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Flat knot 5.1

Min(phi) over symmetries of the knot is: [-4,-2,0,3,3,1,2,3,4,1,2,3,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['5.1']
Arrow polynomial of the knot is: -4K12K2 + 4K12 + 2K1*K3 - K2
Flat knots (up to 7 crossings) with same arrow polynomial are :['5.1', '5.10', '7.328', '7.361', '7.789', '7.2224', '7.2289', '7.2579', '7.2621', '7.2984', '7.2999', '7.3003', '7.3307', '7.3342', '7.4820', '7.5074', '7.5342', '7.6103', '7.6227', '7.6233', '7.6332', '7.6341', '7.7236', '7.7266', '7.7269', '7.7303', '7.7837', '7.8068', '7.8129', '7.8142', '7.8194', '7.9334', '7.9364', '7.9377', '7.9465', '7.9474', '7.9486', '7.9662', '7.9899', '7.9930', '7.11901', '7.12302', '7.12348', '7.12492', '7.13054', '7.13065', '7.13105', '7.13161', '7.13190', '7.13192', '7.13246', '7.13401', '7.13405', '7.13815', '7.13863', '7.14615', '7.14740', '7.14744', '7.15091', '7.15105', '7.15129', '7.15133', '7.15189', '7.15955', '7.16089', '7.17594', '7.17655', '7.17680', '7.31413']
Outer characteristic polynomial of the knot is: t^6+87t^4+45t^2
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.1']
2-strand cable arrow polynomial of the knot is: -128K14 - 512K1*2K2*4 + 192K1*2K2*3 - 320K1*2K2*2 + 336K1*2K2 - 304K12 + 256K1K25K3 + 512K1K2*3K3 + 416K1K2K3 + 96K1K3K4 - 128K26 - 256K2*4K3*2 - 32K2*4K4*2 + 32K2*4K4 - 32K24 + 64K2*3K3K5 + 32K2*3K4K6 - 192K2*2K3*2 - 32K2*2K4*2 + 40K2*2K4 - 8K22K6*2 - 116K2*2 + 32K2K3K5 + 16K2K4K6 - 192K3*2 - 66K4*2 - 4K6**2 + 296
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.1']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.2190', 'vk5.2206', 'vk5.2207', 'vk5.2215', 'vk5.2217', 'vk5.2236', 'vk5.2238', 'vk5.2294', 'vk5.2298', 'vk5.2358', 'vk5.2421', 'vk5.2422']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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,-2,0,3,3,1,2,3,4,1,2,3,1,2,0]

Flat knots (up to 7 crossings) with same phi are :['5.1']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['5.1', '5.10', '7.328', '7.361', '7.789', '7.2224', '7.2289', '7.2579', '7.2621', '7.2984', '7.2999', '7.3003', '7.3307', '7.3342', '7.4820', '7.5074', '7.5342', '7.6103', '7.6227', '7.6233', '7.6332', '7.6341', '7.7236', '7.7266', '7.7269', '7.7303', '7.7837', '7.8068', '7.8129', '7.8142', '7.8194', '7.9334', '7.9364', '7.9377', '7.9465', '7.9474', '7.9486', '7.9662', '7.9899', '7.9930', '7.11901', '7.12302', '7.12348', '7.12492', '7.13054', '7.13065', '7.13105', '7.13161', '7.13190', '7.13192', '7.13246', '7.13401', '7.13405', '7.13815', '7.13863', '7.14615', '7.14740', '7.14744', '7.15091', '7.15105', '7.15129', '7.15133', '7.15189', '7.15955', '7.16089', '7.17594', '7.17655', '7.17680', '7.31413']

Outer characteristic polynomial of the knot is: t^6+87t^4+45t^2

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.1']

2-strand cable arrow polynomial of the knot is: -128*K1**4 - 512*K1**2*K2**4 + 192*K1**2*K2**3 - 320*K1**2*K2**2 + 336*K1**2*K2 - 304*K1**2 + 256*K1*K2**5*K3 + 512*K1*K2**3*K3 + 416*K1*K2*K3 + 96*K1*K3*K4 - 128*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 32*K2**4*K4 - 32*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 192*K2**2*K3**2 - 32*K2**2*K4**2 + 40*K2**2*K4 - 8*K2**2*K6**2 - 116*K2**2 + 32*K2*K3*K5 + 16*K2*K4*K6 - 192*K3**2 - 66*K4**2 - 4*K6**2 + 296

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.1']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.2190', 'vk5.2206', 'vk5.2207', 'vk5.2215', 'vk5.2217', 'vk5.2236', 'vk5.2238', 'vk5.2294', 'vk5.2298', 'vk5.2358', 'vk5.2421', 'vk5.2422']

The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is +.

The reverse -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	O1O2O3O4O5U1U2U3U5U4
R3 orbit	{'O1O2O3O4O5U1U2U3U5U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U2U1U3U4U5
Gauss code of K*	Same
Gauss code of -K*	O1O2O3O4O5U2U1U3U4U5
Diagrammatic symmetry type	+
Flat genus of the diagram	2
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -4 -2 0 3 3],[ 4 0 1 2 4 3],[ 2 -1 0 1 3 2],[ 0 -2 -1 0 2 1],[-3 -4 -3 -2 0 0],[-3 -3 -2 -1 0 0]]
Primitive based matrix	[[ 0 3 3 0 -2 -4],[-3 0 0 -1 -2 -3],[-3 0 0 -2 -3 -4],[ 0 1 2 0 -1 -2],[ 2 2 3 1 0 -1],[ 4 3 4 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-3,0,2,4,0,1,2,3,2,3,4,1,2,1]
Phi over symmetry	[-4,-2,0,3,3,1,2,3,4,1,2,3,1,2,0]
Phi of -K	[-4,-2,0,3,3,1,2,3,4,1,2,3,1,2,0]
Phi of K*	[-3,-3,0,2,4,0,1,2,3,2,3,4,1,2,1]
Phi of -K*	[-4,-2,0,3,3,1,2,3,4,1,2,3,1,2,0]
Symmetry type of based matrix	+
u-polynomial	t^4-2t^3+t^2
Normalized Jones-Krushkal polynomial	-3z-5
Enhanced Jones-Krushkal polynomial	-4w^4z+6w^3z-5w^2z-5w
Inner characteristic polynomial	t^5+49t^3+6t
Outer characteristic polynomial	t^6+87t^4+45t^2
Flat arrow polynomial	-4K12K2 + 4K12 + 2K1*K3 - K2
2-strand cable arrow polynomial	-128K14 - 512K1*2K2*4 + 192K1*2K2*3 - 320K1*2K2*2 + 336K1*2K2 - 304K12 + 256K1K25K3 + 512K1K2*3K3 + 416K1K2K3 + 96K1K3K4 - 128K26 - 256K2*4K3*2 - 32K2*4K4*2 + 32K2*4K4 - 32K24 + 64K2*3K3K5 + 32K2*3K4K6 - 192K2*2K3*2 - 32K2*2K4*2 + 40K2*2K4 - 8K22K6*2 - 116K2*2 + 32K2K3K5 + 16K2K4K6 - 192K3*2 - 66K4*2 - 4K6**2 + 296
Genus of based matrix	1
Fillings of based matrix	[[{1, 5}, {2, 4}, {3}], [{2, 5}, {1, 4}, {3}], [{4, 5}, {1, 3}, {2}], [{4, 5}, {2, 3}, {1}], [{4, 5}, {3}, {1, 2}], [{4, 5}, {3}, {2}, {1}]]
If K is slice	False

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