L11a66

L11a65

L11a67

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Link Presentations

[edit Notes on L11a66's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{4}-4v^{3}+3v^{2}-4v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$14q^{9/2}-17q^{7/2}+16q^{5/2}-{\frac {1}{q^{5/2}}}-14q^{3/2}+{\frac {3}{q^{3/2}}}+q^{17/2}-3q^{15/2}+7q^{13/2}-11q^{11/2}+10{\sqrt {q}}-{\frac {7}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-2z^{5}a^{-1}+4z^{5}a^{-3}-2z^{5}a^{-5}+az^{3}-6z^{3}a^{-1}+7z^{3}a^{-3}-6z^{3}a^{-5}+z^{3}a^{-7}+2az-5za^{-1}+7za^{-3}-6za^{-5}+2za^{-7}+az^{-1}-2a^{-1}z^{-1}+3a^{-3}z^{-1}-3a^{-5}z^{-1}+a^{-7}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-3z^{9}a^{-1}-7z^{9}a^{-3}-4z^{9}a^{-5}-8z^{8}a^{-2}-12z^{8}a^{-4}-7z^{8}a^{-6}-3z^{8}-az^{7}+5z^{7}a^{-1}+10z^{7}a^{-3}-4z^{7}a^{-5}-8z^{7}a^{-7}+35z^{6}a^{-2}+37z^{6}a^{-4}+7z^{6}a^{-6}-6z^{6}a^{-8}+11z^{6}+4az^{5}+11z^{5}a^{-1}+18z^{5}a^{-3}+26z^{5}a^{-5}+12z^{5}a^{-7}-3z^{5}a^{-9}-36z^{4}a^{-2}-30z^{4}a^{-4}+2z^{4}a^{-6}+7z^{4}a^{-8}-z^{4}a^{-10}-12z^{4}-6az^{3}-23z^{3}a^{-1}-31z^{3}a^{-3}-27z^{3}a^{-5}-11z^{3}a^{-7}+2z^{3}a^{-9}+12z^{2}a^{-2}+9z^{2}a^{-4}-5z^{2}a^{-6}-5z^{2}a^{-8}+z^{2}a^{-10}+4z^{2}+4az+11za^{-1}+16za^{-3}+14za^{-5}+5za^{-7}-2a^{-2}+2a^{-6}+a^{-8}-az^{-1}-2a^{-1}z^{-1}-3a^{-3}z^{-1}-3a^{-5}z^{-1}-a^{-7}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     2   2 14                   5 1   -4 12                 6 2     4 10               8 5       -3 8             9 6         3 6           7 8           1 4         7 9             -2 2       5 9               4 0     2 5                 -3 -2   1 5                   4 -4   2                     -2 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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