L11a117

L11a116

L11a118

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a117 at Knotilus!

Link Presentations

[edit Notes on L11a117's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(3t(2)^{2}-2t(2)+3\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{21/2}-2q^{19/2}+4q^{17/2}-6q^{15/2}+8q^{13/2}-10q^{11/2}+10q^{9/2}-9q^{7/2}+6q^{5/2}-5q^{3/2}+2{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-3}-z^{5}a^{-5}-z^{5}a^{-7}+z^{3}a^{-1}-2z^{3}a^{-3}-z^{3}a^{-5}-3z^{3}a^{-7}+z^{3}a^{-9}+2za^{-1}-za^{-3}+za^{-5}-4za^{-7}+2za^{-9}+a^{-1}z^{-1}-a^{-3}z^{-1}+a^{-5}z^{-1}-2a^{-7}z^{-1}+a^{-9}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-2z^{9}a^{-5}-5z^{9}a^{-7}-3z^{9}a^{-9}-2z^{8}a^{-4}-z^{8}a^{-8}-3z^{8}a^{-10}-2z^{7}a^{-3}+5z^{7}a^{-5}+22z^{7}a^{-7}+13z^{7}a^{-9}-2z^{7}a^{-11}-2z^{6}a^{-2}+2z^{6}a^{-4}+9z^{6}a^{-6}+18z^{6}a^{-8}+12z^{6}a^{-10}-z^{6}a^{-12}-z^{5}a^{-1}+z^{5}a^{-3}-8z^{5}a^{-5}-38z^{5}a^{-7}-21z^{5}a^{-9}+7z^{5}a^{-11}+4z^{4}a^{-2}+z^{4}a^{-4}-23z^{4}a^{-6}-39z^{4}a^{-8}-15z^{4}a^{-10}+4z^{4}a^{-12}+3z^{3}a^{-1}+6z^{3}a^{-3}+7z^{3}a^{-5}+25z^{3}a^{-7}+17z^{3}a^{-9}-4z^{3}a^{-11}+15z^{2}a^{-6}+29z^{2}a^{-8}+11z^{2}a^{-10}-3z^{2}a^{-12}-3za^{-1}-3za^{-3}-5za^{-5}-10za^{-7}-5za^{-9}-a^{-2}-4a^{-6}-7a^{-8}-3a^{-10}+a^{-1}z^{-1}+a^{-3}z^{-1}+a^{-5}z^{-1}+2a^{-7}z^{-1}+a^{-9}z^{-1}$ (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                     1   1 18                   3 1   -2 16                 3 1     2 14               5 3       -2 12             5 3         2 10           5 5           0 8         4 5             -1 6       2 5               3 4     3 4                 -1 2   1 4                   3 0   1                     -1 -2 1                       1

Integral Khovanov Homology (db, data source)

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

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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