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Flat knot 6.938

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,0,0,1,2,3,1,0,0,0,0,0,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.938', '7.16073']
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+61t^5+180t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.938', '7.16073']
2-strand cable arrow polynomial of the knot is: -576K14 + 384K1*3K3K4 - 1104K1*2K2*2 + 32K1*2K2K42 - 608K1*2K2K4 + 1640K1*2K2 - 896K12K3*2 - 64K1*2K3K5 - 848K1*2K4*2 - 1476K1*2 + 384K1K23K3 + 480K1K2*2K3K4 - 512K1K22K3 - 192K1K2*2K5 + 32K1K2K3K4*2 - 448K1K2K3K4 - 64K1K2K3K6 + 2984K1K2K3 - 192K1K2K4K5 - 32K1K2K4K7 + 2104K1K3K4 + 664K1K4K5 + 8K1K5K6 + 8K1K6K7 - 32K24K4*2 + 160K2*4K4 - 344K24 + 32K2*3K4K6 - 1040K2*2K3*2 + 32K2*2K4*3 - 872K2*2K4*2 - 32K2*2K4K8 + 1048K2*2K4 - 8K22K6*2 - 1154K2*2 - 96K2K32K4 + 672K2K3K5 + 504K2K4K6 + 8K2K6K8 - 16K3*2K4*2 - 1120K3*2 + 16K3K4K7 - 8K44 + 8K4*2K8 - 828K42 - 120K5*2 - 30K6*2 - 4K7*2 - 2K8**2 + 1516
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.938']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.489', 'vk6.559', 'vk6.631', 'vk6.966', 'vk6.1061', 'vk6.1141', 'vk6.1657', 'vk6.1772', 'vk6.1854', 'vk6.2148', 'vk6.2243', 'vk6.2323', 'vk6.2590', 'vk6.2913', 'vk6.3076', 'vk6.3183', 'vk6.12063', 'vk6.13056', 'vk6.20508', 'vk6.21093', 'vk6.21883', 'vk6.22525', 'vk6.27934', 'vk6.28537', 'vk6.29420', 'vk6.32715', 'vk6.39353', 'vk6.41526', 'vk6.46815', 'vk6.46905', 'vk6.53293', 'vk6.57375']
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,-1,1,1,1,1,0,0,1,2,3,1,0,0,0,0,0,0,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.938', '7.16073']

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+61t^5+180t^3+6t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.938', '7.16073']

2-strand cable arrow polynomial of the knot is: -576*K1**4 + 384*K1**3*K3*K4 - 1104*K1**2*K2**2 + 32*K1**2*K2*K4**2 - 608*K1**2*K2*K4 + 1640*K1**2*K2 - 896*K1**2*K3**2 - 64*K1**2*K3*K5 - 848*K1**2*K4**2 - 1476*K1**2 + 384*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 512*K1*K2**2*K3 - 192*K1*K2**2*K5 + 32*K1*K2*K3*K4**2 - 448*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 2984*K1*K2*K3 - 192*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2104*K1*K3*K4 + 664*K1*K4*K5 + 8*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**4*K4**2 + 160*K2**4*K4 - 344*K2**4 + 32*K2**3*K4*K6 - 1040*K2**2*K3**2 + 32*K2**2*K4**3 - 872*K2**2*K4**2 - 32*K2**2*K4*K8 + 1048*K2**2*K4 - 8*K2**2*K6**2 - 1154*K2**2 - 96*K2*K3**2*K4 + 672*K2*K3*K5 + 504*K2*K4*K6 + 8*K2*K6*K8 - 16*K3**2*K4**2 - 1120*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 828*K4**2 - 120*K5**2 - 30*K6**2 - 4*K7**2 - 2*K8**2 + 1516

