L11a323

L11a322

L11a324

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Link Presentations

[edit Notes on L11a323's Link Presentations]

Planar diagram presentation	X10,1,11,2 X18,11,19,12 X8,9,1,10 X22,19,9,20 X20,6,21,5 X4,22,5,21 X14,7,15,8 X12,4,13,3 X16,14,17,13 X6,15,7,16 X2,18,3,17
Gauss code	{1, -11, 8, -6, 5, -10, 7, -3}, {3, -1, 2, -8, 9, -7, 10, -9, 11, -2, 4, -5, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(u^{2}v^{2}-2u^{2}v+u^{2}-3uv^{2}+5uv-3u+v^{2}-2v+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+9q^{5/2}-16q^{3/2}+21{\sqrt {q}}-{\frac {25}{\sqrt {q}}}+{\frac {24}{q^{3/2}}}-{\frac {22}{q^{5/2}}}+{\frac {16}{q^{7/2}}}-{\frac {9}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z-2a^{3}z^{5}-4a^{3}z^{3}+z^{3}a^{-3}-3a^{3}z+za^{-3}+az^{7}+3az^{5}-2z^{5}a^{-1}+5az^{3}-4z^{3}a^{-1}+4az-3za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-7a^{3}z^{9}-13az^{9}-6z^{9}a^{-1}-10a^{4}z^{8}-19a^{2}z^{8}-7z^{8}a^{-2}-16z^{8}-8a^{5}z^{7}-3a^{3}z^{7}+11az^{7}+2z^{7}a^{-1}-4z^{7}a^{-3}-4a^{6}z^{6}+12a^{4}z^{6}+43a^{2}z^{6}+14z^{6}a^{-2}-z^{6}a^{-4}+42z^{6}-a^{7}z^{5}+11a^{5}z^{5}+23a^{3}z^{5}+20az^{5}+18z^{5}a^{-1}+9z^{5}a^{-3}+5a^{6}z^{4}-4a^{4}z^{4}-27a^{2}z^{4}-8z^{4}a^{-2}+2z^{4}a^{-4}-28z^{4}+a^{7}z^{3}-7a^{5}z^{3}-20a^{3}z^{3}-23az^{3}-18z^{3}a^{-1}-7z^{3}a^{-3}-2a^{6}z^{2}-a^{4}z^{2}+5a^{2}z^{2}+z^{2}a^{-2}-z^{2}a^{-4}+6z^{2}+2a^{5}z+6a^{3}z+8az+6za^{-1}+2za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     3   3 6                   6 1   -5 4                 10 3     7 2               11 6       -5 0             14 10         4 -2           12 13           1 -4         10 12             -2 -6       6 12               6 -8     3 10                 -7 -10   1 6                   5 -12   3                     -3 -14 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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