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Flat knot 6.434

Min(phi) over symmetries of the knot is: [-3,-3,0,1,2,3,0,1,1,3,2,2,1,4,3,1,2,1,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.434']
Arrow polynomial of the knot is: -2K1K2 - 2K1K3 + K1 - 2K22 + K2 + K3 + 2K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.373', '6.434', '6.878', '6.886', '6.952', '6.1160']
Outer characteristic polynomial of the knot is: t^7+98t^5+351t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.434']
2-strand cable arrow polynomial of the knot is: -400K14 + 384K1*3K2K3 + 32K1*3K3K4 - 832K1*3K3 - 544K12K2*2 + 256K1*2K2K32 - 160K1*2K2K4 + 3008K1*2K2 - 2128K12K3*2 - 128K1*2K3K5 - 96K1*2K4*2 - 4144K1*2 + 64K1K23K3 - 736K1K2*2K3 + 384K1K2K33 + 64K1K2K3K42 - 256K1K2K3K4 - 96K1K2K3K6 + 5888K1K2K3 - 32K1K3*2K5 + 2632K1K3K4 + 288K1K4K5 + 16K1K5K6 + 8K1K6K7 - 32K24 - 1520K2*2K3*2 - 40K2*2K4*2 + 400K2*2K4 - 8K22K6*2 - 2954K2*2 - 128K2K32K4 - 64K2K3K4K5 + 1224K2K3K5 + 48K2K4K6 + 16K2K6K8 - 320K3*4 - 144K3*2K4*2 + 208K3*2K6 - 2512K32 + 120K3K4K7 - 8K44 + 16K4*2K8 - 842K42 - 316K5*2 - 46K6*2 - 20K7*2 - 12K8**2 + 3460
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.434']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16990', 'vk6.17233', 'vk6.20208', 'vk6.21492', 'vk6.23396', 'vk6.23705', 'vk6.27396', 'vk6.29018', 'vk6.35457', 'vk6.35899', 'vk6.38815', 'vk6.40996', 'vk6.42893', 'vk6.43194', 'vk6.45570', 'vk6.47347', 'vk6.55159', 'vk6.55406', 'vk6.57050', 'vk6.58157', 'vk6.59539', 'vk6.59882', 'vk6.61556', 'vk6.62732', 'vk6.64969', 'vk6.65176', 'vk6.66669', 'vk6.67504', 'vk6.68261', 'vk6.68417', 'vk6.69320', 'vk6.70076']
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,-3,0,1,2,3,0,1,1,3,2,2,1,4,3,1,2,1,-1,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.434']

Arrow polynomial of the knot is: -2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + K2 + K3 + 2*K4 + 2

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.373', '6.434', '6.878', '6.886', '6.952', '6.1160']

Outer characteristic polynomial of the knot is: t^7+98t^5+351t^3+9t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.434']

2-strand cable arrow polynomial of the knot is: -400*K1**4 + 384*K1**3*K2*K3 + 32*K1**3*K3*K4 - 832*K1**3*K3 - 544*K1**2*K2**2 + 256*K1**2*K2*K3**2 - 160*K1**2*K2*K4 + 3008*K1**2*K2 - 2128*K1**2*K3**2 - 128*K1**2*K3*K5 - 96*K1**2*K4**2 - 4144*K1**2 + 64*K1*K2**3*K3 - 736*K1*K2**2*K3 + 384*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 256*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 5888*K1*K2*K3 - 32*K1*K3**2*K5 + 2632*K1*K3*K4 + 288*K1*K4*K5 + 16*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**4 - 1520*K2**2*K3**2 - 40*K2**2*K4**2 + 400*K2**2*K4 - 8*K2**2*K6**2 - 2954*K2**2 - 128*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1224*K2*K3*K5 + 48*K2*K4*K6 + 16*K2*K6*K8 - 320*K3**4 - 144*K3**2*K4**2 + 208*K3**2*K6 - 2512*K3**2 + 120*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 842*K4**2 - 316*K5**2 - 46*K6**2 - 20*K7**2 - 12*K8**2 + 3460

