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Flat knot 6.194

Min(phi) over symmetries of the knot is: [-4,-3,0,1,3,3,0,3,1,4,5,1,0,2,3,0,1,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.194']
Arrow polynomial of the knot is: -2K12 - 2K1K2 - 2K1K3 + K1 + 2K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.62', '6.176', '6.181', '6.194', '6.228', '6.267', '6.268', '6.449']
Outer characteristic polynomial of the knot is: t^7+114t^5+207t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.194']
2-strand cable arrow polynomial of the knot is: -144K14 + 384K1*3K2K3 - 704K1*3K3 - 256K12K2*2K3*2 - 896K1*2K2*2 + 384K1*2K2K32 + 32K1*2K2K3K5 - 192K12K2K4 + 3080K1*2K2 - 1456K12K3*2 - 32K1*2K3K5 - 3872K1*2 + 288K1K23K3 + 224K1K2*2K3K4 - 1248K1K22K3 - 32K1K2*2K5 + 128K1K2K33 - 544K1K2K3K4 - 32K1K2K3K6 + 5896K1K2K3 + 2104K1K3K4 + 64K1K4K5 + 8K1K5K6 + 16K1K6K7 - 72K24 - 1792K2*2K3*2 - 72K2*2K4*2 + 728K2*2K4 - 8K22K6*2 - 2986K2*2 - 96K2K32K4 + 1256K2K3K5 + 56K2K4K6 + 8K2K6K8 - 128K3*4 - 64K3*2K4*2 + 112K3*2K6 - 2368K32 + 64K3K4K7 - 716K42 - 160K5*2 - 46K6*2 - 24K7*2 - 2K8**2 + 3108
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.194']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16930', 'vk6.17173', 'vk6.20222', 'vk6.21517', 'vk6.23326', 'vk6.23621', 'vk6.27422', 'vk6.29033', 'vk6.35364', 'vk6.35787', 'vk6.38832', 'vk6.41025', 'vk6.42841', 'vk6.43121', 'vk6.45597', 'vk6.47357', 'vk6.55084', 'vk6.55336', 'vk6.57061', 'vk6.58186', 'vk6.59479', 'vk6.59770', 'vk6.61582', 'vk6.62755', 'vk6.64926', 'vk6.65134', 'vk6.66680', 'vk6.67519', 'vk6.68221', 'vk6.68364', 'vk6.69331', 'vk6.70083']
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,-3,0,1,3,3,0,3,1,4,5,1,0,2,3,0,1,2,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.194']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.62', '6.176', '6.181', '6.194', '6.228', '6.267', '6.268', '6.449']

Outer characteristic polynomial of the knot is: t^7+114t^5+207t^3+9t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.194']

2-strand cable arrow polynomial of the knot is: -144*K1**4 + 384*K1**3*K2*K3 - 704*K1**3*K3 - 256*K1**2*K2**2*K3**2 - 896*K1**2*K2**2 + 384*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 192*K1**2*K2*K4 + 3080*K1**2*K2 - 1456*K1**2*K3**2 - 32*K1**2*K3*K5 - 3872*K1**2 + 288*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1248*K1*K2**2*K3 - 32*K1*K2**2*K5 + 128*K1*K2*K3**3 - 544*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5896*K1*K2*K3 + 2104*K1*K3*K4 + 64*K1*K4*K5 + 8*K1*K5*K6 + 16*K1*K6*K7 - 72*K2**4 - 1792*K2**2*K3**2 - 72*K2**2*K4**2 + 728*K2**2*K4 - 8*K2**2*K6**2 - 2986*K2**2 - 96*K2*K3**2*K4 + 1256*K2*K3*K5 + 56*K2*K4*K6 + 8*K2*K6*K8 - 128*K3**4 - 64*K3**2*K4**2 + 112*K3**2*K6 - 2368*K3**2 + 64*K3*K4*K7 - 716*K4**2 - 160*K5**2 - 46*K6**2 - 24*K7**2 - 2*K8**2 + 3108

