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Flat knot 6.492

Min(phi) over symmetries of the knot is: [-4,-1,0,1,2,2,0,3,1,4,4,2,1,1,2,0,2,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.492']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 6K12 - 4K1K2 - 2K1K3 - K1 + 2K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.94', '6.482', '6.492']
Outer characteristic polynomial of the knot is: t^7+71t^5+162t^3+15t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.492']
2-strand cable arrow polynomial of the knot is: -64K14K2*2 + 128K1*4K2 - 208K14 + 64K1*3K2K3 - 192K1*3K3 - 768K12K2*4 - 512K1*2K2*3K4 + 3680K12K2*3 - 64K1*2K2*2K3*2 + 448K1*2K2*2K4 - 7984K12K2*2 + 64K1*2K2K32 - 736K1*2K2K4 + 5832K1*2K2 - 176K12K3*2 - 4016K1*2 + 384K1K24K3K4 - 768K1K24K3 - 256K1K2*3K3K4 + 3392K1K23K3 + 832K1K2*2K3K4 - 2272K1K22K3 + 32K1K2*2K4K5 - 576K1K22K5 - 448K1K2K3K4 - 32K1K2K3K6 + 6344K1K2K3 + 672K1K3K4 + 32K1K4K5 - 32K2*6 - 384K2*4K3*2 - 288K2*4K4*2 + 1184K2*4K4 - 4920K24 + 288K2*3K3K5 + 64K2*3K4K6 - 64K2*3K6 - 1808K22K3*2 - 704K2*2K4*2 + 2912K2*2K4 - 64K22K5*2 - 8K2*2K6*2 - 698K2*2 + 616K2K3K5 + 112K2K4K6 + 8K2K5K7 - 1452K32 - 464K4*2 - 36K5*2 - 6K6**2 + 2910
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.492']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70461', 'vk6.70478', 'vk6.70524', 'vk6.70599', 'vk6.70631', 'vk6.70658', 'vk6.70751', 'vk6.70839', 'vk6.70912', 'vk6.70942', 'vk6.71005', 'vk6.71110', 'vk6.71151', 'vk6.71168', 'vk6.71240', 'vk6.71299', 'vk6.71318', 'vk6.71335', 'vk6.73548', 'vk6.74356', 'vk6.75005', 'vk6.75303', 'vk6.76574', 'vk6.76638', 'vk6.76987', 'vk6.78283', 'vk6.79400', 'vk6.79938', 'vk6.81497', 'vk6.86868', 'vk6.88056', 'vk6.89229']
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: [-4,-1,0,1,2,2,0,3,1,4,4,2,1,1,2,0,2,1,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.492']

Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 6*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 2*K2 + K3 + 3

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.94', '6.482', '6.492']

Outer characteristic polynomial of the knot is: t^7+71t^5+162t^3+15t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.492']

2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 128*K1**4*K2 - 208*K1**4 + 64*K1**3*K2*K3 - 192*K1**3*K3 - 768*K1**2*K2**4 - 512*K1**2*K2**3*K4 + 3680*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 + 448*K1**2*K2**2*K4 - 7984*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 736*K1**2*K2*K4 + 5832*K1**2*K2 - 176*K1**2*K3**2 - 4016*K1**2 + 384*K1*K2**4*K3*K4 - 768*K1*K2**4*K3 - 256*K1*K2**3*K3*K4 + 3392*K1*K2**3*K3 + 832*K1*K2**2*K3*K4 - 2272*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 576*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6344*K1*K2*K3 + 672*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 - 384*K2**4*K3**2 - 288*K2**4*K4**2 + 1184*K2**4*K4 - 4920*K2**4 + 288*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 1808*K2**2*K3**2 - 704*K2**2*K4**2 + 2912*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 698*K2**2 + 616*K2*K3*K5 + 112*K2*K4*K6 + 8*K2*K5*K7 - 1452*K3**2 - 464*K4**2 - 36*K5**2 - 6*K6**2 + 2910

