L11a455

L11a454

L11a456

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a455 at Knotilus!

Link Presentations

[edit Notes on L11a455's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X20,12,21,11 X18,8,19,7 X16,10,17,9 X8,18,9,17 X22,16,13,15 X12,20,5,19 X10,22,11,21 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -6, 5, -9, 3, -8}, {11, -2, 7, -5, 6, -4, 8, -3, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{3}w-uv^{3}+2uv^{2}w^{2}-4uv^{2}w+2uv^{2}+uvw^{3}-4uvw^{2}+3uvw-uw^{3}+2uw^{2}-2v^{3}w+v^{3}-3v^{2}w^{2}+4v^{2}w-v^{2}-2vw^{3}+4vw^{2}-2vw+w^{3}-w^{2}}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{7}+3q^{6}-5q^{5}+10q^{4}+q^{-4}-11q^{3}-2q^{-3}+13q^{2}+5q^{-2}-13q-8q^{-1}+12$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}-2z^{2}a^{2}+z^{4}-2z^{2}+z^{-2}+2z^{4}a^{-2}-2a^{-2}z^{-2}-3a^{-2}+z^{4}a^{-4}+a^{-4}z^{-2}+2a^{-4}-z^{2}a^{-6}$ (db)
Kauffman polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +6 z^9 a^{-3} +3 z^9 a^{-5} +7 z^8 a^{-2} +5 z^8 a^{-4} +3 z^8 a^{-6} +5 z^8+4 a z^7-14 z^7 a^{-3} -9 z^7 a^{-5} +z^7 a^{-7} +3 a^2 z^6-30 z^6 a^{-2} -29 z^6 a^{-4} -13 z^6 a^{-6} -11 z^6+2 a^3 z^5-4 a z^5-16 z^5 a^{-1} -2 z^5 a^{-3} +4 z^5 a^{-5} -4 z^5 a^{-7} +a^4 z^4-2 a^2 z^4+39 z^4 a^{-2} +39 z^4 a^{-4} +18 z^4 a^{-6} +15 z^4-2 a^3 z^3+2 a z^3+23 z^3 a^{-1} +17 z^3 a^{-3} +2 z^3 a^{-5} +4 z^3 a^{-7} -2 a^4 z^2-29 z^2 a^{-2} -26 z^2 a^{-4} -9 z^2 a^{-6} -10 z^2-11 z a^{-1} -11 z a^{-3} +a^4+13 a^{-2} +8 a^{-4} +5+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-2} - z^{-2} } (db)

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums $\chi$ (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over $r$).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 15                       1 -1 13                     2   2 11                   3 1   -2 9                 7 2     5 7               6 5       -1 5             7 5         2 3           6 6           0 1         6 7             -1 -1       3 7               4 -3     2 5                 -3 -5     3                   3 -7 1 2                     -1 -9 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-1} $i=1$ $r=-4$ ${\mathbb {Z} }$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} $r=-3$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}} ${\mathbb {Z} }^{3}$ $r=0$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}} ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{6}} $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{7}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3} ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{7}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=6} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{2}} ${\mathbb {Z} }^{2}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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