L11n233

L11n232

L11n234

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Link Presentations

[edit Notes on L11n233's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X9,18,10,19 X6,13,7,14 X14,7,15,8 X8,15,1,16 X17,22,18,9 X21,16,22,17 X19,4,20,5 X5,20,6,21
Gauss code	{1, -2, 3, 10, -11, -5, 6, -7}, {-4, -1, 2, -3, 5, -6, 7, 9, -8, 4, -10, 11, -9, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-u^{3}v^{5}+u^{2}v^{4}-3u^{2}v^{3}+2u^{2}v^{2}-u^{2}v-uv^{4}+2uv^{3}-3uv^{2}+uv-1}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}-{\frac {1}{q^{13/2}}}-{\frac {1}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {4}{q^{19/2}}}+{\frac {5}{q^{21/2}}}-{\frac {5}{q^{23/2}}}+{\frac {4}{q^{25/2}}}-{\frac {3}{q^{27/2}}}+{\frac {1}{q^{29/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-2a^{13}z^{3}-4a^{13}z-a^{13}z^{-1}+a^{11}z^{7}+9a^{11}z^{5}+23a^{11}z^{3}+19a^{11}z+3a^{11}z^{-1}-a^{9}z^{9}-9a^{9}z^{7}-28a^{9}z^{5}-37a^{9}z^{3}-19a^{9}z-2a^{9}z^{-1}$ (db)
Kauffman polynomial	$a^{18}z^{4}-a^{18}z^{2}+3a^{17}z^{5}-5a^{17}z^{3}+a^{17}z+3a^{16}z^{6}-4a^{16}z^{4}+a^{15}z^{7}+3a^{15}z^{5}-7a^{15}z^{3}+3a^{15}z+3a^{14}z^{6}-2a^{14}z^{4}-a^{14}z^{2}+a^{14}+4a^{13}z^{5}-6a^{13}z^{3}+5a^{13}z-a^{13}z^{-1}+a^{12}z^{8}-9a^{12}z^{6}+26a^{12}z^{4}-21a^{12}z^{2}+3a^{12}+a^{11}z^{9}-10a^{11}z^{7}+32a^{11}z^{5}-41a^{11}z^{3}+22a^{11}z-3a^{11}z^{-1}+a^{10}z^{8}-9a^{10}z^{6}+23a^{10}z^{4}-19a^{10}z^{2}+3a^{10}+a^{9}z^{9}-9a^{9}z^{7}+28a^{9}z^{5}-37a^{9}z^{3}+19a^{9}z-2a^{9}z^{-1}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -8                       1 1 -10                       1 1 -12                   1     1 -14               2         2 -16             2 1 1       -2 -18           3 2           1 -20         2 2 1           -1 -22       3 3               0 -24     2 3                 1 -26   1 2                   -1 -28   2                     2 -30 1                       -1

Integral Khovanov Homology (db, data source)

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

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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