L11a56

L11a55

L11a57

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a56 at Knotilus!

Link Presentations

[edit Notes on L11a56's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{4}-2t(2)^{3}+3t(2)^{2}-2t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{23/2}-3q^{21/2}+6q^{19/2}-9q^{17/2}+13q^{15/2}-14q^{13/2}+13q^{11/2}-12q^{9/2}+8q^{7/2}-6q^{5/2}+2q^{3/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$z^{5}a^{-9}+3z^{3}a^{-9}+2za^{-9}+a^{-9}z^{-1}-z^{7}a^{-7}-4z^{5}a^{-7}-5z^{3}a^{-7}-3za^{-7}-a^{-7}z^{-1}-z^{7}a^{-5}-4z^{5}a^{-5}-5z^{3}a^{-5}-4za^{-5}-2a^{-5}z^{-1}+z^{5}a^{-3}+4z^{3}a^{-3}+5za^{-3}+2a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-2z^{9}a^{-5}-6z^{9}a^{-7}-4z^{9}a^{-9}-2z^{8}a^{-4}-3z^{8}a^{-6}-7z^{8}a^{-8}-6z^{8}a^{-10}-z^{7}a^{-3}+3z^{7}a^{-5}+15z^{7}a^{-7}+5z^{7}a^{-9}-6z^{7}a^{-11}+7z^{6}a^{-4}+17z^{6}a^{-6}+25z^{6}a^{-8}+10z^{6}a^{-10}-5z^{6}a^{-12}+5z^{5}a^{-3}+9z^{5}a^{-5}-6z^{5}a^{-7}+z^{5}a^{-9}+8z^{5}a^{-11}-3z^{5}a^{-13}-5z^{4}a^{-4}-15z^{4}a^{-6}-23z^{4}a^{-8}-6z^{4}a^{-10}+6z^{4}a^{-12}-z^{4}a^{-14}-9z^{3}a^{-3}-16z^{3}a^{-5}-2z^{3}a^{-7}-3z^{3}a^{-9}-5z^{3}a^{-11}+3z^{3}a^{-13}-3z^{2}a^{-4}+4z^{2}a^{-6}+10z^{2}a^{-8}-z^{2}a^{-10}-3z^{2}a^{-12}+z^{2}a^{-14}+7za^{-3}+8za^{-5}+za^{-11}+3a^{-4}-3a^{-8}+a^{-12}-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-7}z^{-1}+a^{-9}z^{-1}$ (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                   4 1   -3 18                 5 2     3 16               8 4       -4 14             6 5         1 12           7 8           1 10         5 6             -1 8       3 7               4 6     3 5                 -2 4   1 5                   4 2   1                     -1 0 1                       1

Integral Khovanov Homology (db, data source)

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

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.

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