L11a371

L11a370

L11a372

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a371 at Knotilus!

Link Presentations

[edit Notes on L11a371's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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