L11a111

L11a110

L11a112

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a111 at Knotilus!

Link Presentations

[edit Notes on L11a111's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(2v^{4}-v^{3}+2v^{2}-v+2\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{23/2}-2q^{21/2}+4q^{19/2}-6q^{17/2}+9q^{15/2}-10q^{13/2}+9q^{11/2}-9q^{9/2}+6q^{7/2}-5q^{5/2}+2q^{3/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$z^{5}a^{-9}+4z^{3}a^{-9}+4za^{-9}+2a^{-9}z^{-1}-z^{7}a^{-7}-5z^{5}a^{-7}-9z^{3}a^{-7}-9za^{-7}-4a^{-7}z^{-1}-z^{7}a^{-5}-4z^{5}a^{-5}-3z^{3}a^{-5}+za^{-5}+a^{-5}z^{-1}+z^{5}a^{-3}+4z^{3}a^{-3}+4za^{-3}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-2z^{9}a^{-5}-5z^{9}a^{-7}-3z^{9}a^{-9}-2z^{8}a^{-4}-z^{8}a^{-8}-3z^{8}a^{-10}-z^{7}a^{-3}+6z^{7}a^{-5}+21z^{7}a^{-7}+11z^{7}a^{-9}-3z^{7}a^{-11}+8z^{6}a^{-4}+11z^{6}a^{-6}+12z^{6}a^{-8}+6z^{6}a^{-10}-3z^{6}a^{-12}+5z^{5}a^{-3}-z^{5}a^{-5}-35z^{5}a^{-7}-22z^{5}a^{-9}+5z^{5}a^{-11}-2z^{5}a^{-13}-7z^{4}a^{-4}-18z^{4}a^{-6}-22z^{4}a^{-8}-4z^{4}a^{-10}+6z^{4}a^{-12}-z^{4}a^{-14}-8z^{3}a^{-3}-2z^{3}a^{-5}+35z^{3}a^{-7}+22z^{3}a^{-9}-4z^{3}a^{-11}+3z^{3}a^{-13}-z^{2}a^{-4}+15z^{2}a^{-6}+20z^{2}a^{-8}-4z^{2}a^{-10}-6z^{2}a^{-12}+2z^{2}a^{-14}+5za^{-3}-3za^{-5}-17za^{-7}-10za^{-9}-za^{-11}+a^{-4}-5a^{-6}-6a^{-8}+a^{-10}+2a^{-12}-a^{-3}z^{-1}+a^{-5}z^{-1}+4a^{-7}z^{-1}+2a^{-9}z^{-1}$ (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 24                       1 -1 22                     1   1 20                   3 1   -2 18                 3 1     2 16               6 3       -3 14             4 3         1 12           5 6           1 10         4 4             0 8       2 5               3 6     3 4                 -1 4   1 4                   3 2   1                     -1 0 1                       1

Integral Khovanov Homology (db, data source)

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

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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