L11a158

L11a157

L11a159

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a158 at Knotilus!

Link Presentations

[edit Notes on L11a158's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X2738 X16,12,17,11 X14,5,15,6 X4,15,5,16 X20,18,21,17 X18,13,19,14 X12,19,13,20 X6,22,1,21
Gauss code	{1, -4, 2, -7, 6, -11}, {4, -1, 3, -2, 5, -10, 9, -6, 7, -5, 8, -9, 10, -8, 11, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in 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 u} , $v$, $w$, ...)	$-{\frac {u^{2}v^{4}-4u^{2}v^{3}+6u^{2}v^{2}-4u^{2}v+u^{2}-2uv^{4}+7uv^{3}-9uv^{2}+7uv-2u+v^{4}-4v^{3}+6v^{2}-4v+1}{uv^{2}}}$ (db)
Jones polynomial	$-4q^{9/2}+{\frac {3}{q^{9/2}}}+8q^{7/2}-{\frac {7}{q^{7/2}}}-13q^{5/2}+{\frac {11}{q^{5/2}}}+17q^{3/2}-{\frac {16}{q^{3/2}}}+q^{11/2}-{\frac {1}{q^{11/2}}}-19{\sqrt {q}}+{\frac {18}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+9az^{3}-7z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-6a^{3}z+8az-5za^{-1}+za^{-3}+a^{5}z^{-1}-2a^{3}z^{-1}+2az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+a^{5}z^{7}-4a^{5}z^{5}+4z^{5}a^{-5}+6a^{5}z^{3}-2z^{3}a^{-5}-4a^{5}z+a^{5}z^{-1}+3a^{4}z^{8}-11a^{4}z^{6}+8z^{6}a^{-4}+13a^{4}z^{4}-7z^{4}a^{-4}-5a^{4}z^{2}+2z^{2}a^{-4}+3a^{3}z^{9}-4a^{3}z^{7}+11z^{7}a^{-3}-13a^{3}z^{5}-14z^{5}a^{-3}+26a^{3}z^{3}+7z^{3}a^{-3}-14a^{3}z-2za^{-3}+2a^{3}z^{-1}+a^{2}z^{10}+10a^{2}z^{8}+10z^{8}a^{-2}-40a^{2}z^{6}-11z^{6}a^{-2}+40a^{2}z^{4}-z^{4}a^{-2}-12a^{2}z^{2}+2z^{2}a^{-2}+a^{2}+8az^{9}+5z^{9}a^{-1}-8az^{7}+8z^{7}a^{-1}-30az^{5}-39z^{5}a^{-1}+44az^{3}+33z^{3}a^{-1}-18az-10za^{-1}+2az^{-1}+a^{-1}z^{-1}+z^{10}+17z^{8}-48z^{6}+34z^{4}-7z^{2}$ (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 $j$, alternation over $r$).

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     3   3 8                   5 1   -4 6                 8 3     5 4               9 5       -4 2             10 8         2 0           9 10           1 -2         7 9             -2 -4       5 10               5 -6     2 6                 -4 -8   1 5                   4 -10   2                     -2 -12 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} $r=-6$ ${\mathbb {Z} }$ $r=-5$ 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} 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}} $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ 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}^{2}} $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ 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^{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}^{7}} $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ 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}^{5}\oplus{\mathbb Z}_2^{8}} ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ 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}\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=5} ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a157 L11a159