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.434']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16990', 'vk6.17233', 'vk6.20208', 'vk6.21492', 'vk6.23396', 'vk6.23705', 'vk6.27396', 'vk6.29018', 'vk6.35457', 'vk6.35899', 'vk6.38815', 'vk6.40996', 'vk6.42893', 'vk6.43194', 'vk6.45570', 'vk6.47347', 'vk6.55159', 'vk6.55406', 'vk6.57050', 'vk6.58157', 'vk6.59539', 'vk6.59882', 'vk6.61556', 'vk6.62732', 'vk6.64969', 'vk6.65176', 'vk6.66669', 'vk6.67504', 'vk6.68261', 'vk6.68417', 'vk6.69320', 'vk6.70076']

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	O1O2O3O4O5U6U2O6U1U3U5U4
R3 orbit	{'O1O2O3O4O5U6U2O6U1U3U5U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U2U1U3U5O6U4U6
Gauss code of K*	O1O2O3O4U1U5U2U4U3O6O5U6
Gauss code of -K*	O1O2O3O4U5O6O5U2U1U3U6U4
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 -2 0 3 3 -1],[ 3 0 1 2 4 3 2],[ 2 -1 0 0 2 1 2],[ 0 -2 0 0 2 1 0],[-3 -4 -2 -2 0 0 -3],[-3 -3 -1 -1 0 0 -3],[ 1 -2 -2 0 3 3 0]]
Primitive based matrix	[[ 0 3 3 0 -1 -2 -3],[-3 0 0 -1 -3 -1 -3],[-3 0 0 -2 -3 -2 -4],[ 0 1 2 0 0 0 -2],[ 1 3 3 0 0 -2 -2],[ 2 1 2 0 2 0 -1],[ 3 3 4 2 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-3,0,1,2,3,0,1,3,1,3,2,3,2,4,0,0,2,2,2,1]
Phi over symmetry	[-3,-3,0,1,2,3,0,1,1,3,2,2,1,4,3,1,2,1,-1,0,0]
Phi of -K	[-3,-2,-1,0,3,3,0,0,1,2,3,-1,2,3,4,1,1,1,1,2,0]
Phi of K*	[-3,-3,0,1,2,3,0,1,1,3,2,2,1,4,3,1,2,1,-1,0,0]
Phi of -K*	[-3,-2,-1,0,3,3,1,2,2,3,4,2,0,1,2,0,3,3,1,2,0]
Symmetry type of based matrix	c
u-polynomial	-t^3+t^2+t
Normalized Jones-Krushkal polynomial	5z^2+22z+25
Enhanced Jones-Krushkal polynomial	-2w^4z^2+7w^3z^2-2w^3z+24w^2z+25w
Inner characteristic polynomial	t^6+66t^4+194t^2+1
Outer characteristic polynomial	t^7+98t^5+351t^3+9t
Flat arrow polynomial	-2K1K2 - 2K1K3 + K1 - 2K22 + K2 + K3 + 2K4 + 2
2-strand cable arrow polynomial	-400K14 + 384K1*3K2K3 + 32K1*3K3K4 - 832K1*3K3 - 544K12K2*2 + 256K1*2K2K32 - 160K1*2K2K4 + 3008K1*2K2 - 2128K12K3*2 - 128K1*2K3K5 - 96K1*2K4*2 - 4144K1*2 + 64K1K23K3 - 736K1K2*2K3 + 384K1K2K33 + 64K1K2K3K42 - 256K1K2K3K4 - 96K1K2K3K6 + 5888K1K2K3 - 32K1K3*2K5 + 2632K1K3K4 + 288K1K4K5 + 16K1K5K6 + 8K1K6K7 - 32K24 - 1520K2*2K3*2 - 40K2*2K4*2 + 400K2*2K4 - 8K22K6*2 - 2954K2*2 - 128K2K32K4 - 64K2K3K4K5 + 1224K2K3K5 + 48K2K4K6 + 16K2K6K8 - 320K3*4 - 144K3*2K4*2 + 208K3*2K6 - 2512K32 + 120K3K4K7 - 8K44 + 16K4*2K8 - 842K42 - 316K5*2 - 46K6*2 - 20K7*2 - 12K8**2 + 3460
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}], [{2, 6}, {5}, {4}, {3}, {1}], [{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}, {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}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]]
If K is slice	False

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