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Flat knot 6.300

Min(phi) over symmetries of the knot is: [-4,-2,0,0,3,3,0,2,3,3,4,1,2,2,2,0,1,2,2,3,0]
Flat knots (up to 7 crossings) with same phi are :['6.300']
Arrow polynomial of the knot is: 4K12K2 - 2K1K3 - K2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.64', '6.74', '6.106', '6.178', '6.300', '6.397', '6.479', '6.481', '6.500']
Outer characteristic polynomial of the knot is: t^7+107t^5+85t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.300']
2-strand cable arrow polynomial of the knot is: 768K14K2 - 1472K14 + 768K1*3K2K3 - 256K1*3K3 + 480K12K2*3 + 384K1*2K2*2K4 - 4352K12K2*2 - 1152K1*2K2K4 + 4392K1*2K2 - 576K12K3*2 - 416K1*2K4*2 - 2616K1*2 + 192K1K22K3K4 - 832K1K22K3 + 96K1K2*2K4K5 - 448K1K22K5 - 416K1K2K3K4 + 4656K1K2K3 - 128K1K2K4K5 + 1584K1K3K4 + 672K1K4K5 - 32K2*4K4*2 + 128K2*4K4 - 576K24 + 32K2*3K3K5 + 32K2*3K4K6 - 64K2*3K6 - 192K22K3*2 - 336K2*2K4*2 + 1496K2*2K4 - 64K22K5*2 - 8K2*2K6*2 - 2512K2*2 - 32K2K32K4 + 552K2K3K5 + 272K2K4K6 - 1336K32 - 938K4*2 - 288K5*2 - 48K6**2 + 2584
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.300']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.74158', 'vk6.74161', 'vk6.74758', 'vk6.74761', 'vk6.76292', 'vk6.76297', 'vk6.76826', 'vk6.76829', 'vk6.79182', 'vk6.79191', 'vk6.79650', 'vk6.79659', 'vk6.80669', 'vk6.80677', 'vk6.81042', 'vk6.81047', 'vk6.82880', 'vk6.82904', 'vk6.83111', 'vk6.83341', 'vk6.83400', 'vk6.84184', 'vk6.84295', 'vk6.85558', 'vk6.85615', 'vk6.85884', 'vk6.86199', 'vk6.86637', 'vk6.87501', 'vk6.88162', 'vk6.88580', 'vk6.89387']
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,-2,0,0,3,3,0,2,3,3,4,1,2,2,2,0,1,2,2,3,0]

Flat knots (up to 7 crossings) with same phi are :['6.300']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.64', '6.74', '6.106', '6.178', '6.300', '6.397', '6.479', '6.481', '6.500']

Outer characteristic polynomial of the knot is: t^7+107t^5+85t^3+6t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.300']

2-strand cable arrow polynomial of the knot is: 768*K1**4*K2 - 1472*K1**4 + 768*K1**3*K2*K3 - 256*K1**3*K3 + 480*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 4352*K1**2*K2**2 - 1152*K1**2*K2*K4 + 4392*K1**2*K2 - 576*K1**2*K3**2 - 416*K1**2*K4**2 - 2616*K1**2 + 192*K1*K2**2*K3*K4 - 832*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 448*K1*K2**2*K5 - 416*K1*K2*K3*K4 + 4656*K1*K2*K3 - 128*K1*K2*K4*K5 + 1584*K1*K3*K4 + 672*K1*K4*K5 - 32*K2**4*K4**2 + 128*K2**4*K4 - 576*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 192*K2**2*K3**2 - 336*K2**2*K4**2 + 1496*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 2512*K2**2 - 32*K2*K3**2*K4 + 552*K2*K3*K5 + 272*K2*K4*K6 - 1336*K3**2 - 938*K4**2 - 288*K5**2 - 48*K6**2 + 2584

