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Flat knot 6.1171

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,0,1,3,4,0,0,1,1,1,1,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1171']
Arrow polynomial of the knot is: 8K12K2 - 8K12 - 2K1K2 - 4K1K3 + K1 + 2K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1171']
Outer characteristic polynomial of the knot is: t^7+62t^5+58t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1171']
2-strand cable arrow polynomial of the knot is: -640K14K2*2 + 1248K1*4K2 - 1584K14 + 384K1*3K2K3 - 608K1*3K3 - 256K12K2*4 + 2240K1*2K2*3 - 64K1*2K2*2K3*2 - 512K1*2K2*2K4*2 + 800K1*2K2*2K4 - 7456K12K2*2 + 64K1*2K2K32 + 192K1*2K2K42 - 1728K1*2K2K4 + 7880K1*2K2 - 176K12K3*2 - 528K1*2K4*2 - 5328K1*2 + 928K1K23K3 + 992K1K2*2K3K4 - 2208K1K22K3 + 576K1K2*2K4K5 - 736K1K22K5 + 32K1K2K3K4*2 - 928K1K2K3K4 + 7752K1K2K3 - 160K1K2K4K5 + 1832K1K3K4 + 656K1K4K5 - 192K24K4*2 + 832K2*4K4 - 2688K24 + 64K2*3K3K5 + 128K2*3K4K6 - 160K2*3K6 - 1056K22K3*2 - 1256K2*2K4*2 + 3496K2*2K4 - 192K22K5*2 - 16K2*2K6*2 - 3546K2*2 - 32K2K32K4 - 32K2K3K4K5 + 808K2K3K5 + 328K2K4K6 + 8K2K5K7 - 2004K3*2 - 1256K4*2 - 228K5*2 - 14K6**2 + 4342
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1171']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4196', 'vk6.4275', 'vk6.5450', 'vk6.5562', 'vk6.7561', 'vk6.7646', 'vk6.9065', 'vk6.9144', 'vk6.18240', 'vk6.18575', 'vk6.24712', 'vk6.25125', 'vk6.36839', 'vk6.37302', 'vk6.44075', 'vk6.44414', 'vk6.48508', 'vk6.48587', 'vk6.49200', 'vk6.49306', 'vk6.50297', 'vk6.50371', 'vk6.51062', 'vk6.51093', 'vk6.56035', 'vk6.56309', 'vk6.60584', 'vk6.60923', 'vk6.65701', 'vk6.65995', 'vk6.68746', 'vk6.68954']
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,-1,1,2,2,0,0,1,3,4,0,0,1,1,1,1,2,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.1171']

Arrow polynomial of the knot is: 8*K1**2*K2 - 8*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + 3

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1171']

Outer characteristic polynomial of the knot is: t^7+62t^5+58t^3+10t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1171']

2-strand cable arrow polynomial of the knot is: -640*K1**4*K2**2 + 1248*K1**4*K2 - 1584*K1**4 + 384*K1**3*K2*K3 - 608*K1**3*K3 - 256*K1**2*K2**4 + 2240*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 512*K1**2*K2**2*K4**2 + 800*K1**2*K2**2*K4 - 7456*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 192*K1**2*K2*K4**2 - 1728*K1**2*K2*K4 + 7880*K1**2*K2 - 176*K1**2*K3**2 - 528*K1**2*K4**2 - 5328*K1**2 + 928*K1*K2**3*K3 + 992*K1*K2**2*K3*K4 - 2208*K1*K2**2*K3 + 576*K1*K2**2*K4*K5 - 736*K1*K2**2*K5 + 32*K1*K2*K3*K4**2 - 928*K1*K2*K3*K4 + 7752*K1*K2*K3 - 160*K1*K2*K4*K5 + 1832*K1*K3*K4 + 656*K1*K4*K5 - 192*K2**4*K4**2 + 832*K2**4*K4 - 2688*K2**4 + 64*K2**3*K3*K5 + 128*K2**3*K4*K6 - 160*K2**3*K6 - 1056*K2**2*K3**2 - 1256*K2**2*K4**2 + 3496*K2**2*K4 - 192*K2**2*K5**2 - 16*K2**2*K6**2 - 3546*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 808*K2*K3*K5 + 328*K2*K4*K6 + 8*K2*K5*K7 - 2004*K3**2 - 1256*K4**2 - 228*K5**2 - 14*K6**2 + 4342

