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Flat knot 6.4

Min(phi) over symmetries of the knot is: [-5,-3,0,0,4,4,1,2,3,4,5,1,2,3,4,0,1,2,2,3,0]
Flat knots (up to 7 crossings) with same phi are :['6.4']
Arrow polynomial of the knot is: -2K1K4 + K3 + K5 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.4', '6.12', '6.15', '6.48']
Outer characteristic polynomial of the knot is: t^7+169t^5+131t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.4']
2-strand cable arrow polynomial of the knot is: -1152K14 + 1536K1*3K2K3 - 1024K1*2K2*2K3*2 - 2624K1*2K2*2 - 1600K1*2K2K4 + 3664K1*2K2 - 1472K12K3*2 - 192K1*2K3K5 - 128K1*2K4*2 - 3104K1*2 + 640K1K23K3 + 1408K1K2*2K3K4 - 960K1K22K3 - 192K1K2*2K5 + 64K1K2*2K6K7 + 512K1K2K3*3 - 192K1K2K3K6 + 4624K1K2K3 - 128K1K2K4K5 - 64K1K2K4K7 - 64K1K2K5K6 + 2560K1K3K4 + 656K1K4K5 + 32K1K6K7 - 2K10*2 + 8K10K2K8 - 224K24 + 64K2*3K3K5 - 1088K2*2K3*2 - 576K2*2K4*2 + 1440K2*2K4 - 64K22K5*2 - 32K2*2K6*2 - 64K2*2K7*2 - 8K2*2K8*2 - 2788K2*2 - 320K2K32K4 - 64K2K3K4K5 + 736K2K3K5 + 384K2K4K6 + 240K2K5K7 + 8K2K6K8 + 16K2K7K9 - 128K3*4 + 96K3*2K6 - 1720K32 + 48K3K4K7 + 16K3K5K8 - 1280K4*2 - 472K5*2 - 66K6*2 - 96K7*2 - 12K8**2 + 3146
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.4']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.82810', 'vk6.82814', 'vk6.82930', 'vk6.82940', 'vk6.82941', 'vk6.82976', 'vk6.83244', 'vk6.83300', 'vk6.83302', 'vk6.83910', 'vk6.86060', 'vk6.86136', 'vk6.86138', 'vk6.86784', 'vk6.86787', 'vk6.86817', 'vk6.86819', 'vk6.89804', 'vk6.89902', 'vk6.90097']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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,-3,0,0,4,4,1,2,3,4,5,1,2,3,4,0,1,2,2,3,0]

Flat knots (up to 7 crossings) with same phi are :['6.4']

Arrow polynomial of the knot is: -2*K1*K4 + K3 + K5 + 1

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.4', '6.12', '6.15', '6.48']

Outer characteristic polynomial of the knot is: t^7+169t^5+131t^3+7t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.4']

2-strand cable arrow polynomial of the knot is: -1152*K1**4 + 1536*K1**3*K2*K3 - 1024*K1**2*K2**2*K3**2 - 2624*K1**2*K2**2 - 1600*K1**2*K2*K4 + 3664*K1**2*K2 - 1472*K1**2*K3**2 - 192*K1**2*K3*K5 - 128*K1**2*K4**2 - 3104*K1**2 + 640*K1*K2**3*K3 + 1408*K1*K2**2*K3*K4 - 960*K1*K2**2*K3 - 192*K1*K2**2*K5 + 64*K1*K2**2*K6*K7 + 512*K1*K2*K3**3 - 192*K1*K2*K3*K6 + 4624*K1*K2*K3 - 128*K1*K2*K4*K5 - 64*K1*K2*K4*K7 - 64*K1*K2*K5*K6 + 2560*K1*K3*K4 + 656*K1*K4*K5 + 32*K1*K6*K7 - 2*K10**2 + 8*K10*K2*K8 - 224*K2**4 + 64*K2**3*K3*K5 - 1088*K2**2*K3**2 - 576*K2**2*K4**2 + 1440*K2**2*K4 - 64*K2**2*K5**2 - 32*K2**2*K6**2 - 64*K2**2*K7**2 - 8*K2**2*K8**2 - 2788*K2**2 - 320*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 736*K2*K3*K5 + 384*K2*K4*K6 + 240*K2*K5*K7 + 8*K2*K6*K8 + 16*K2*K7*K9 - 128*K3**4 + 96*K3**2*K6 - 1720*K3**2 + 48*K3*K4*K7 + 16*K3*K5*K8 - 1280*K4**2 - 472*K5**2 - 66*K6**2 - 96*K7**2 - 12*K8**2 + 3146

