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Flat knot 6.867

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,0,0,1,2,3,1,0,1,1,0,-1,-1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.867', '7.28076']
Arrow polynomial of the knot is: 4K12K2 - 2K12 - 2K1K2 - 2K1K3 + K1 - 2K2**2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.128', '6.408', '6.452', '6.532', '6.867', '6.917', '6.938', '6.1164', '6.1173', '6.1174']
Outer characteristic polynomial of the knot is: t^7+33t^5+46t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.867', '7.28076']
2-strand cable arrow polynomial of the knot is: -3136K14 + 1408K1*3K2K3 + 256K1*3K3K4 - 864K1*3K3 + 640K12K2*2K4 - 5808K12K2*2 + 32K1*2K2K42 - 1568K1*2K2K4 + 6560K1*2K2 - 1728K12K3*2 - 32K1*2K3K5 - 976K1*2K4*2 - 1988K1*2 + 1568K1K23K3 + 384K1K2*2K3K4 - 480K1K22K3 - 672K1K2*2K5 + 32K1K2K3K4*2 - 896K1K2K3K4 - 32K1K2K3K6 + 6408K1K2K3 - 224K1K2K4K5 - 32K1K2K4K7 + 2488K1K3K4 + 752K1K4K5 + 16K1K5K6 + 8K1K6K7 - 32K24K4*2 + 704K2*4K4 - 1944K24 + 32K2*3K4K6 - 160K2*3K6 - 1232K22K3*2 + 32K2*2K4*3 - 1128K2*2K4*2 - 32K2*2K4K8 + 2032K2*2K4 - 8K22K6*2 - 1674K2*2 - 64K2K32K4 + 800K2K3K5 + 592K2K4K6 + 8K2K6K8 - 16K3*2K4*2 - 1488K3*2 + 16K3K4K7 - 8K44 + 8K4*2K8 - 924K42 - 128K5*2 - 38K6*2 - 4K7*2 - 2K8**2 + 2540
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.867']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.49', 'vk6.98', 'vk6.193', 'vk6.248', 'vk6.385', 'vk6.796', 'vk6.805', 'vk6.1250', 'vk6.1339', 'vk6.1392', 'vk6.1542', 'vk6.2023', 'vk6.2412', 'vk6.2431', 'vk6.2674', 'vk6.2979', 'vk6.10445', 'vk6.10462', 'vk6.10690', 'vk6.10877', 'vk6.14646', 'vk6.16257', 'vk6.19175', 'vk6.25741', 'vk6.25888', 'vk6.30136', 'vk6.30373', 'vk6.30500', 'vk6.33421', 'vk6.33583', 'vk6.53793', 'vk6.63408']
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,-1,1,1,1,1,0,0,1,2,3,1,0,1,1,0,-1,-1,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.867', '7.28076']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.128', '6.408', '6.452', '6.532', '6.867', '6.917', '6.938', '6.1164', '6.1173', '6.1174']

Outer characteristic polynomial of the knot is: t^7+33t^5+46t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.867', '7.28076']

2-strand cable arrow polynomial of the knot is: -3136*K1**4 + 1408*K1**3*K2*K3 + 256*K1**3*K3*K4 - 864*K1**3*K3 + 640*K1**2*K2**2*K4 - 5808*K1**2*K2**2 + 32*K1**2*K2*K4**2 - 1568*K1**2*K2*K4 + 6560*K1**2*K2 - 1728*K1**2*K3**2 - 32*K1**2*K3*K5 - 976*K1**2*K4**2 - 1988*K1**2 + 1568*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 480*K1*K2**2*K3 - 672*K1*K2**2*K5 + 32*K1*K2*K3*K4**2 - 896*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6408*K1*K2*K3 - 224*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2488*K1*K3*K4 + 752*K1*K4*K5 + 16*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**4*K4**2 + 704*K2**4*K4 - 1944*K2**4 + 32*K2**3*K4*K6 - 160*K2**3*K6 - 1232*K2**2*K3**2 + 32*K2**2*K4**3 - 1128*K2**2*K4**2 - 32*K2**2*K4*K8 + 2032*K2**2*K4 - 8*K2**2*K6**2 - 1674*K2**2 - 64*K2*K3**2*K4 + 800*K2*K3*K5 + 592*K2*K4*K6 + 8*K2*K6*K8 - 16*K3**2*K4**2 - 1488*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 924*K4**2 - 128*K5**2 - 38*K6**2 - 4*K7**2 - 2*K8**2 + 2540

