L11a112

L11a111

L11a113

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a112 at Knotilus!

Link Presentations

[edit Notes on L11a112's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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