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.938']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.489', 'vk6.559', 'vk6.631', 'vk6.966', 'vk6.1061', 'vk6.1141', 'vk6.1657', 'vk6.1772', 'vk6.1854', 'vk6.2148', 'vk6.2243', 'vk6.2323', 'vk6.2590', 'vk6.2913', 'vk6.3076', 'vk6.3183', 'vk6.12063', 'vk6.13056', 'vk6.20508', 'vk6.21093', 'vk6.21883', 'vk6.22525', 'vk6.27934', 'vk6.28537', 'vk6.29420', 'vk6.32715', 'vk6.39353', 'vk6.41526', 'vk6.46815', 'vk6.46905', 'vk6.53293', 'vk6.57375']

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	O1O2O3O4U5U6O5O6U1U4U3U2
R3 orbit	{'O1O2O3O4U5U6O5O6U1U4U3U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U3U2U1U4O5O6U5U6
Gauss code of K*	O1O2O3O4U1U4U3U2O5O6U5U6
Gauss code of -K*	O1O2O3O4U5U6O5O6U3U2U1U4
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 2 4],[-1 -3 0 0 0 -2 0],[-1 -2 0 0 0 -2 0],[-1 -1 0 0 0 -2 0],[ 1 -2 2 2 2 0 1],[-1 -4 0 0 0 -1 0]]
Primitive based matrix	[[ 0 1 1 1 1 -1 -3],[-1 0 0 0 0 -1 -4],[-1 0 0 0 0 -2 -1],[-1 0 0 0 0 -2 -2],[-1 0 0 0 0 -2 -3],[ 1 1 2 2 2 0 -2],[ 3 4 1 2 3 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,-1,1,3,0,0,0,1,4,0,0,2,1,0,2,2,2,3,2]
Phi over symmetry	[-3,-1,1,1,1,1,0,0,1,2,3,1,0,0,0,0,0,0,0,0,0]
Phi of -K	[-3,-1,1,1,1,1,0,0,1,2,3,1,0,0,0,0,0,0,0,0,0]
Phi of K*	[-1,-1,-1,-1,1,3,0,0,0,0,1,0,0,0,2,0,0,3,1,0,0]
Phi of -K*	[-3,-1,1,1,1,1,2,1,2,3,4,2,2,2,1,0,0,0,0,0,0]
Symmetry type of based matrix	c
u-polynomial	t^3-3t
Normalized Jones-Krushkal polynomial	z^2+6z+9
Enhanced Jones-Krushkal polynomial	-4w^4z^2+5w^3z^2-12w^3z+18w^2z+9w
Inner characteristic polynomial	t^6+47t^4+134t^2
Outer characteristic polynomial	t^7+61t^5+180t^3+6t
Flat arrow polynomial	4K12K2 - 2K12 - 2K1K2 - 2K1K3 + K1 - 2K2**2 + K3 + K4 + 2
2-strand cable arrow polynomial	-576K14 + 384K1*3K3K4 - 1104K1*2K2*2 + 32K1*2K2K42 - 608K1*2K2K4 + 1640K1*2K2 - 896K12K3*2 - 64K1*2K3K5 - 848K1*2K4*2 - 1476K1*2 + 384K1K23K3 + 480K1K2*2K3K4 - 512K1K22K3 - 192K1K2*2K5 + 32K1K2K3K4*2 - 448K1K2K3K4 - 64K1K2K3K6 + 2984K1K2K3 - 192K1K2K4K5 - 32K1K2K4K7 + 2104K1K3K4 + 664K1K4K5 + 8K1K5K6 + 8K1K6K7 - 32K24K4*2 + 160K2*4K4 - 344K24 + 32K2*3K4K6 - 1040K2*2K3*2 + 32K2*2K4*3 - 872K2*2K4*2 - 32K2*2K4K8 + 1048K2*2K4 - 8K22K6*2 - 1154K2*2 - 96K2K32K4 + 672K2K3K5 + 504K2K4K6 + 8K2K6K8 - 16K3*2K4*2 - 1120K3*2 + 16K3K4K7 - 8K44 + 8K4*2K8 - 828K42 - 120K5*2 - 30K6*2 - 4K7*2 - 2K8**2 + 1516
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}], [{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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