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.4']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.82810', 'vk6.82814', 'vk6.82930', 'vk6.82940', 'vk6.82941', 'vk6.82976', 'vk6.83244', 'vk6.83300', 'vk6.83302', 'vk6.83910', 'vk6.86060', 'vk6.86136', 'vk6.86138', 'vk6.86784', 'vk6.86787', 'vk6.86817', 'vk6.86819', 'vk6.89804', 'vk6.89902', 'vk6.90097']

The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is +.

The reverse -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	O1O2O3O4O5O6U1U2U4U3U6U5
R3 orbit	{'O1O2O3O4O5O6U1U2U4U3U6U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U2U1U4U3U5U6
Gauss code of K*	Same
Gauss code of -K*	O1O2O3O4O5O6U2U1U4U3U5U6
Diagrammatic symmetry type	+
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 -3 0 0 4 4],[ 5 0 1 3 2 5 4],[ 3 -1 0 2 1 4 3],[ 0 -3 -2 0 0 3 2],[ 0 -2 -1 0 0 2 1],[-4 -5 -4 -3 -2 0 0],[-4 -4 -3 -2 -1 0 0]]
Primitive based matrix	[[ 0 4 4 0 0 -3 -5],[-4 0 0 -1 -2 -3 -4],[-4 0 0 -2 -3 -4 -5],[ 0 1 2 0 0 -1 -2],[ 0 2 3 0 0 -2 -3],[ 3 3 4 1 2 0 -1],[ 5 4 5 2 3 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,-4,0,0,3,5,0,1,2,3,4,2,3,4,5,0,1,2,2,3,1]
Phi over symmetry	[-5,-3,0,0,4,4,1,2,3,4,5,1,2,3,4,0,1,2,2,3,0]
Phi of -K	[-5,-3,0,0,4,4,1,2,3,4,5,1,2,3,4,0,1,2,2,3,0]
Phi of K*	[-4,-4,0,0,3,5,0,1,2,3,4,2,3,4,5,0,1,2,2,3,1]
Phi of -K*	[-5,-3,0,0,4,4,1,2,3,4,5,1,2,3,4,0,1,2,2,3,0]
Symmetry type of based matrix	+
u-polynomial	t^5-2t^4+t^3
Normalized Jones-Krushkal polynomial	9z^2+29z+23
Enhanced Jones-Krushkal polynomial	9w^3z^2+29w^2z+23w
Inner characteristic polynomial	t^6+103t^4+27t^2+1
Outer characteristic polynomial	t^7+169t^5+131t^3+7t
Flat arrow polynomial	-2K1K4 + K3 + K5 + 1
2-strand cable arrow polynomial	-1152K14 + 1536K1*3K2K3 - 1024K1*2K2*2K3*2 - 2624K1*2K2*2 - 1600K1*2K2K4 + 3664K1*2K2 - 1472K12K3*2 - 192K1*2K3K5 - 128K1*2K4*2 - 3104K1*2 + 640K1K23K3 + 1408K1K2*2K3K4 - 960K1K22K3 - 192K1K2*2K5 + 64K1K2*2K6K7 + 512K1K2K3*3 - 192K1K2K3K6 + 4624K1K2K3 - 128K1K2K4K5 - 64K1K2K4K7 - 64K1K2K5K6 + 2560K1K3K4 + 656K1K4K5 + 32K1K6K7 - 2K10*2 + 8K10K2K8 - 224K24 + 64K2*3K3K5 - 1088K2*2K3*2 - 576K2*2K4*2 + 1440K2*2K4 - 64K22K5*2 - 32K2*2K6*2 - 64K2*2K7*2 - 8K2*2K8*2 - 2788K2*2 - 320K2K32K4 - 64K2K3K4K5 + 736K2K3K5 + 384K2K4K6 + 240K2K5K7 + 8K2K6K8 + 16K2K7K9 - 128K3*4 + 96K3*2K6 - 1720K32 + 48K3K4K7 + 16K3K5K8 - 1280K4*2 - 472K5*2 - 66K6*2 - 96K7*2 - 12K8**2 + 3146
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}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {3}, {1, 2}]]
If K is slice	False

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