L11a129

L11a128

L11a130

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a129 at Knotilus!

Link Presentations

[edit Notes on L11a129's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2t(1)t(2)^{3}-3t(2)^{3}-6t(1)t(2)^{2}+6t(2)^{2}+6t(1)t(2)-6t(2)-3t(1)+2}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{5/2}+3q^{3/2}-5{\sqrt {q}}+{\frac {7}{\sqrt {q}}}-{\frac {9}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {11}{q^{7/2}}}+{\frac {8}{q^{9/2}}}-{\frac {7}{q^{11/2}}}+{\frac {4}{q^{13/2}}}-{\frac {2}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{9}z^{-1}+3za^{7}+3a^{7}z^{-1}-3z^{3}a^{5}-5za^{5}-2a^{5}z^{-1}+z^{5}a^{3}+z^{3}a^{3}+z^{5}a+2z^{3}a+za-z^{3}a^{-1}-za^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{10}+2z^{2}a^{10}-a^{10}-2z^{5}a^{9}+3z^{3}a^{9}-2za^{9}+a^{9}z^{-1}-2z^{6}a^{8}-z^{4}a^{8}+5z^{2}a^{8}-3a^{8}-2z^{7}a^{7}-2z^{5}a^{7}+7z^{3}a^{7}-7za^{7}+3a^{7}z^{-1}-2z^{8}a^{6}+z^{4}a^{6}+3z^{2}a^{6}-3a^{6}-2z^{9}a^{5}+3z^{7}a^{5}-4z^{5}a^{5}+7z^{3}a^{5}-5za^{5}+2a^{5}z^{-1}-z^{10}a^{4}-z^{8}a^{4}+8z^{6}a^{4}-7z^{4}a^{4}+2z^{2}a^{4}-5z^{9}a^{3}+17z^{7}a^{3}-16z^{5}a^{3}+6z^{3}a^{3}-za^{3}-z^{10}a^{2}-2z^{8}a^{2}+19z^{6}a^{2}-23z^{4}a^{2}+7z^{2}a^{2}-3z^{9}a+11z^{7}a-8z^{5}a-z^{3}a-3z^{8}+13z^{6}-15z^{4}+5z^{2}-z^{7}a^{-1}+4z^{5}a^{-1}-4z^{3}a^{-1}+za^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                       1 1 4                     2   -2 2                   3 1   2 0                 4 2     -2 -2               5 3       2 -4             6 5         -1 -6           5 4           1 -8         3 6             3 -10       4 5               -1 -12     1 4                 3 -14   1 3                   -2 -16   1                     1 -18 1                       -1

Integral Khovanov Homology (db, data source)

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

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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