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Flat knot 6.1057

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,2,2,2,1,0,1,1,0,0,0,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1057']
Arrow polynomial of the knot is: 4K13 - 2K1*2 - 6K1K2 + K2 + 2K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.362', '6.624', '6.789', '6.859', '6.882', '6.975', '6.989', '6.1048', '6.1057', '6.1158']
Outer characteristic polynomial of the knot is: t^7+36t^5+68t^3+19t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1057']
2-strand cable arrow polynomial of the knot is: -128K14K2*2 + 32K1*4K2 - 144K14 + 288K1*3K2K3 - 32K1*3K3 - 1152K12K2*4 + 1728K1*2K2*3 - 64K1*2K2*2K3*2 + 32K1*2K2*2K4 - 6128K12K2*2 - 288K1*2K2K4 + 4280K1*2K2 - 224K12K3*2 - 3048K1*2 + 2048K1K23K3 + 32K1K2*2K3K4 - 928K1K22K3 - 160K1K2*2K5 + 64K1K2K33 - 224K1K2K3K4 - 32K1K2K3K6 + 5872K1K2K3 + 448K1K3K4 - 32K2*6 + 64K2*4K4 - 1576K24 - 1008K2*2K3*2 - 56K2*2K4*2 + 872K2*2K4 - 1484K22 + 352K2K3K5 + 24K2K4K6 - 16K3*4 + 8K3*2K6 - 1596K32 - 174K4*2 - 12K5*2 - 4K6**2 + 2364
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1057']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71570', 'vk6.71681', 'vk6.72097', 'vk6.72307', 'vk6.74035', 'vk6.74597', 'vk6.76081', 'vk6.76793', 'vk6.77188', 'vk6.77287', 'vk6.77486', 'vk6.77648', 'vk6.79021', 'vk6.79599', 'vk6.80559', 'vk6.81010', 'vk6.81098', 'vk6.81147', 'vk6.81166', 'vk6.81206', 'vk6.81310', 'vk6.81456', 'vk6.82266', 'vk6.83500', 'vk6.83832', 'vk6.83976', 'vk6.85397', 'vk6.86323', 'vk6.87113', 'vk6.88011', 'vk6.88332', 'vk6.88956']
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,-1,0,1,1,2,1,1,2,2,2,1,0,1,1,0,0,0,1,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.1057']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.362', '6.624', '6.789', '6.859', '6.882', '6.975', '6.989', '6.1048', '6.1057', '6.1158']

Outer characteristic polynomial of the knot is: t^7+36t^5+68t^3+19t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1057']

2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 32*K1**4*K2 - 144*K1**4 + 288*K1**3*K2*K3 - 32*K1**3*K3 - 1152*K1**2*K2**4 + 1728*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 + 32*K1**2*K2**2*K4 - 6128*K1**2*K2**2 - 288*K1**2*K2*K4 + 4280*K1**2*K2 - 224*K1**2*K3**2 - 3048*K1**2 + 2048*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 928*K1*K2**2*K3 - 160*K1*K2**2*K5 + 64*K1*K2*K3**3 - 224*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5872*K1*K2*K3 + 448*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 1576*K2**4 - 1008*K2**2*K3**2 - 56*K2**2*K4**2 + 872*K2**2*K4 - 1484*K2**2 + 352*K2*K3*K5 + 24*K2*K4*K6 - 16*K3**4 + 8*K3**2*K6 - 1596*K3**2 - 174*K4**2 - 12*K5**2 - 4*K6**2 + 2364

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1057']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71570', 'vk6.71681', 'vk6.72097', 'vk6.72307', 'vk6.74035', 'vk6.74597', 'vk6.76081', 'vk6.76793', 'vk6.77188', 'vk6.77287', 'vk6.77486', 'vk6.77648', 'vk6.79021', 'vk6.79599', 'vk6.80559', 'vk6.81010', 'vk6.81098', 'vk6.81147', 'vk6.81166', 'vk6.81206', 'vk6.81310', 'vk6.81456', 'vk6.82266', 'vk6.83500', 'vk6.83832', 'vk6.83976', 'vk6.85397', 'vk6.86323', 'vk6.87113', 'vk6.88011', 'vk6.88332', 'vk6.88956']

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	O1O2O3O4U5U1O6U2O5U6U3U4
R3 orbit	{'O1O2O3O4U5U1O6U2O5U6U3U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U1U2U5O6U3O5U4U6
Gauss code of K*	O1O2O3U4U5U2U3O6O4U1O5U6
Gauss code of -K*	O1O2O3U4O5U3O6O4U1U2U5U6
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 3 -1 0],[ 2 0 1 1 2 1 0],[ 1 -1 0 1 2 1 0],[-1 -1 -1 0 1 0 -1],[-3 -2 -2 -1 0 -2 -1],[ 1 -1 -1 0 2 0 0],[ 0 0 0 1 1 0 0]]
Primitive based matrix	[[ 0 3 1 0 -1 -1 -2],[-3 0 -1 -1 -2 -2 -2],[-1 1 0 -1 0 -1 -1],[ 0 1 1 0 0 0 0],[ 1 2 0 0 0 -1 -1],[ 1 2 1 0 1 0 -1],[ 2 2 1 0 1 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,0,1,1,2,1,1,2,2,2,1,0,1,1,0,0,0,1,1,1]
Phi over symmetry	[-3,-1,0,1,1,2,1,1,2,2,2,1,0,1,1,0,0,0,1,1,1]
Phi of -K	[-2,-1,-1,0,1,3,0,0,2,2,3,-1,1,1,2,1,2,2,0,2,1]
Phi of K*	[-3,-1,0,1,1,2,1,2,2,2,3,0,1,2,2,1,1,2,1,0,0]
Phi of -K*	[-2,-1,-1,0,1,3,1,1,0,1,2,-1,0,0,2,0,1,2,1,1,1]
Symmetry type of based matrix	c
u-polynomial	-t^3+t^2+t
Normalized Jones-Krushkal polynomial	4z^2+17z+19
Enhanced Jones-Krushkal polynomial	-2w^4z^2+6w^3z^2-8w^3z+25w^2z+19w
Inner characteristic polynomial	t^6+20t^4+19t^2+4
Outer characteristic polynomial	t^7+36t^5+68t^3+19t
Flat arrow polynomial	4K13 - 2K1*2 - 6K1K2 + K2 + 2K3 + 2
2-strand cable arrow polynomial	-128K14K2*2 + 32K1*4K2 - 144K14 + 288K1*3K2K3 - 32K1*3K3 - 1152K12K2*4 + 1728K1*2K2*3 - 64K1*2K2*2K3*2 + 32K1*2K2*2K4 - 6128K12K2*2 - 288K1*2K2K4 + 4280K1*2K2 - 224K12K3*2 - 3048K1*2 + 2048K1K23K3 + 32K1K2*2K3K4 - 928K1K22K3 - 160K1K2*2K5 + 64K1K2K33 - 224K1K2K3K4 - 32K1K2K3K6 + 5872K1K2K3 + 448K1K3K4 - 32K2*6 + 64K2*4K4 - 1576K24 - 1008K2*2K3*2 - 56K2*2K4*2 + 872K2*2K4 - 1484K22 + 352K2K3K5 + 24K2K4K6 - 16K3*4 + 8K3*2K6 - 1596K32 - 174K4*2 - 12K5*2 - 4K6**2 + 2364
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}], [{4, 6}, {5}, {3}, {2}, {1}], [{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}, {2, 5}, {4}, {3}, {1}], [{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}, {4}, {3}, {1, 2}]]
If K is slice	False

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