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.300']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.74158', 'vk6.74161', 'vk6.74758', 'vk6.74761', 'vk6.76292', 'vk6.76297', 'vk6.76826', 'vk6.76829', 'vk6.79182', 'vk6.79191', 'vk6.79650', 'vk6.79659', 'vk6.80669', 'vk6.80677', 'vk6.81042', 'vk6.81047', 'vk6.82880', 'vk6.82904', 'vk6.83111', 'vk6.83341', 'vk6.83400', 'vk6.84184', 'vk6.84295', 'vk6.85558', 'vk6.85615', 'vk6.85884', 'vk6.86199', 'vk6.86637', 'vk6.87501', 'vk6.88162', 'vk6.88580', 'vk6.89387']

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	O1O2O3O4O5U2U1O6U4U3U6U5
R3 orbit	{'O1O2O3O4O5U2U1O6U4U3U6U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U1U6U3U2O6U5U4
Gauss code of K*	O1O2O3O4U5U6U2U1U4O6O5U3
Gauss code of -K*	O1O2O3O4U2O5O6U1U4U3U6U5
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 -3 0 0 4 2],[ 3 0 0 3 2 4 2],[ 3 0 0 2 1 3 2],[ 0 -3 -2 0 0 3 2],[ 0 -2 -1 0 0 2 1],[-4 -4 -3 -3 -2 0 0],[-2 -2 -2 -2 -1 0 0]]
Primitive based matrix	[[ 0 4 2 0 0 -3 -3],[-4 0 0 -2 -3 -3 -4],[-2 0 0 -1 -2 -2 -2],[ 0 2 1 0 0 -1 -2],[ 0 3 2 0 0 -2 -3],[ 3 3 2 1 2 0 0],[ 3 4 2 2 3 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,-2,0,0,3,3,0,2,3,3,4,1,2,2,2,0,1,2,2,3,0]
Phi over symmetry	[-4,-2,0,0,3,3,0,2,3,3,4,1,2,2,2,0,1,2,2,3,0]
Phi of -K	[-3,-3,0,0,2,4,0,0,1,3,3,1,2,3,4,0,0,1,1,2,2]
Phi of K*	[-4,-2,0,0,3,3,2,1,2,3,4,0,1,3,3,0,0,1,1,2,0]
Phi of -K*	[-3,-3,0,0,2,4,0,1,2,2,3,2,3,2,4,0,1,2,2,3,0]
Symmetry type of based matrix	c
u-polynomial	-t^4+2t^3-t^2
Normalized Jones-Krushkal polynomial	9z^2+28z+21
Enhanced Jones-Krushkal polynomial	9w^3z^2+28w^2z+21w
Inner characteristic polynomial	t^6+69t^4+26t^2+1
Outer characteristic polynomial	t^7+107t^5+85t^3+6t
Flat arrow polynomial	4K12K2 - 2K1K3 - K2
2-strand cable arrow polynomial	768K14K2 - 1472K14 + 768K1*3K2K3 - 256K1*3K3 + 480K12K2*3 + 384K1*2K2*2K4 - 4352K12K2*2 - 1152K1*2K2K4 + 4392K1*2K2 - 576K12K3*2 - 416K1*2K4*2 - 2616K1*2 + 192K1K22K3K4 - 832K1K22K3 + 96K1K2*2K4K5 - 448K1K22K5 - 416K1K2K3K4 + 4656K1K2K3 - 128K1K2K4K5 + 1584K1K3K4 + 672K1K4K5 - 32K2*4K4*2 + 128K2*4K4 - 576K24 + 32K2*3K3K5 + 32K2*3K4K6 - 64K2*3K6 - 192K22K3*2 - 336K2*2K4*2 + 1496K2*2K4 - 64K22K5*2 - 8K2*2K6*2 - 2512K2*2 - 32K2K32K4 + 552K2K3K5 + 272K2K4K6 - 1336K32 - 938K4*2 - 288K5*2 - 48K6**2 + 2584
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {3}, {1, 2}]]
If K is slice	False

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