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Flat knot 6.1283

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,1,2,3,1,1,1,1,1,1,1,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1283']
Arrow polynomial of the knot is: -10K12 - 6K1K2 + 3K1 + 5K2 + 3K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.335', '6.1283', '6.1316']
Outer characteristic polynomial of the knot is: t^7+48t^5+54t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1283']
2-strand cable arrow polynomial of the knot is: -64K16 + 1440K1*4K2 - 5344K14 + 960K1*3K2K3 - 1792K1*3K3 + 288K12K2*2K4 - 5168K12K2*2 - 992K1*2K2K4 + 11936K1*2K2 - 1600K12K3*2 - 64K1*2K3K5 - 224K1*2K4*2 - 7228K1*2 + 160K1K23K3 - 512K1K2*2K3 - 256K1K2*2K5 - 448K1K2K3K4 - 64K1K2K3K6 + 9136K1K2K3 + 2392K1K3K4 + 520K1K4K5 + 40K1K5K6 - 264K24 - 336K2*2K3*2 - 88K2*2K4*2 + 1096K2*2K4 - 6138K22 + 672K2K3K5 + 136K2K4K6 + 40K3*2K6 - 3168K32 - 1062K4*2 - 324K5*2 - 70K6**2 + 6388
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1283']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19999', 'vk6.20078', 'vk6.21269', 'vk6.21360', 'vk6.27050', 'vk6.27139', 'vk6.28753', 'vk6.28828', 'vk6.38447', 'vk6.38544', 'vk6.40634', 'vk6.40741', 'vk6.45331', 'vk6.45440', 'vk6.47098', 'vk6.47182', 'vk6.56814', 'vk6.56899', 'vk6.57946', 'vk6.58037', 'vk6.61332', 'vk6.61425', 'vk6.62506', 'vk6.62582', 'vk6.66534', 'vk6.66607', 'vk6.67321', 'vk6.67398', 'vk6.69180', 'vk6.69255', 'vk6.69929', 'vk6.69996']
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,-1,0,1,1,2,0,2,1,2,3,1,1,1,1,1,1,1,-1,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.1283']

Arrow polynomial of the knot is: -10*K1**2 - 6*K1*K2 + 3*K1 + 5*K2 + 3*K3 + 6

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.335', '6.1283', '6.1316']

Outer characteristic polynomial of the knot is: t^7+48t^5+54t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1283']

2-strand cable arrow polynomial of the knot is: -64*K1**6 + 1440*K1**4*K2 - 5344*K1**4 + 960*K1**3*K2*K3 - 1792*K1**3*K3 + 288*K1**2*K2**2*K4 - 5168*K1**2*K2**2 - 992*K1**2*K2*K4 + 11936*K1**2*K2 - 1600*K1**2*K3**2 - 64*K1**2*K3*K5 - 224*K1**2*K4**2 - 7228*K1**2 + 160*K1*K2**3*K3 - 512*K1*K2**2*K3 - 256*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 9136*K1*K2*K3 + 2392*K1*K3*K4 + 520*K1*K4*K5 + 40*K1*K5*K6 - 264*K2**4 - 336*K2**2*K3**2 - 88*K2**2*K4**2 + 1096*K2**2*K4 - 6138*K2**2 + 672*K2*K3*K5 + 136*K2*K4*K6 + 40*K3**2*K6 - 3168*K3**2 - 1062*K4**2 - 324*K5**2 - 70*K6**2 + 6388

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1283']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19999', 'vk6.20078', 'vk6.21269', 'vk6.21360', 'vk6.27050', 'vk6.27139', 'vk6.28753', 'vk6.28828', 'vk6.38447', 'vk6.38544', 'vk6.40634', 'vk6.40741', 'vk6.45331', 'vk6.45440', 'vk6.47098', 'vk6.47182', 'vk6.56814', 'vk6.56899', 'vk6.57946', 'vk6.58037', 'vk6.61332', 'vk6.61425', 'vk6.62506', 'vk6.62582', 'vk6.66534', 'vk6.66607', 'vk6.67321', 'vk6.67398', 'vk6.69180', 'vk6.69255', 'vk6.69929', 'vk6.69996']

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	O1O2O3U1O4O5U6U2O6U4U3U5
R3 orbit	{'O1O2O3U1O4O5U6U2O6U4U3U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U1U5O6U2U6O4O5U3
Gauss code of K*	O1O2O3U4U5U2O4U1U3O6O5U6
Gauss code of -K*	O1O2O3U4O5O4U1U3O6U2U5U6
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 -2 -1 1 0 3 -1],[ 2 0 1 2 1 2 1],[ 1 -1 0 1 0 2 1],[-1 -2 -1 0 0 2 -1],[ 0 -1 0 0 0 1 0],[-3 -2 -2 -2 -1 0 -3],[ 1 -1 -1 1 0 3 0]]
Primitive based matrix	[[ 0 3 1 0 -1 -1 -2],[-3 0 -2 -1 -2 -3 -2],[-1 2 0 0 -1 -1 -2],[ 0 1 0 0 0 0 -1],[ 1 2 1 0 0 1 -1],[ 1 3 1 0 -1 0 -1],[ 2 2 2 1 1 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,0,1,1,2,2,1,2,3,2,0,1,1,2,0,0,1,-1,1,1]
Phi over symmetry	[-3,-1,0,1,1,2,0,2,1,2,3,1,1,1,1,1,1,1,-1,0,0]
Phi of -K	[-2,-1,-1,0,1,3,0,0,1,1,3,-1,1,1,2,1,1,1,1,2,0]
Phi of K*	[-3,-1,0,1,1,2,0,2,1,2,3,1,1,1,1,1,1,1,-1,0,0]
Phi of -K*	[-2,-1,-1,0,1,3,1,1,1,2,2,-1,0,1,3,0,1,2,0,1,2]
Symmetry type of based matrix	c
u-polynomial	-t^3+t^2+t
Normalized Jones-Krushkal polynomial	z^2+22z+41
Enhanced Jones-Krushkal polynomial	w^3z^2+22w^2z+41w
Inner characteristic polynomial	t^6+32t^4+29t^2+1
Outer characteristic polynomial	t^7+48t^5+54t^3+5t
Flat arrow polynomial	-10K12 - 6K1K2 + 3K1 + 5K2 + 3K3 + 6
2-strand cable arrow polynomial	-64K16 + 1440K1*4K2 - 5344K14 + 960K1*3K2K3 - 1792K1*3K3 + 288K12K2*2K4 - 5168K12K2*2 - 992K1*2K2K4 + 11936K1*2K2 - 1600K12K3*2 - 64K1*2K3K5 - 224K1*2K4*2 - 7228K1*2 + 160K1K23K3 - 512K1K2*2K3 - 256K1K2*2K5 - 448K1K2K3K4 - 64K1K2K3K6 + 9136K1K2K3 + 2392K1K3K4 + 520K1K4K5 + 40K1K5K6 - 264K24 - 336K2*2K3*2 - 88K2*2K4*2 + 1096K2*2K4 - 6138K22 + 672K2K3K5 + 136K2K4K6 + 40K3*2K6 - 3168K32 - 1062K4*2 - 324K5*2 - 70K6**2 + 6388
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {5}, {4}, {1, 2}]]
If K is slice	False

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