L11a131

L11a130

L11a132

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a131 at Knotilus!

Link Presentations

[edit Notes on L11a131's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(3v^{2}-2v+3\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{21/2}-3q^{19/2}+5q^{17/2}-7q^{15/2}+9q^{13/2}-10q^{11/2}+10q^{9/2}-8q^{7/2}+5q^{5/2}-4q^{3/2}+{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-9}+za^{-9}-z^{5}a^{-7}-2z^{3}a^{-7}-za^{-7}-z^{5}a^{-5}-z^{3}a^{-5}+za^{-5}+a^{-5}z^{-1}-z^{5}a^{-3}-3z^{3}a^{-3}-4za^{-3}-3a^{-3}z^{-1}+z^{3}a^{-1}+3za^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-z^{9}a^{-5}-4z^{9}a^{-7}-3z^{9}a^{-9}-z^{8}a^{-4}+3z^{8}a^{-6}-4z^{8}a^{-10}-z^{7}a^{-3}+z^{7}a^{-5}+15z^{7}a^{-7}+10z^{7}a^{-9}-3z^{7}a^{-11}-z^{6}a^{-2}-8z^{6}a^{-6}+6z^{6}a^{-8}+14z^{6}a^{-10}-z^{6}a^{-12}-z^{5}a^{-1}-z^{5}a^{-3}-3z^{5}a^{-5}-24z^{5}a^{-7}-11z^{5}a^{-9}+10z^{5}a^{-11}+z^{4}a^{-2}-z^{4}a^{-4}+6z^{4}a^{-6}-7z^{4}a^{-8}-12z^{4}a^{-10}+3z^{4}a^{-12}+4z^{3}a^{-1}+7z^{3}a^{-3}+3z^{3}a^{-5}+13z^{3}a^{-7}+7z^{3}a^{-9}-6z^{3}a^{-11}+3z^{2}a^{-2}+5z^{2}a^{-4}-z^{2}a^{-6}+z^{2}a^{-8}+3z^{2}a^{-10}-z^{2}a^{-12}-5za^{-1}-7za^{-3}-2za^{-5}-2za^{-7}-2za^{-9}-3a^{-2}-3a^{-4}-a^{-6}+2a^{-1}z^{-1}+3a^{-3}z^{-1}+a^{-5}z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 22                       1 -1 20                     2   2 18                   3 1   -2 16                 4 2     2 14               5 3       -2 12             5 4         1 10           5 5           0 8         3 5             -2 6       2 5               3 4     2 3                 -1 2   1 4                   3 0                         0 -2 1                       1

Integral Khovanov Homology (db, data source)

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

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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