L11n118

L11n117

L11n119

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n118 at Knotilus!

Link Presentations

[edit Notes on L11n118's Link Presentations]

Planar diagram presentation	X6172 X3,15,4,14 X9,16,10,17 X11,21,12,20 X21,9,22,8 X7,19,8,18 X19,13,20,12 X15,10,16,11 X17,5,18,22 X2536 X13,1,14,4
Gauss code	{1, -10, -2, 11}, {10, -1, -6, 5, -3, 8, -4, 7, -11, 2, -8, 3, -9, 6, -7, 4, -5, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-2)(v-1)(2v-1)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-10q^{9/2}+11q^{7/2}-13q^{5/2}+11q^{3/2}-{\frac {2}{q^{3/2}}}+q^{15/2}-3q^{13/2}+7q^{11/2}-9{\sqrt {q}}+{\frac {5}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$2z^{5}a^{-3}-4z^{3}a^{-1}+6z^{3}a^{-3}-3z^{3}a^{-5}+2az-6za^{-1}+8za^{-3}-5za^{-5}+za^{-7}+az^{-1}-2a^{-1}z^{-1}+3a^{-3}z^{-1}-3a^{-5}z^{-1}+a^{-7}z^{-1}$ (db)
Kauffman polynomial	$-2z^{9}a^{-3}-2z^{9}a^{-5}-6z^{8}a^{-2}-10z^{8}a^{-4}-4z^{8}a^{-6}-5z^{7}a^{-1}-6z^{7}a^{-3}-4z^{7}a^{-5}-3z^{7}a^{-7}+17z^{6}a^{-2}+27z^{6}a^{-4}+8z^{6}a^{-6}-z^{6}a^{-8}-z^{6}+13z^{5}a^{-1}+30z^{5}a^{-3}+25z^{5}a^{-5}+8z^{5}a^{-7}-24z^{4}a^{-2}-22z^{4}a^{-4}+z^{4}a^{-6}+3z^{4}a^{-8}-4z^{4}-3az^{3}-23z^{3}a^{-1}-39z^{3}a^{-3}-25z^{3}a^{-5}-6z^{3}a^{-7}+11z^{2}a^{-2}+7z^{2}a^{-4}-3z^{2}a^{-6}-3z^{2}a^{-8}+4z^{2}+4az+13za^{-1}+18za^{-3}+12za^{-5}+3za^{-7}-2a^{-2}+2a^{-6}+a^{-8}-az^{-1}-2a^{-1}z^{-1}-3a^{-3}z^{-1}-3a^{-5}z^{-1}-a^{-7}z^{-1}$ (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 χ 16                   1 -1 14                 2   2 12               5 1   -4 10             5 2     3 8           6 5       -1 6         7 5         2 4       4 6           2 2     5 7             -2 0   2 6               4 -2   3                 -3 -4 2                   2

Integral Khovanov Homology (db, data source)

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

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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