L11a215

L11a214

L11a216

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a215 at Knotilus!

Link Presentations

[edit Notes on L11a215's Link Presentations]

Planar diagram presentation	X8192 X12,4,13,3 X22,12,7,11 X20,16,21,15 X18,10,19,9 X10,20,11,19 X14,22,15,21 X16,6,17,5 X2738 X4,14,5,13 X6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 5, -6, 3, -2, 10, -7, 4, -8, 11, -5, 6, -4, 7, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2u^{2}v^{4}-4u^{2}v^{3}+5u^{2}v^{2}-4u^{2}v+u^{2}-3uv^{4}+6uv^{3}-7uv^{2}+6uv-3u+v^{4}-4v^{3}+5v^{2}-4v+2}{uv^{2}}}$ (db)
Jones polynomial	$-16q^{9/2}+11q^{7/2}-8q^{5/2}+3q^{3/2}+q^{23/2}-3q^{21/2}+7q^{19/2}-12q^{17/2}+16q^{15/2}-18q^{13/2}+18q^{11/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-5}-z^{7}a^{-7}+z^{5}a^{-3}-3z^{5}a^{-5}-4z^{5}a^{-7}+z^{5}a^{-9}+3z^{3}a^{-3}-z^{3}a^{-5}-7z^{3}a^{-7}+3z^{3}a^{-9}+3za^{-3}+2za^{-5}-7za^{-7}+3za^{-9}+a^{-3}z^{-1}-2a^{-7}z^{-1}+a^{-9}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-6}-2z^{10}a^{-8}-4z^{9}a^{-5}-11z^{9}a^{-7}-7z^{9}a^{-9}-3z^{8}a^{-4}-4z^{8}a^{-6}-11z^{8}a^{-8}-10z^{8}a^{-10}-z^{7}a^{-3}+10z^{7}a^{-5}+30z^{7}a^{-7}+10z^{7}a^{-9}-9z^{7}a^{-11}+10z^{6}a^{-4}+29z^{6}a^{-6}+45z^{6}a^{-8}+20z^{6}a^{-10}-6z^{6}a^{-12}+4z^{5}a^{-3}-z^{5}a^{-5}-22z^{5}a^{-7}+14z^{5}a^{-11}-3z^{5}a^{-13}-9z^{4}a^{-4}-33z^{4}a^{-6}-50z^{4}a^{-8}-19z^{4}a^{-10}+6z^{4}a^{-12}-z^{4}a^{-14}-6z^{3}a^{-3}-6z^{3}a^{-5}+11z^{3}a^{-7}-2z^{3}a^{-9}-11z^{3}a^{-11}+2z^{3}a^{-13}+z^{2}a^{-4}+15z^{2}a^{-6}+27z^{2}a^{-8}+9z^{2}a^{-10}-3z^{2}a^{-12}+z^{2}a^{-14}+4za^{-3}-7za^{-7}-za^{-9}+2za^{-11}+a^{-4}-3a^{-6}-5a^{-8}-2a^{-10}-a^{-3}z^{-1}+2a^{-7}z^{-1}+a^{-9}z^{-1}$ (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 24                       1 -1 22                     2   2 20                   5 1   -4 18                 7 2     5 16               9 5       -4 14             9 7         2 12           9 9           0 10         7 9             -2 8       5 10               5 6     3 6                 -3 4   1 6                   5 2   2                     -2 0 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=4$ $i=6$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=6$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=9$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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