L11a361

L11a360

L11a362

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a361 at Knotilus!

Link Presentations

[edit Notes on L11a361'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 X20,10,21,9 X18,8,19,7 X8,20,9,19 X10,18,1,17
Gauss code	{1, -5, 2, -6, 4, -7, 9, -10, 8, -11}, {5, -1, 3, -2, 6, -4, 11, -9, 10, -8, 7, -3}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

L11a360 L11a362