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.194']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16930', 'vk6.17173', 'vk6.20222', 'vk6.21517', 'vk6.23326', 'vk6.23621', 'vk6.27422', 'vk6.29033', 'vk6.35364', 'vk6.35787', 'vk6.38832', 'vk6.41025', 'vk6.42841', 'vk6.43121', 'vk6.45597', 'vk6.47357', 'vk6.55084', 'vk6.55336', 'vk6.57061', 'vk6.58186', 'vk6.59479', 'vk6.59770', 'vk6.61582', 'vk6.62755', 'vk6.64926', 'vk6.65134', 'vk6.66680', 'vk6.67519', 'vk6.68221', 'vk6.68364', 'vk6.69331', 'vk6.70083']

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	O1O2O3O4O5U2O6U1U6U3U5U4
R3 orbit	{'O1O2O3O4O5U2O6U1U6U3U5U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U2U1U3U6U5O6U4
Gauss code of K*	O1O2O3O4O5U1U6U3U5U4O6U2
Gauss code of -K*	O1O2O3O4O5U4O6U2U1U3U6U5
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 -3 0 3 3 1],[ 4 0 0 3 5 4 1],[ 3 0 0 1 3 2 0],[ 0 -3 -1 0 2 1 0],[-3 -5 -3 -2 0 0 0],[-3 -4 -2 -1 0 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix	[[ 0 3 3 1 0 -3 -4],[-3 0 0 0 -1 -2 -4],[-3 0 0 0 -2 -3 -5],[-1 0 0 0 0 0 -1],[ 0 1 2 0 0 -1 -3],[ 3 2 3 0 1 0 0],[ 4 4 5 1 3 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-3,-1,0,3,4,0,0,1,2,4,0,2,3,5,0,0,1,1,3,0]
Phi over symmetry	[-4,-3,0,1,3,3,0,3,1,4,5,1,0,2,3,0,1,2,0,0,0]
Phi of -K	[-4,-3,0,1,3,3,1,1,4,2,3,2,4,3,4,1,1,2,2,2,0]
Phi of K*	[-3,-3,-1,0,3,4,0,2,1,3,2,2,2,4,3,1,4,4,2,1,1]
Phi of -K*	[-4,-3,0,1,3,3,0,3,1,4,5,1,0,2,3,0,1,2,0,0,0]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-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+70t^4+53t^2+1
Outer characteristic polynomial	t^7+114t^5+207t^3+9t
Flat arrow polynomial	-2K12 - 2K1K2 - 2K1K3 + K1 + 2K2 + K3 + K4 + 2
2-strand cable arrow polynomial	-144K14 + 384K1*3K2K3 - 704K1*3K3 - 256K12K2*2K3*2 - 896K1*2K2*2 + 384K1*2K2K32 + 32K1*2K2K3K5 - 192K12K2K4 + 3080K1*2K2 - 1456K12K3*2 - 32K1*2K3K5 - 3872K1*2 + 288K1K23K3 + 224K1K2*2K3K4 - 1248K1K22K3 - 32K1K2*2K5 + 128K1K2K33 - 544K1K2K3K4 - 32K1K2K3K6 + 5896K1K2K3 + 2104K1K3K4 + 64K1K4K5 + 8K1K5K6 + 16K1K6K7 - 72K24 - 1792K2*2K3*2 - 72K2*2K4*2 + 728K2*2K4 - 8K22K6*2 - 2986K2*2 - 96K2K32K4 + 1256K2K3K5 + 56K2K4K6 + 8K2K6K8 - 128K3*4 - 64K3*2K4*2 + 112K3*2K6 - 2368K32 + 64K3K4K7 - 716K42 - 160K5*2 - 46K6*2 - 24K7*2 - 2K8**2 + 3108
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}], [{1, 6}, {5}, {4}, {3}, {2}], [{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}, {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}, {3, 5}, {4}, {2}, {1}], [{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}]]
If K is slice	False

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