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.492']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70461', 'vk6.70478', 'vk6.70524', 'vk6.70599', 'vk6.70631', 'vk6.70658', 'vk6.70751', 'vk6.70839', 'vk6.70912', 'vk6.70942', 'vk6.71005', 'vk6.71110', 'vk6.71151', 'vk6.71168', 'vk6.71240', 'vk6.71299', 'vk6.71318', 'vk6.71335', 'vk6.73548', 'vk6.74356', 'vk6.75005', 'vk6.75303', 'vk6.76574', 'vk6.76638', 'vk6.76987', 'vk6.78283', 'vk6.79400', 'vk6.79938', 'vk6.81497', 'vk6.86868', 'vk6.88056', 'vk6.89229']

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	O1O2O3O4O5U1U4U5O6U2U3U6
R3 orbit	{'O1O2O3O4O5U1U4U5O6U2U3U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U6U3U4O6U1U2U5
Gauss code of K*	O1O2O3U4O5O6O4U1U5U6U2U3
Gauss code of -K*	O1O2O3U1O4O5O6U4U5U2U3U6
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 -4 -1 1 0 2 2],[ 4 0 3 4 1 2 2],[ 1 -3 0 1 -1 1 2],[-1 -4 -1 0 -1 1 1],[ 0 -1 1 1 0 1 0],[-2 -2 -1 -1 -1 0 0],[-2 -2 -2 -1 0 0 0]]
Primitive based matrix	[[ 0 2 2 1 0 -1 -4],[-2 0 0 -1 0 -2 -2],[-2 0 0 -1 -1 -1 -2],[-1 1 1 0 -1 -1 -4],[ 0 0 1 1 0 1 -1],[ 1 2 1 1 -1 0 -3],[ 4 2 2 4 1 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-1,0,1,4,0,1,0,2,2,1,1,1,2,1,1,4,-1,1,3]
Phi over symmetry	[-4,-1,0,1,2,2,0,3,1,4,4,2,1,1,2,0,2,1,0,0,0]
Phi of -K	[-4,-1,0,1,2,2,0,3,1,4,4,2,1,1,2,0,2,1,0,0,0]
Phi of K*	[-2,-2,-1,0,1,4,0,0,1,2,4,0,2,1,4,0,1,1,2,3,0]
Phi of -K*	[-4,-1,0,1,2,2,3,1,4,2,2,-1,1,1,2,1,1,0,1,1,0]
Symmetry type of based matrix	c
u-polynomial	t^4-2t^2
Normalized Jones-Krushkal polynomial	5z^2+18z+17
Enhanced Jones-Krushkal polynomial	-4w^4z^2+9w^3z^2-8w^3z+26w^2z+17w
Inner characteristic polynomial	t^6+45t^4+56t^2
Outer characteristic polynomial	t^7+71t^5+162t^3+15t
Flat arrow polynomial	4K13 + 4K1*2K2 - 6K12 - 4K1K2 - 2K1K3 - K1 + 2K2 + K3 + 3
2-strand cable arrow polynomial	-64K14K2*2 + 128K1*4K2 - 208K14 + 64K1*3K2K3 - 192K1*3K3 - 768K12K2*4 - 512K1*2K2*3K4 + 3680K12K2*3 - 64K1*2K2*2K3*2 + 448K1*2K2*2K4 - 7984K12K2*2 + 64K1*2K2K32 - 736K1*2K2K4 + 5832K1*2K2 - 176K12K3*2 - 4016K1*2 + 384K1K24K3K4 - 768K1K24K3 - 256K1K2*3K3K4 + 3392K1K23K3 + 832K1K2*2K3K4 - 2272K1K22K3 + 32K1K2*2K4K5 - 576K1K22K5 - 448K1K2K3K4 - 32K1K2K3K6 + 6344K1K2K3 + 672K1K3K4 + 32K1K4K5 - 32K2*6 - 384K2*4K3*2 - 288K2*4K4*2 + 1184K2*4K4 - 4920K24 + 288K2*3K3K5 + 64K2*3K4K6 - 64K2*3K6 - 1808K22K3*2 - 704K2*2K4*2 + 2912K2*2K4 - 64K22K5*2 - 8K2*2K6*2 - 698K2*2 + 616K2K3K5 + 112K2K4K6 + 8K2K5K7 - 1452K32 - 464K4*2 - 36K5*2 - 6K6**2 + 2910
Genus of based matrix	2
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{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}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {1, 5}, {4}, {3}, {2}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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