2K3K4 + 6024K1K2K3 + 856K1K3K4 + 64K1K4K5 - 64K26 + 96K2*4K4 - 976K24 - 32K2*3K6 - 544K22K3*2 - 40K2*2K4*2 + 1112K2*2K4 - 3686K22 - 32K2K32K4 + 336K2K3K5 + 24K2K4K6 - 1904K32 - 372K4*2 - 56K5*2 - 2K6**2 + 3938
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.977']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71373', 'vk6.71434', 'vk6.71899', 'vk6.71960', 'vk6.72456', 'vk6.72586', 'vk6.72705', 'vk6.72818', 'vk6.72882', 'vk6.73017', 'vk6.74241', 'vk6.74369', 'vk6.74434', 'vk6.74870', 'vk6.75048', 'vk6.76623', 'vk6.76921', 'vk6.77030', 'vk6.77404', 'vk6.77757', 'vk6.77809', 'vk6.79291', 'vk6.79415', 'vk6.79765', 'vk6.79833', 'vk6.79890', 'vk6.80861', 'vk6.80918', 'vk6.81369', 'vk6.85520', 'vk6.87226', 'vk6.89261']
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

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,0,2,2,2,4,0,1,0,1,0,2,1,1,1,-2]

Flat knots (up to 7 crossings) with same phi are :['6.977']

Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 6*K1*K2 - 3*K1 + 4*K2 + K3 + 5

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.238', '6.431', '6.945', '6.977', '6.981', '6.997', '6.1050', '6.1070', '6.1098', '6.1376']

Outer characteristic polynomial of the knot is: t^7+55t^5+43t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.977']

2-strand cable arrow polynomial of the knot is: 160*K1**4*K2 - 1120*K1**4 + 128*K1**3*K2*K3 - 832*K1**3*K3 + 544*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 3328*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 224*K1**2*K2*K4 + 7048*K1**2*K2 - 544*K1**2*K3**2 - 5680*K1**2 + 416*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1184*K1*K2**2*K3 - 128*K1*K2**2*K5 + 32*K1*K2*K3**3 - 224*K1*K2*K3*K4 + 6024*K1*K2*K3 + 856*K1*K3*K4 + 64*K1*K4*K5 - 64*K2**6 + 96*K2**4*K4 - 976*K2**4 - 32*K2**3*K6 - 544*K2**2*K3**2 - 40*K2**2*K4**2 + 1112*K2**2*K4 - 3686*K2**2 - 32*K2*K3**2*K4 + 336*K2*K3*K5 + 24*K2*K4*K6 - 1904*K3**2 - 372*K4**2 - 56*K5**2 - 2*K6**2 + 3938

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.977']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71373', 'vk6.71434', 'vk6.71899', 'vk6.71960', 'vk6.72456', 'vk6.72586', 'vk6.72705', 'vk6.72818', 'vk6.72882', 'vk6.73017', 'vk6.74241', 'vk6.74369', 'vk6.74434', 'vk6.74870', 'vk6.75048', 'vk6.76623', 'vk6.76921', 'vk6.77030', 'vk6.77404', 'vk6.77757', 'vk6.77809', 'vk6.79291', 'vk6.79415', 'vk6.79765', 'vk6.79833', 'vk6.79890', 'vk6.80861', 'vk6.80918', 'vk6.81369', 'vk6.85520', 'vk6.87226', 'vk6.89261']

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	O1O2O3O4U1U3O5U2O6U5U4U6
R3 orbit	{'O1O2O3O4U1U3O5U2O6U5U4U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U1U6O5U3O6U2U4
Gauss code of K*	O1O2O3U4U5U6U2O4O6U1O5U3
Gauss code of -K*	O1O2O3U1O4U3O5O6U2U5U4U6
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 0 2 0 2],[ 3 0 2 1 3 1 1],[ 1 -2 0 0 3 1 2],[ 0 -1 0 0 1 0 1],[-2 -3 -3 -1 0 0 2],[ 0 -1 -1 0 0 0 1],[-2 -1 -2 -1 -2 -1 0]]
Primitive based matrix	[[ 0 2 2 0 0 -1 -3],[-2 0 2 0 -1 -3 -3],[-2 -2 0 -1 -1 -2 -1],[ 0 0 1 0 0 -1 -1],[ 0 1 1 0 0 0 -1],[ 1 3 2 1 0 0 -2],[ 3 3 1 1 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,0,0,1,3,-2,0,1,3,3,1,1,2,1,0,1,1,0,1,2]
Phi over symmetry	[-3,-1,0,0,2,2,0,2,2,2,4,0,1,0,1,0,2,1,1,1,-2]
Phi of -K	[-3,-1,0,0,2,2,0,2,2,2,4,0,1,0,1,0,2,1,1,1,-2]
Phi of K*	[-2,-2,0,0,1,3,-2,1,1,1,4,1,2,0,2,0,1,2,0,2,0]
Phi of -K*	[-3,-1,0,0,2,2,2,1,1,1,3,0,1,2,3,0,1,1,1,0,-2]
Symmetry type of based matrix	c
u-polynomial	t^3-2t^2+t
Normalized Jones-Krushkal polynomial	2z^2+19z+31
Enhanced Jones-Krushkal polynomial	2w^3z^2+19w^2z+31w
Inner characteristic polynomial	t^6+37t^4+11t^2
Outer characteristic polynomial	t^7+55t^5+43t^3+4t
Flat arrow polynomial	8K13 - 8K1*2 - 6K1K2 - 3K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial	160K14K2 - 1120K14 + 128K1*3K2K3 - 832K1*3K3 + 544K12K2*3 - 64K1*2K2*2K3*2 - 3328K1*2K2*2 + 192K1*2K2K32 - 224K1*2K2K4 + 7048K1*2K2 - 544K12K3*2 - 5680K1*2 + 416K1K23K3 + 64K1K2*2K3K4 - 1184K1K22K3 - 128K1K2*2K5 + 32K1K2K33 - 224K1K2K3K4 + 6024K1K2K3 + 856K1K3K4 + 64K1K4K5 - 64K26 + 96K2*4K4 - 976K24 - 32K2*3K6 - 544K22K3*2 - 40K2*2K4*2 + 1112K2*2K4 - 3686K22 - 32K2K32K4 + 336K2K3K5 + 24K2K4K6 - 1904K32 - 372K4*2 - 56K5*2 - 2K6**2 + 3938
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice	False

Contact