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.867']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.49', 'vk6.98', 'vk6.193', 'vk6.248', 'vk6.385', 'vk6.796', 'vk6.805', 'vk6.1250', 'vk6.1339', 'vk6.1392', 'vk6.1542', 'vk6.2023', 'vk6.2412', 'vk6.2431', 'vk6.2674', 'vk6.2979', 'vk6.10445', 'vk6.10462', 'vk6.10690', 'vk6.10877', 'vk6.14646', 'vk6.16257', 'vk6.19175', 'vk6.25741', 'vk6.25888', 'vk6.30136', 'vk6.30373', 'vk6.30500', 'vk6.33421', 'vk6.33583', 'vk6.53793', 'vk6.63408']

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	O1O2O3O4U1U4O5O6U5U6U3U2
R3 orbit	{'O1O2O3O4U1U4O5O6U5U6U3U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U3U2U5U6O5O6U1U4
Gauss code of K*	O1O2O3O4U5U4U3U6O5O6U1U2
Gauss code of -K*	O1O2O3O4U3U4O5O6U5U2U1U6
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 1 1 -1 1],[ 3 0 3 2 1 0 0],[-1 -3 0 0 0 -1 1],[-1 -2 0 0 0 -1 1],[-1 -1 0 0 0 0 0],[ 1 0 1 1 0 0 1],[-1 0 -1 -1 0 -1 0]]
Primitive based matrix	[[ 0 1 1 1 1 -1 -3],[-1 0 1 0 0 -1 -2],[-1 -1 0 0 -1 -1 0],[-1 0 0 0 0 0 -1],[-1 0 1 0 0 -1 -3],[ 1 1 1 0 1 0 0],[ 3 2 0 1 3 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,-1,1,3,-1,0,0,1,2,0,1,1,0,0,0,1,1,3,0]
Phi over symmetry	[-3,-1,1,1,1,1,0,0,1,2,3,1,0,1,1,0,-1,-1,0,0,0]
Phi of -K	[-3,-1,1,1,1,1,2,1,2,3,4,1,1,2,1,0,0,-1,0,-1,0]
Phi of K*	[-1,-1,-1,-1,1,3,-1,-1,0,1,4,0,0,1,1,0,1,2,2,3,2]
Phi of -K*	[-3,-1,1,1,1,1,0,0,1,2,3,1,0,1,1,0,-1,-1,0,0,0]
Symmetry type of based matrix	c
u-polynomial	t^3-3t
Normalized Jones-Krushkal polynomial	7z^2+24z+21
Enhanced Jones-Krushkal polynomial	-4w^4z^2+11w^3z^2+24w^2z+21w
Inner characteristic polynomial	t^6+19t^4+20t^2
Outer characteristic polynomial	t^7+33t^5+46t^3+8t
Flat arrow polynomial	4K12K2 - 2K12 - 2K1K2 - 2K1K3 + K1 - 2K2**2 + K3 + K4 + 2
2-strand cable arrow polynomial	-3136K14 + 1408K1*3K2K3 + 256K1*3K3K4 - 864K1*3K3 + 640K12K2*2K4 - 5808K12K2*2 + 32K1*2K2K42 - 1568K1*2K2K4 + 6560K1*2K2 - 1728K12K3*2 - 32K1*2K3K5 - 976K1*2K4*2 - 1988K1*2 + 1568K1K23K3 + 384K1K2*2K3K4 - 480K1K22K3 - 672K1K2*2K5 + 32K1K2K3K4*2 - 896K1K2K3K4 - 32K1K2K3K6 + 6408K1K2K3 - 224K1K2K4K5 - 32K1K2K4K7 + 2488K1K3K4 + 752K1K4K5 + 16K1K5K6 + 8K1K6K7 - 32K24K4*2 + 704K2*4K4 - 1944K24 + 32K2*3K4K6 - 160K2*3K6 - 1232K22K3*2 + 32K2*2K4*3 - 1128K2*2K4*2 - 32K2*2K4K8 + 2032K2*2K4 - 8K22K6*2 - 1674K2*2 - 64K2K32K4 + 800K2K3K5 + 592K2K4K6 + 8K2K6K8 - 16K3*2K4*2 - 1488K3*2 + 16K3K4K7 - 8K44 + 8K4*2K8 - 924K42 - 128K5*2 - 38K6*2 - 4K7*2 - 2K8**2 + 2540
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice	False

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