L10a18

L10a17

L10a19

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Link Presentations

[edit Notes on L10a18's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)^{3}\left(v^{2}+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	${\frac {10}{q^{9/2}}}-{\frac {11}{q^{7/2}}}+{\frac {9}{q^{5/2}}}+q^{3/2}-{\frac {9}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {2}{q^{15/2}}}+{\frac {5}{q^{13/2}}}-{\frac {8}{q^{11/2}}}-3{\sqrt {q}}+{\frac {5}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{3}\right)-3a^{7}z-2a^{7}z^{-1}+2a^{5}z^{5}+8a^{5}z^{3}+11a^{5}z+5a^{5}z^{-1}-a^{3}z^{7}-5a^{3}z^{5}-10a^{3}z^{3}-10a^{3}z-3a^{3}z^{-1}+az^{5}+3az^{3}+2az$ (db)
Kauffman polynomial	$-z^{4}a^{10}+2z^{2}a^{10}-a^{10}-2z^{5}a^{9}+2z^{3}a^{9}-3z^{6}a^{8}+2z^{4}a^{8}-4z^{7}a^{7}+6z^{5}a^{7}-8z^{3}a^{7}+5za^{7}-2a^{7}z^{-1}-3z^{8}a^{6}+z^{6}a^{6}+6z^{4}a^{6}-11z^{2}a^{6}+5a^{6}-z^{9}a^{5}-7z^{7}a^{5}+24z^{5}a^{5}-29z^{3}a^{5}+17za^{5}-5a^{5}z^{-1}-6z^{8}a^{4}+11z^{6}a^{4}+2z^{4}a^{4}-11z^{2}a^{4}+5a^{4}-z^{9}a^{3}-6z^{7}a^{3}+26z^{5}a^{3}-29z^{3}a^{3}+16za^{3}-3a^{3}z^{-1}-3z^{8}a^{2}+6z^{6}a^{2}+2z^{4}a^{2}-4z^{2}a^{2}-3z^{7}a+10z^{5}a-10z^{3}a+4za-z^{6}+3z^{4}-2z^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                     1 -1 2                   2   2 0                 3 1   -2 -2               6 2     4 -4             5 5       0 -6           6 4         2 -8         4 5           1 -10       4 6             -2 -12     1 4               3 -14   1 4                 -3 -16   1                   1 -18 1                     -1

Integral Khovanov Homology (db, data source)

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

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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