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1171']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4196', 'vk6.4275', 'vk6.5450', 'vk6.5562', 'vk6.7561', 'vk6.7646', 'vk6.9065', 'vk6.9144', 'vk6.18240', 'vk6.18575', 'vk6.24712', 'vk6.25125', 'vk6.36839', 'vk6.37302', 'vk6.44075', 'vk6.44414', 'vk6.48508', 'vk6.48587', 'vk6.49200', 'vk6.49306', 'vk6.50297', 'vk6.50371', 'vk6.51062', 'vk6.51093', 'vk6.56035', 'vk6.56309', 'vk6.60584', 'vk6.60923', 'vk6.65701', 'vk6.65995', 'vk6.68746', 'vk6.68954']

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	O1O2O3O4U1U5U4O5O6U2U3U6
R3 orbit	{'O1O2O3O4U1U5U4O5O6U2U3U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U2U3O5O6U1U6U4
Gauss code of K*	O1O2O3U2U4O5O6O4U1U5U6U3
Gauss code of -K*	O1O2O3U1U4O5O4O6U5U2U3U6
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 2 -1 2],[ 3 0 2 3 1 2 2],[ 1 -2 0 1 1 0 2],[-1 -3 -1 0 1 -2 1],[-2 -1 -1 -1 0 -2 0],[ 1 -2 0 2 2 0 2],[-2 -2 -2 -1 0 -2 0]]
Primitive based matrix	[[ 0 2 2 1 -1 -1 -3],[-2 0 0 -1 -1 -2 -1],[-2 0 0 -1 -2 -2 -2],[-1 1 1 0 -1 -2 -3],[ 1 1 2 1 0 0 -2],[ 1 2 2 2 0 0 -2],[ 3 1 2 3 2 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-1,1,1,3,0,1,1,2,1,1,2,2,2,1,2,3,0,2,2]
Phi over symmetry	[-3,-1,-1,1,2,2,0,0,1,3,4,0,0,1,1,1,1,2,0,0,0]
Phi of -K	[-3,-1,-1,1,2,2,0,0,1,3,4,0,0,1,1,1,1,2,0,0,0]
Phi of K*	[-2,-2,-1,1,1,3,0,0,1,1,3,0,1,2,4,0,1,1,0,0,0]
Phi of -K*	[-3,-1,-1,1,2,2,2,2,3,1,2,0,1,1,2,2,2,2,1,1,0]
Symmetry type of based matrix	c
u-polynomial	t^3-2t^2+t
Normalized Jones-Krushkal polynomial	8z^2+29z+27
Enhanced Jones-Krushkal polynomial	8w^3z^2+29w^2z+27w
Inner characteristic polynomial	t^6+42t^4+30t^2+4
Outer characteristic polynomial	t^7+62t^5+58t^3+10t
Flat arrow polynomial	8K12K2 - 8K12 - 2K1K2 - 4K1K3 + K1 + 2K2 + K3 + 3
2-strand cable arrow polynomial	-640K14K2*2 + 1248K1*4K2 - 1584K14 + 384K1*3K2K3 - 608K1*3K3 - 256K12K2*4 + 2240K1*2K2*3 - 64K1*2K2*2K3*2 - 512K1*2K2*2K4*2 + 800K1*2K2*2K4 - 7456K12K2*2 + 64K1*2K2K32 + 192K1*2K2K42 - 1728K1*2K2K4 + 7880K1*2K2 - 176K12K3*2 - 528K1*2K4*2 - 5328K1*2 + 928K1K23K3 + 992K1K2*2K3K4 - 2208K1K22K3 + 576K1K2*2K4K5 - 736K1K22K5 + 32K1K2K3K4*2 - 928K1K2K3K4 + 7752K1K2K3 - 160K1K2K4K5 + 1832K1K3K4 + 656K1K4K5 - 192K24K4*2 + 832K2*4K4 - 2688K24 + 64K2*3K3K5 + 128K2*3K4K6 - 160K2*3K6 - 1056K22K3*2 - 1256K2*2K4*2 + 3496K2*2K4 - 192K22K5*2 - 16K2*2K6*2 - 3546K2*2 - 32K2K32K4 - 32K2K3K4K5 + 808K2K3K5 + 328K2K4K6 + 8K2K5K7 - 2004K3*2 - 1256K4*2 - 228K5*2 - 14K6**2 + 4342
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice	False

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