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Flat knot 6.25

Min(phi) over symmetries of the knot is: [-5,-2,-1,2,3,3,1,3,2,4,5,1,1,2,3,1,2,3,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.25']
Arrow polynomial of the knot is: 4K1K2*2 - 4K1K2 + K1 - 2K2*K3 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.25', '6.29']
Outer characteristic polynomial of the knot is: t^7+136t^5+105t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.25']
2-strand cable arrow polynomial of the knot is: -144K14 - 288K1*3K3 - 480K12K2*2 - 480K1*2K2K4 + 2784K1*2K2 - 16K12K3*2 - 240K1*2K4*2 - 32K1*2K4K6 - 3640K1*2 - 416K1K22K3 + 224K1K2K3K4*2 - 704K1K2K3K4 + 3136K1K2K3 - 32K1K2K4K7 + 1720K1K3K4 + 568K1K4K5 + 48K1K5K6 - 32K2*4 + 96K2*2K3*2K4 - 224K22K3*2 - 32K2*2K4*4 + 64K2*2K4*3 - 528K2*2K4*2 + 1816K2*2K4 - 3482K22 - 192K2K32K4 - 96K2K3K4K5 + 512K2K3K5 + 32K2K43K6 - 64K2K4*2K6 + 520K2K4K6 + 24K2K6K8 - 192K32K4*2 + 16K3*2K6 - 1732K32 + 80K3K4K7 - 48K44 - 8K4*2K6*2 + 48K4*2K8 - 1448K42 - 276K5*2 - 134K6*2 - 16K8**2 + 3286
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.25']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73318', 'vk6.73459', 'vk6.74030', 'vk6.74589', 'vk6.75206', 'vk6.75463', 'vk6.76065', 'vk6.76780', 'vk6.78195', 'vk6.78425', 'vk6.79009', 'vk6.79583', 'vk6.80016', 'vk6.80167', 'vk6.80546', 'vk6.81000', 'vk6.81883', 'vk6.82356', 'vk6.82378', 'vk6.82599', 'vk6.83622', 'vk6.83665', 'vk6.84301', 'vk6.84361', 'vk6.84475', 'vk6.84590', 'vk6.84626', 'vk6.85229', 'vk6.85594', 'vk6.86755', 'vk6.88705', 'vk6.88975']
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: [-5,-2,-1,2,3,3,1,3,2,4,5,1,1,2,3,1,2,3,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.25']

Arrow polynomial of the knot is: 4*K1*K2**2 - 4*K1*K2 + K1 - 2*K2*K3 + K3 + 1

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.25', '6.29']

Outer characteristic polynomial of the knot is: t^7+136t^5+105t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.25']

2-strand cable arrow polynomial of the knot is: -144*K1**4 - 288*K1**3*K3 - 480*K1**2*K2**2 - 480*K1**2*K2*K4 + 2784*K1**2*K2 - 16*K1**2*K3**2 - 240*K1**2*K4**2 - 32*K1**2*K4*K6 - 3640*K1**2 - 416*K1*K2**2*K3 + 224*K1*K2*K3*K4**2 - 704*K1*K2*K3*K4 + 3136*K1*K2*K3 - 32*K1*K2*K4*K7 + 1720*K1*K3*K4 + 568*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**4 + 96*K2**2*K3**2*K4 - 224*K2**2*K3**2 - 32*K2**2*K4**4 + 64*K2**2*K4**3 - 528*K2**2*K4**2 + 1816*K2**2*K4 - 3482*K2**2 - 192*K2*K3**2*K4 - 96*K2*K3*K4*K5 + 512*K2*K3*K5 + 32*K2*K4**3*K6 - 64*K2*K4**2*K6 + 520*K2*K4*K6 + 24*K2*K6*K8 - 192*K3**2*K4**2 + 16*K3**2*K6 - 1732*K3**2 + 80*K3*K4*K7 - 48*K4**4 - 8*K4**2*K6**2 + 48*K4**2*K8 - 1448*K4**2 - 276*K5**2 - 134*K6**2 - 16*K8**2 + 3286

