Invariant Table Check a Knot Higher Crossing Crossref Virtual Knots Please cite FlatKnotInfo

Flat knot 6.1492

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,0,1,2,0,1,0,1,0,1,1,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1492', '6.1698', '7.39287', '7.42910']
Arrow polynomial of the knot is: 4K13 - 6K1*2 - 4K1K2 - K1 + 3K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.361', '6.460', '6.555', '6.651', '6.753', '6.782', '6.1029', '6.1197', '6.1200', '6.1232', '6.1236', '6.1278', '6.1281', '6.1343', '6.1380', '6.1385', '6.1389', '6.1484', '6.1492', '6.1493', '6.1527', '6.1533', '6.1550', '6.1553', '6.1557', '6.1576', '6.1578', '6.1582', '6.1586', '6.1674', '6.1698', '6.1754', '6.1759', '6.1775', '6.1791', '6.1798', '6.1800', '6.1805', '6.1822', '6.1826', '6.1839', '6.1844', '6.1845']
Outer characteristic polynomial of the knot is: t^7+20t^5+32t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1492', '6.1698', '7.39287']
2-strand cable arrow polynomial of the knot is: -512K16 - 512K1*4K2*2 + 2912K1*4K2 - 6816K14 + 800K1*3K2K3 - 1600K1*3K3 - 192K12K2*4 + 1888K1*2K2*3 + 192K1*2K2*2K4 - 6960K12K2*2 - 1344K1*2K2K4 + 9040K1*2K2 - 320K12K3*2 - 48K1*2K4*2 - 1068K1*2 + 416K1K23K3 - 576K1K2*2K3 - 256K1K2*2K5 - 64K1K2K3K4 + 5240K1K2K3 + 520K1K3K4 + 64K1K4K5 - 32K2*6 + 64K2*4K4 - 1048K24 - 32K2*3K6 - 192K22K3*2 - 16K2*2K4*2 + 888K2*2K4 - 1926K22 + 136K2K3K5 + 16K2K4K6 - 724K3*2 - 186K4*2 - 24K5*2 - 2K6**2 + 2272
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1492']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.15', 'vk6.24', 'vk6.33', 'vk6.147', 'vk6.154', 'vk6.162', 'vk6.171', 'vk6.1200', 'vk6.1209', 'vk6.1298', 'vk6.1308', 'vk6.1317', 'vk6.2354', 'vk6.2390', 'vk6.2397', 'vk6.2961', 'vk6.3528', 'vk6.3553', 'vk6.6904', 'vk6.6929', 'vk6.6937', 'vk6.6960', 'vk6.15379', 'vk6.15386', 'vk6.15498', 'vk6.33437', 'vk6.33448', 'vk6.33492', 'vk6.33505', 'vk6.33602', 'vk6.49933', 'vk6.53744']
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: [-2,-1,0,1,1,1,0,1,0,1,2,0,1,0,1,0,1,1,0,-1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.1492', '6.1698', '7.39287', '7.42910']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.361', '6.460', '6.555', '6.651', '6.753', '6.782', '6.1029', '6.1197', '6.1200', '6.1232', '6.1236', '6.1278', '6.1281', '6.1343', '6.1380', '6.1385', '6.1389', '6.1484', '6.1492', '6.1493', '6.1527', '6.1533', '6.1550', '6.1553', '6.1557', '6.1576', '6.1578', '6.1582', '6.1586', '6.1674', '6.1698', '6.1754', '6.1759', '6.1775', '6.1791', '6.1798', '6.1800', '6.1805', '6.1822', '6.1826', '6.1839', '6.1844', '6.1845']

Outer characteristic polynomial of the knot is: t^7+20t^5+32t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1492', '6.1698', '7.39287']

