L11a153

L11a152

L11a154

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a153 at Knotilus!

Link Presentations

[edit Notes on L11a153's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{5}-2u^{2}v^{4}+3u^{2}v^{3}-3u^{2}v^{2}+2u^{2}v+uv^{6}-3uv^{5}+6uv^{4}-7uv^{3}+6uv^{2}-3uv+u+2v^{5}-3v^{4}+3v^{3}-2v^{2}+v}{uv^{3}}}$ (db)
Jones polynomial	$-3q^{9/2}+{\frac {2}{q^{9/2}}}+7q^{7/2}-{\frac {6}{q^{7/2}}}-11q^{5/2}+{\frac {9}{q^{5/2}}}+14q^{3/2}-{\frac {13}{q^{3/2}}}+q^{11/2}-{\frac {1}{q^{11/2}}}-16{\sqrt {q}}+{\frac {15}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-az^{7}-z^{7}a^{-1}+a^{3}z^{5}-5az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}+4a^{3}z^{3}-11az^{3}-5z^{3}a^{-1}+3z^{3}a^{-3}+6a^{3}z-12az+3za^{-3}+3a^{3}z^{-1}-5az^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-3a^{3}z^{9}-7az^{9}-4z^{9}a^{-1}-2a^{4}z^{8}-5a^{2}z^{8}-7z^{8}a^{-2}-10z^{8}-a^{5}z^{7}+11a^{3}z^{7}+19az^{7}-z^{7}a^{-1}-8z^{7}a^{-3}+7a^{4}z^{6}+26a^{2}z^{6}+8z^{6}a^{-2}-6z^{6}a^{-4}+33z^{6}+5a^{5}z^{5}-16a^{3}z^{5}-22az^{5}+14z^{5}a^{-1}+12z^{5}a^{-3}-3z^{5}a^{-5}-6a^{4}z^{4}-33a^{2}z^{4}+7z^{4}a^{-4}-z^{4}a^{-6}-35z^{4}-8a^{5}z^{3}+17a^{3}z^{3}+29az^{3}-8z^{3}a^{-1}-10z^{3}a^{-3}+2z^{3}a^{-5}+19a^{2}z^{2}-3z^{2}a^{-2}-5z^{2}a^{-4}+z^{2}a^{-6}+22z^{2}+4a^{5}z-12a^{3}z-21az-2za^{-1}+3za^{-3}-5a^{2}+a^{-4}-5+3a^{3}z^{-1}+5az^{-1}+2a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     2   2 8                   5 1   -4 6                 6 2     4 4               8 5       -3 2             8 6         2 0           8 9           1 -2         5 7             -2 -4       4 8               4 -6     2 5                 -3 -8     4                   4 -10 1 2                     -1 -12 1                       1

Integral Khovanov Homology (db, data source)

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

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.

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