L11a340

L11a339

L11a341

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a340 at Knotilus!

Link Presentations

[edit Notes on L11a340's Link Presentations]

Planar diagram presentation	X10,1,11,2 X14,4,15,3 X22,5,9,6 X6,9,7,10 X20,12,21,11 X18,14,19,13 X12,20,13,19 X16,8,17,7 X4,16,5,15 X8,18,1,17 X2,21,3,22
Gauss code	{1, -11, 2, -9, 3, -4, 8, -10}, {4, -1, 5, -7, 6, -2, 9, -8, 10, -6, 7, -5, 11, -3}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	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{(u-1) (v-1) \left(2 u^2 v^2-2 u v^2+7 u v-2 u+2\right)}{u^{3/2} v^{3/2}}} (db)
Jones polynomial	$-12q^{9/2}+16q^{7/2}-{\frac {1}{q^{7/2}}}-20q^{5/2}+{\frac {3}{q^{5/2}}}+19q^{3/2}-{\frac {8}{q^{3/2}}}+q^{15/2}-3q^{13/2}+7q^{11/2}-17{\sqrt {q}}+{\frac {13}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}-z^{7}a^{-3}+az^{5}-4z^{5}a^{-1}-4z^{5}a^{-3}+z^{5}a^{-5}+3az^{3}-7z^{3}a^{-1}-6z^{3}a^{-3}+3z^{3}a^{-5}+4az-5za^{-1}-2za^{-3}+3za^{-5}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}a^{-4}-5z^{9}a^{-1}-10z^{9}a^{-3}-5z^{9}a^{-5}-7z^{8}a^{-2}-5z^{8}a^{-4}-5z^{8}a^{-6}-7z^{8}-6az^{7}+z^{7}a^{-1}+21z^{7}a^{-3}+11z^{7}a^{-5}-3z^{7}a^{-7}-3a^{2}z^{6}+19z^{6}a^{-2}+20z^{6}a^{-4}+13z^{6}a^{-6}-z^{6}a^{-8}+10z^{6}-a^{3}z^{5}+10az^{5}+12z^{5}a^{-1}-17z^{5}a^{-3}-10z^{5}a^{-5}+8z^{5}a^{-7}+4a^{2}z^{4}-16z^{4}a^{-2}-20z^{4}a^{-4}-11z^{4}a^{-6}+3z^{4}a^{-8}-6z^{4}+2a^{3}z^{3}-8az^{3}-17z^{3}a^{-1}+6z^{3}a^{-3}+8z^{3}a^{-5}-5z^{3}a^{-7}-a^{2}z^{2}+4z^{2}a^{-2}+7z^{2}a^{-4}+5z^{2}a^{-6}-2z^{2}a^{-8}+3z^{2}-a^{3}z+5az+8za^{-1}-2za^{-3}-3za^{-5}+za^{-7}+1-az^{-1}-a^{-1}z^{-1}$ (db)

Khovanov Homology

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

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     2   2 12                   5 1   -4 10                 7 2     5 8               9 5       -4 6             11 7         4 4           8 9           1 2         9 11             -2 0       6 10               4 -2     2 7                 -5 -4   1 6                   5 -6   2                     -2 -8 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ 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 i=2} $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=5$ 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^{5}} ${\mathbb {Z} }^{5}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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