L11a79

L11a78

L11a80

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a79 at Knotilus!

Link Presentations

[edit Notes on L11a79's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {3t(1)t(2)^{3}-7t(2)^{3}-11t(1)t(2)^{2}+14t(2)^{2}+14t(1)t(2)-11t(2)-7t(1)+3}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-15q^{9/2}+20q^{7/2}-{\frac {1}{q^{7/2}}}-23q^{5/2}+{\frac {3}{q^{5/2}}}+22q^{3/2}-{\frac {8}{q^{3/2}}}+q^{15/2}-4q^{13/2}+10q^{11/2}-20{\sqrt {q}}+{\frac {13}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-7}-3z^{3}a^{-5}-za^{-5}+a^{-5}z^{-1}+2z^{5}a^{-3}+2z^{3}a^{-3}+a^{3}z+za^{-3}-a^{-3}z^{-1}+z^{5}a^{-1}-2az^{3}-3z^{3}a^{-1}+az-5za^{-1}+2az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-2z^{4}a^{-8}+z^{2}a^{-8}+4z^{7}a^{-7}-8z^{5}a^{-7}+4z^{3}a^{-7}-za^{-7}+8z^{8}a^{-6}-19z^{6}a^{-6}+15z^{4}a^{-6}-10z^{2}a^{-6}+4a^{-6}+7z^{9}a^{-5}-9z^{7}a^{-5}-7z^{5}a^{-5}+9z^{3}a^{-5}-2za^{-5}-a^{-5}z^{-1}+2z^{10}a^{-4}+16z^{8}a^{-4}-54z^{6}a^{-4}+59z^{4}a^{-4}-34z^{2}a^{-4}+9a^{-4}+13z^{9}a^{-3}-21z^{7}a^{-3}+a^{3}z^{5}+2z^{5}a^{-3}-2a^{3}z^{3}+9z^{3}a^{-3}+a^{3}z-2za^{-3}-a^{-3}z^{-1}+2z^{10}a^{-2}+15z^{8}a^{-2}+3a^{2}z^{6}-44z^{6}a^{-2}-4a^{2}z^{4}+46z^{4}a^{-2}+a^{2}z^{2}-20z^{2}a^{-2}+4a^{-2}+6z^{9}a^{-1}+6az^{7}-2z^{7}a^{-1}-10az^{5}-10z^{5}a^{-1}+10az^{3}+16z^{3}a^{-1}-7az-9za^{-1}+2az^{-1}+2a^{-1}z^{-1}+7z^{8}-7z^{6}+4z^{2}-2$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     3   3 12                   7 1   -6 10                 8 3     5 8               12 7       -5 6             11 8         3 4           11 12           1 2         9 11             -2 0       5 12               7 -2     3 8                 -5 -4     5                   5 -6 1 3                     -2 -8 1                       1

Integral Khovanov Homology (db, data source)

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

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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