L11n123

L11n122

L11n124

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Link Presentations

[edit Notes on L11n123's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X9,16,10,17 X20,11,21,12 X8,21,9,22 X18,7,19,8 X12,19,13,20 X15,10,16,11 X22,17,5,18 X2536 X4,13,1,14
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 $u$, $v$, $w$, ...)	$-{\frac {t(2)^{5}+3t(1)t(2)^{4}-3t(2)^{4}-4t(1)t(2)^{3}+4t(2)^{3}+4t(1)t(2)^{2}-4t(2)^{2}-3t(1)t(2)+3t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {11}{q^{13/2}}}+{\frac {10}{q^{15/2}}}-{\frac {9}{q^{17/2}}}+{\frac {7}{q^{19/2}}}-{\frac {3}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{11}-2a^{11}z^{-1}+4z^{3}a^{9}+10za^{9}+5a^{9}z^{-1}-3z^{5}a^{7}-11z^{3}a^{7}-11za^{7}-3a^{7}z^{-1}-z^{5}a^{5}-2z^{3}a^{5}$ (db)
Kauffman polynomial	$-z^{4}a^{14}+2z^{2}a^{14}-a^{14}-3z^{5}a^{13}+3z^{3}a^{13}-z^{8}a^{12}+z^{6}a^{12}-6z^{4}a^{12}+6z^{2}a^{12}-z^{9}a^{11}-z^{5}a^{11}-2z^{3}a^{11}+4za^{11}-2a^{11}z^{-1}-5z^{8}a^{10}+10z^{6}a^{10}-8z^{4}a^{10}-4z^{2}a^{10}+5a^{10}-z^{9}a^{9}-6z^{7}a^{9}+20z^{5}a^{9}-25z^{3}a^{9}+15za^{9}-5a^{9}z^{-1}-4z^{8}a^{8}+6z^{6}a^{8}+2z^{4}a^{8}-8z^{2}a^{8}+5a^{8}-6z^{7}a^{7}+17z^{5}a^{7}-18z^{3}a^{7}+11za^{7}-3a^{7}z^{-1}-3z^{6}a^{6}+5z^{4}a^{6}-z^{5}a^{5}+2z^{3}a^{5}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                   1 1 -6                 3 1 -2 -8               4     4 -10             4 3     -1 -12           7 4       3 -14         4 5         1 -16       5 6           -1 -18     2 4             2 -20   1 5               -4 -22   2                 2 -24 1                   -1

Integral Khovanov Homology (db, data source)

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

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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