L11a484

L11a483

L11a485

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Link Presentations

[edit Notes on L11a484's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X12,6,13,5 X8493 X16,10,17,9 X22,17,19,18 X20,12,21,11 X10,20,11,19 X18,21,5,22 X2,14,3,13
Gauss code	{1, -11, 5, -3}, {9, -8, 10, -7}, {4, -1, 2, -5, 6, -9, 8, -4, 11, -2, 3, -6, 7, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)(w-1)^{3}\left(w^{2}+1\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$q^{7}-4q^{6}+8q^{5}-13q^{4}-q^{-4}+18q^{3}+4q^{-3}-20q^{2}-8q^{-2}+21q+14q^{-1}-16$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-4}+3z^{4}a^{-4}+2z^{2}a^{-4}-z^{8}a^{-2}-5z^{6}a^{-2}-a^{2}z^{4}-9z^{4}a^{-2}-2a^{2}z^{2}-6z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{6}+7z^{4}+6z^{2}-2z^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-8}+4z^{5}a^{-7}-2z^{3}a^{-7}+8z^{6}a^{-6}-7z^{4}a^{-6}+z^{2}a^{-6}+11z^{7}a^{-5}-13z^{5}a^{-5}+4z^{3}a^{-5}-za^{-5}+12z^{8}a^{-4}-20z^{6}a^{-4}+12z^{4}a^{-4}-5z^{2}a^{-4}+8z^{9}a^{-3}+a^{3}z^{7}-6z^{7}a^{-3}-3a^{3}z^{5}-18z^{5}a^{-3}+3a^{3}z^{3}+20z^{3}a^{-3}-a^{3}z-4za^{-3}+2z^{10}a^{-2}+4a^{2}z^{8}+18z^{8}a^{-2}-14a^{2}z^{6}-67z^{6}a^{-2}+16a^{2}z^{4}+66z^{4}a^{-2}-8a^{2}z^{2}-21z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+5az^{9}+13z^{9}a^{-1}-12az^{7}-30z^{7}a^{-1}-az^{5}+z^{5}a^{-1}+12az^{3}+23z^{3}a^{-1}-4az-6za^{-1}-2az^{-1}-2a^{-1}z^{-1}+2z^{10}+10z^{8}-53z^{6}+62z^{4}-23z^{2}+2z^{-2}+1$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                       1 1 13                     3   -3 11                   5 1   4 9                 8 3     -5 7               10 5       5 5             10 8         -2 3           11 10           1 1         9 14             5 -1       5 7               -2 -3     3 9                 6 -5   1 5                   -4 -7   3                     3 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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