L11a384

L11a383

L11a385

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a384 at Knotilus!

Link Presentations

[edit Notes on L11a384's Link Presentations]

Planar diagram presentation	X12,1,13,2 X14,8,15,7 X16,4,17,3 X20,6,21,5 X18,10,19,9 X10,18,1,17 X4,20,5,19 X22,14,11,13 X6,16,7,15 X2,11,3,12 X8,22,9,21
Gauss code	{1, -10, 3, -7, 4, -9, 2, -11, 5, -6}, {10, -1, 8, -2, 9, -3, 6, -5, 7, -4, 11, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{4}v^{2}-u^{4}v+3u^{3}v^{3}-8u^{3}v^{2}+5u^{3}v-u^{3}+u^{2}v^{4}-8u^{2}v^{3}+13u^{2}v^{2}-8u^{2}v+u^{2}-uv^{4}+5uv^{3}-8uv^{2}+3uv-v^{3}+v^{2}}{u^{2}v^{2}}}$ (db)
Jones polynomial	$22q^{9/2}-19q^{7/2}+14q^{5/2}-8q^{3/2}+q^{21/2}-4q^{19/2}+9q^{17/2}-15q^{15/2}+19q^{13/2}-23q^{11/2}+3{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-9}-z^{5}a^{-7}+z^{3}a^{-7}+2za^{-7}-a^{-7}z^{-1}-3z^{5}a^{-5}-6z^{3}a^{-5}-3za^{-5}+a^{-5}z^{-1}-z^{5}a^{-3}+z^{3}a^{-3}+3za^{-3}+z^{3}a^{-1}+za^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-6}-2z^{10}a^{-8}-6z^{9}a^{-5}-12z^{9}a^{-7}-6z^{9}a^{-9}-8z^{8}a^{-4}-13z^{8}a^{-6}-12z^{8}a^{-8}-7z^{8}a^{-10}-6z^{7}a^{-3}+2z^{7}a^{-5}+18z^{7}a^{-7}+6z^{7}a^{-9}-4z^{7}a^{-11}-3z^{6}a^{-2}+13z^{6}a^{-4}+33z^{6}a^{-6}+33z^{6}a^{-8}+15z^{6}a^{-10}-z^{6}a^{-12}-z^{5}a^{-1}+9z^{5}a^{-3}+8z^{5}a^{-5}-4z^{5}a^{-7}+7z^{5}a^{-9}+9z^{5}a^{-11}+4z^{4}a^{-2}-13z^{4}a^{-4}-28z^{4}a^{-6}-21z^{4}a^{-8}-8z^{4}a^{-10}+2z^{4}a^{-12}+2z^{3}a^{-1}-7z^{3}a^{-3}-8z^{3}a^{-5}+2z^{3}a^{-7}-5z^{3}a^{-9}-6z^{3}a^{-11}-z^{2}a^{-2}+6z^{2}a^{-4}+10z^{2}a^{-6}+5z^{2}a^{-8}+z^{2}a^{-10}-z^{2}a^{-12}-za^{-1}+3za^{-3}-za^{-5}-5za^{-7}+za^{-9}+za^{-11}-a^{-6}+a^{-5}z^{-1}+a^{-7}z^{-1}$ (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                     3   3 18                   6 1   -5 16                 9 3     6 14               11 7       -4 12             12 8         4 10           10 11           1 8         9 12             -3 6       5 10               5 4     3 9                 -6 2   1 6                   5 0   2                     -2 -2 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=2$ $i=4$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=5$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=6$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=9$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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