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Flat knot 6.210

Min(phi) over symmetries of the knot is: [-4,0,1,3,2,2,3,1,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.210']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 6K12 - 6K1K2 - 2K1K3 + 2K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.91', '6.202', '6.210']
Outer characteristic polynomial of the knot is: t^5+49t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.210']
2-strand cable arrow polynomial of the knot is: -128K16 + 256K1*4K2*3 - 768K1*4K2*2 + 992K1*4K2 - 1008K14 + 160K1*3K2K3 - 384K1*2K2*4 + 704K1*2K2*3 - 1808K1*2K2*2 + 1752K1*2K2 - 80K12K3*2 - 32K1*2K4*2 - 968K1*2 + 288K1K23K3 + 1272K1K2K3 + 248K1K3K4 + 136K1K4K5 + 24K1K5K6 - 32K26 - 64K2*4K3*2 - 32K2*4K4*2 + 192K2*4K4 - 664K24 + 128K2*3K3K5 + 32K2*3K4K6 - 320K2*2K3*2 - 216K2*2K4*2 + 440K2*2K4 - 80K22K5*2 - 8K2*2K6*2 - 712K2*2 + 312K2K3K5 + 120K2K4K6 + 16K2K5K7 - 480K32 - 332K4*2 - 168K5*2 - 40K6**2 + 1250
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.210']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11033', 'vk6.11111', 'vk6.12199', 'vk6.12306', 'vk6.14100', 'vk6.14101', 'vk6.14305', 'vk6.14306', 'vk6.15535', 'vk6.15536', 'vk6.16023', 'vk6.16024', 'vk6.16273', 'vk6.22572', 'vk6.22576', 'vk6.22721', 'vk6.22820', 'vk6.26110', 'vk6.26531', 'vk6.30602', 'vk6.30697', 'vk6.34053', 'vk6.34105', 'vk6.34526', 'vk6.34551', 'vk6.34555', 'vk6.38115', 'vk6.38140', 'vk6.42253', 'vk6.44637', 'vk6.44764', 'vk6.51842', 'vk6.54067', 'vk6.54521', 'vk6.54553', 'vk6.56584', 'vk6.56638', 'vk6.59006', 'vk6.64519', 'vk6.64520']
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: [-4,0,1,3,2,2,3,1,2,1]

Flat knots (up to 7 crossings) with same phi are :['6.210']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.91', '6.202', '6.210']

Outer characteristic polynomial of the knot is: t^5+49t^3+11t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.210']

2-strand cable arrow polynomial of the knot is: -128*K1**6 + 256*K1**4*K2**3 - 768*K1**4*K2**2 + 992*K1**4*K2 - 1008*K1**4 + 160*K1**3*K2*K3 - 384*K1**2*K2**4 + 704*K1**2*K2**3 - 1808*K1**2*K2**2 + 1752*K1**2*K2 - 80*K1**2*K3**2 - 32*K1**2*K4**2 - 968*K1**2 + 288*K1*K2**3*K3 + 1272*K1*K2*K3 + 248*K1*K3*K4 + 136*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 664*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 320*K2**2*K3**2 - 216*K2**2*K4**2 + 440*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 712*K2**2 + 312*K2*K3*K5 + 120*K2*K4*K6 + 16*K2*K5*K7 - 480*K3**2 - 332*K4**2 - 168*K5**2 - 40*K6**2 + 1250

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.210']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11033', 'vk6.11111', 'vk6.12199', 'vk6.12306', 'vk6.14100', 'vk6.14101', 'vk6.14305', 'vk6.14306', 'vk6.15535', 'vk6.15536', 'vk6.16023', 'vk6.16024', 'vk6.16273', 'vk6.22572', 'vk6.22576', 'vk6.22721', 'vk6.22820', 'vk6.26110', 'vk6.26531', 'vk6.30602', 'vk6.30697', 'vk6.34053', 'vk6.34105', 'vk6.34526', 'vk6.34551', 'vk6.34555', 'vk6.38115', 'vk6.38140', 'vk6.42253', 'vk6.44637', 'vk6.44764', 'vk6.51842', 'vk6.54067', 'vk6.54521', 'vk6.54553', 'vk6.56584', 'vk6.56638', 'vk6.59006', 'vk6.64519', 'vk6.64520']

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	O1O2O3O4O5U2O6U4U6U3U1U5
R3 orbit	{'O1O2O3O4O5U2O6U4U6U3U1U5', 'O1O2O3O4O5U2U3O6U4U6U1U5'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4O5U1U5U3U6U2O6U4
Gauss code of K*	O1O2O3O4O5U4U6U3U1U5O6U2
Gauss code of -K*	O1O2O3O4O5U4O6U1U5U3U6U2
Diagrammatic symmetry type	c
Flat genus of the diagram	2
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -1 -3 0 -1 4 1],[ 1 0 -2 1 0 4 1],[ 3 2 0 2 1 3 1],[ 0 -1 -2 0 -1 2 1],[ 1 0 -1 1 0 2 1],[-4 -4 -3 -2 -2 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix	[[ 0 4 0 -1 -3],[-4 0 -2 -2 -3],[ 0 2 0 -1 -2],[ 1 2 1 0 -1],[ 3 3 2 1 0]]
If based matrix primitive	False
Phi of primitive based matrix	[-4,0,1,3,2,2,3,1,2,1]
Phi over symmetry	[-4,0,1,3,2,2,3,1,2,1]
Phi of -K	[-3,-1,0,4,1,1,4,0,3,2]
Phi of K*	[-4,0,1,3,2,3,4,0,1,1]
Phi of -K*	[-3,-1,0,4,1,2,3,1,2,2]
Symmetry type of based matrix	c
u-polynomial	-t^4+t^3+t
Normalized Jones-Krushkal polynomial	9z+19
Enhanced Jones-Krushkal polynomial	-4w^3z+13w^2z+19w
Inner characteristic polynomial	t^4+23t^2+1
Outer characteristic polynomial	t^5+49t^3+11t
Flat arrow polynomial	4K13 + 4K1*2K2 - 6K12 - 6K1K2 - 2K1K3 + 2K2 + 2*K3 + 3
2-strand cable arrow polynomial	-128K16 + 256K1*4K2*3 - 768K1*4K2*2 + 992K1*4K2 - 1008K14 + 160K1*3K2K3 - 384K1*2K2*4 + 704K1*2K2*3 - 1808K1*2K2*2 + 1752K1*2K2 - 80K12K3*2 - 32K1*2K4*2 - 968K1*2 + 288K1K23K3 + 1272K1K2K3 + 248K1K3K4 + 136K1K4K5 + 24K1K5K6 - 32K26 - 64K2*4K3*2 - 32K2*4K4*2 + 192K2*4K4 - 664K24 + 128K2*3K3K5 + 32K2*3K4K6 - 320K2*2K3*2 - 216K2*2K4*2 + 440K2*2K4 - 80K22K5*2 - 8K2*2K6*2 - 712K2*2 + 312K2K3K5 + 120K2K4K6 + 16K2K5K7 - 480K32 - 332K4*2 - 168K5*2 - 40K6**2 + 1250
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {4}, {1, 3}, {2}]]
If K is slice	False

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