L11a10

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L11a11

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a10 at Knotilus!

Link Presentations

[edit Notes on L11a10's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X12,6,13,5 X8493 X16,10,17,9 X22,14,5,13 X10,16,11,15 X14,22,15,21 X20,12,21,11 X2,18,3,17
Gauss code	{1, -11, 5, -3}, {4, -1, 2, -5, 6, -8, 10, -4, 7, -9, 8, -6, 11, -2, 3, -10, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, 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 w} , ...)	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 \frac{(t(1)-1) (t(2)-1) \left(4 t(2)^2-7 t(2)+4\right)}{\sqrt{t(1)} t(2)^{3/2}}} (db)
Jones polynomial	$-q^{17/2}+4q^{15/2}-7q^{13/2}+12q^{11/2}-16q^{9/2}+18q^{7/2}-20q^{5/2}+16q^{3/2}-13{\sqrt {q}}+{\frac {8}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {1}{q^{5/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT 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^3 a^{-7} + a^{-7} z^{-1} +z^5 a^{-5} -2 z a^{-5} -2 a^{-5} z^{-1} +2 z^5 a^{-3} +3 z^3 a^{-3} +2 z a^{-3} +z^5 a^{-1} -a z^3+ a^{-1} z^{-1} } (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^7 a^{-9} -3 z^5 a^{-9} +2 z^3 a^{-9} +4 z^8 a^{-8} -15 z^6 a^{-8} +16 z^4 a^{-8} -3 z^2 a^{-8} -2 a^{-8} +5 z^9 a^{-7} -16 z^7 a^{-7} +13 z^5 a^{-7} -2 z^3 a^{-7} + a^{-7} z^{-1} +2 z^{10} a^{-6} +5 z^8 a^{-6} -32 z^6 a^{-6} +32 z^4 a^{-6} -3 z^2 a^{-6} -5 a^{-6} +11 z^9 a^{-5} -27 z^7 a^{-5} +11 z^5 a^{-5} +5 z^3 a^{-5} -3 z a^{-5} +2 a^{-5} z^{-1} +2 z^{10} a^{-4} +11 z^8 a^{-4} -37 z^6 a^{-4} +25 z^4 a^{-4} -z^2 a^{-4} -3 a^{-4} +6 z^9 a^{-3} +z^7 a^{-3} -24 z^5 a^{-3} +19 z^3 a^{-3} -4 z a^{-3} +10 z^8 a^{-2} -12 z^6 a^{-2} +a^2 z^4+z^4 a^{-2} + a^{-2} +11 z^7 a^{-1} +4 a z^5-15 z^5 a^{-1} -2 a z^3+8 z^3 a^{-1} -z a^{-1} - a^{-1} z^{-1} +8 z^6-7 z^4+z^2} (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 18                       1 1 16                     3   -3 14                   4 1   3 12                 8 3     -5 10               8 4       4 8             10 8         -2 6           10 8           2 4         6 10             4 2       7 10               -3 0     3 8                 5 -2   1 5                   -4 -4   3                     3 -6 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ 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 r=-1} ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ 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=0} ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{7}$ $r=1$ 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}^{10}\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}} 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=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 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}} 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}^{10}} $r=3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ 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} ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ 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}^{8}} 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}^{3}\oplus{\mathbb Z}_2^{4}} ${\mathbb {Z} }^{4}$ $r=7$ 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}\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=8} ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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