2-strand cable arrow polynomial of the knot is: -512*K1**6 - 512*K1**4*K2**2 + 2912*K1**4*K2 - 6816*K1**4 + 800*K1**3*K2*K3 - 1600*K1**3*K3 - 192*K1**2*K2**4 + 1888*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 6960*K1**2*K2**2 - 1344*K1**2*K2*K4 + 9040*K1**2*K2 - 320*K1**2*K3**2 - 48*K1**2*K4**2 - 1068*K1**2 + 416*K1*K2**3*K3 - 576*K1*K2**2*K3 - 256*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 5240*K1*K2*K3 + 520*K1*K3*K4 + 64*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1048*K2**4 - 32*K2**3*K6 - 192*K2**2*K3**2 - 16*K2**2*K4**2 + 888*K2**2*K4 - 1926*K2**2 + 136*K2*K3*K5 + 16*K2*K4*K6 - 724*K3**2 - 186*K4**2 - 24*K5**2 - 2*K6**2 + 2272

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1492']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.15', 'vk6.24', 'vk6.33', 'vk6.147', 'vk6.154', 'vk6.162', 'vk6.171', 'vk6.1200', 'vk6.1209', 'vk6.1298', 'vk6.1308', 'vk6.1317', 'vk6.2354', 'vk6.2390', 'vk6.2397', 'vk6.2961', 'vk6.3528', 'vk6.3553', 'vk6.6904', 'vk6.6929', 'vk6.6937', 'vk6.6960', 'vk6.15379', 'vk6.15386', 'vk6.15498', 'vk6.33437', 'vk6.33448', 'vk6.33492', 'vk6.33505', 'vk6.33602', 'vk6.49933', 'vk6.53744']

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	O1O2O3U1O4U3U4O5O6U5U6U2
R3 orbit	{'O1O2O3U1O4U3U4O5O6U5U6U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3U2U4U5O4O5U6U1O6U3
Gauss code of K*	O1O2O3U4U3U5O4U6O5O6U1U2
Gauss code of -K*	O1O2O3U2U3O4O5U4O6U5U1U6
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 0 1 -1 1],[ 2 0 2 1 1 0 0],[-1 -2 0 -1 1 -1 1],[ 0 -1 1 0 1 0 0],[-1 -1 -1 -1 0 0 0],[ 1 0 1 0 0 0 1],[-1 0 -1 0 0 -1 0]]
Primitive based matrix	[[ 0 1 1 1 0 -1 -2],[-1 0 1 1 -1 -1 -2],[-1 -1 0 0 0 -1 0],[-1 -1 0 0 -1 0 -1],[ 0 1 0 1 0 0 -1],[ 1 1 1 0 0 0 0],[ 2 2 0 1 1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,0,1,2,-1,-1,1,1,2,0,0,1,0,1,0,1,0,1,0]
Phi over symmetry	[-2,-1,0,1,1,1,0,1,0,1,2,0,1,0,1,0,1,1,0,-1,-1]
Phi of -K	[-2,-1,0,1,1,1,1,1,1,2,3,1,1,2,1,0,0,1,-1,-1,0]
Phi of K*	[-1,-1,-1,0,1,2,-1,0,0,2,2,1,0,1,1,1,1,3,1,1,1]
Phi of -K*	[-2,-1,0,1,1,1,0,1,0,1,2,0,1,0,1,0,1,1,0,-1,-1]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	3z^2+22z+33
Enhanced Jones-Krushkal polynomial	3w^3z^2+22w^2z+33w
Inner characteristic polynomial	t^6+12t^4+15t^2
Outer characteristic polynomial	t^7+20t^5+32t^3+5t
Flat arrow polynomial	4K13 - 6K1*2 - 4K1K2 - K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	-512K16 - 512K1*4K2*2 + 2912K1*4K2 - 6816K14 + 800K1*3K2K3 - 1600K1*3K3 - 192K12K2*4 + 1888K1*2K2*3 + 192K1*2K2*2K4 - 6960K12K2*2 - 1344K1*2K2K4 + 9040K1*2K2 - 320K12K3*2 - 48K1*2K4*2 - 1068K1*2 + 416K1K23K3 - 576K1K2*2K3 - 256K1K2*2K5 - 64K1K2K3K4 + 5240K1K2K3 + 520K1K3K4 + 64K1K4K5 - 32K2*6 + 64K2*4K4 - 1048K24 - 32K2*3K6 - 192K22K3*2 - 16K2*2K4*2 + 888K2*2K4 - 1926K22 + 136K2K3K5 + 16K2K4K6 - 724K3*2 - 186K4*2 - 24K5*2 - 2K6**2 + 2272
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}]]
If K is slice	False

Contact