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.25']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73318', 'vk6.73459', 'vk6.74030', 'vk6.74589', 'vk6.75206', 'vk6.75463', 'vk6.76065', 'vk6.76780', 'vk6.78195', 'vk6.78425', 'vk6.79009', 'vk6.79583', 'vk6.80016', 'vk6.80167', 'vk6.80546', 'vk6.81000', 'vk6.81883', 'vk6.82356', 'vk6.82378', 'vk6.82599', 'vk6.83622', 'vk6.83665', 'vk6.84301', 'vk6.84361', 'vk6.84475', 'vk6.84590', 'vk6.84626', 'vk6.85229', 'vk6.85594', 'vk6.86755', 'vk6.88705', 'vk6.88975']

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	O1O2O3O4O5O6U1U3U6U2U5U4
R3 orbit	{'O1O2O3O4O5O6U1U3U6U2U5U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U3U2U5U1U4U6
Gauss code of K*	O1O2O3O4O5O6U1U4U2U6U5U3
Gauss code of -K*	O1O2O3O4O5O6U4U2U1U5U3U6
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 -5 -1 -2 3 3 2],[ 5 0 3 1 5 4 2],[ 1 -3 0 -1 3 2 1],[ 2 -1 1 0 3 2 1],[-3 -5 -3 -3 0 0 0],[-3 -4 -2 -2 0 0 0],[-2 -2 -1 -1 0 0 0]]
Primitive based matrix	[[ 0 3 3 2 -1 -2 -5],[-3 0 0 0 -2 -2 -4],[-3 0 0 0 -3 -3 -5],[-2 0 0 0 -1 -1 -2],[ 1 2 3 1 0 -1 -3],[ 2 2 3 1 1 0 -1],[ 5 4 5 2 3 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-3,-2,1,2,5,0,0,2,2,4,0,3,3,5,1,1,2,1,3,1]
Phi over symmetry	[-5,-2,-1,2,3,3,1,3,2,4,5,1,1,2,3,1,2,3,0,0,0]
Phi of -K	[-5,-2,-1,2,3,3,2,1,5,3,4,0,3,2,3,2,1,2,1,1,0]
Phi of K*	[-3,-3,-2,1,2,5,0,1,1,2,3,1,2,3,4,2,3,5,0,1,2]
Phi of -K*	[-5,-2,-1,2,3,3,1,3,2,4,5,1,1,2,3,1,2,3,0,0,0]
Symmetry type of based matrix	c
u-polynomial	t^5-2t^3+t
Normalized Jones-Krushkal polynomial	6z^2+23z+23
Enhanced Jones-Krushkal polynomial	6w^3z^2+23w^2z+23w
Inner characteristic polynomial	t^6+84t^4+11t^2
Outer characteristic polynomial	t^7+136t^5+105t^3+4t
Flat arrow polynomial	4K1K2*2 - 4K1K2 + K1 - 2K2*K3 + K3 + 1
2-strand cable arrow polynomial	-144K14 - 288K1*3K3 - 480K12K2*2 - 480K1*2K2K4 + 2784K1*2K2 - 16K12K3*2 - 240K1*2K4*2 - 32K1*2K4K6 - 3640K1*2 - 416K1K22K3 + 224K1K2K3K4*2 - 704K1K2K3K4 + 3136K1K2K3 - 32K1K2K4K7 + 1720K1K3K4 + 568K1K4K5 + 48K1K5K6 - 32K2*4 + 96K2*2K3*2K4 - 224K22K3*2 - 32K2*2K4*4 + 64K2*2K4*3 - 528K2*2K4*2 + 1816K2*2K4 - 3482K22 - 192K2K32K4 - 96K2K3K4K5 + 512K2K3K5 + 32K2K43K6 - 64K2K4*2K6 + 520K2K4K6 + 24K2K6K8 - 192K32K4*2 + 16K3*2K6 - 1732K32 + 80K3K4K7 - 48K44 - 8K4*2K6*2 + 48K4*2K8 - 1448K42 - 276K5*2 - 134K6*2 - 16K8**2 + 3286
Genus of based matrix	1
Fillings of based matrix	[[{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}]]
If K is slice	False

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