L11a126

L11a125

L11a127

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a126 at Knotilus!

Link Presentations

[edit Notes on L11a126's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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