L11a483

L11a482

L11a484

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Link Presentations

[edit Notes on L11a483's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(w-1)^{2}\left(2vw^{2}-2vw+v+w^{3}-2w^{2}+2w\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$q^{9}-4q^{8}+9q^{7}-15q^{6}+22q^{5}-25q^{4}+27q^{3}-22q^{2}+18q-11+5q^{-1}-q^{-2}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-8}+a^{-8}-3z^{4}a^{-6}-6z^{2}a^{-6}+a^{-6}z^{-2}-3a^{-6}+2z^{6}a^{-4}+6z^{4}a^{-4}+8z^{2}a^{-4}-2a^{-4}z^{-2}+2a^{-4}+z^{6}a^{-2}-3z^{2}a^{-2}+a^{-2}z^{-2}-a^{-2}-z^{4}+1$ (db)
Kauffman polynomial	$2z^{10}a^{-4}+2z^{10}a^{-6}+9z^{9}a^{-3}+15z^{9}a^{-5}+6z^{9}a^{-7}+14z^{8}a^{-2}+26z^{8}a^{-4}+19z^{8}a^{-6}+7z^{8}a^{-8}+11z^{7}a^{-1}-17z^{7}a^{-5}-2z^{7}a^{-7}+4z^{7}a^{-9}-22z^{6}a^{-2}-70z^{6}a^{-4}-59z^{6}a^{-6}-15z^{6}a^{-8}+z^{6}a^{-10}+5z^{6}+az^{5}-14z^{5}a^{-1}-22z^{5}a^{-3}-18z^{5}a^{-5}-20z^{5}a^{-7}-9z^{5}a^{-9}+14z^{4}a^{-2}+65z^{4}a^{-4}+60z^{4}a^{-6}+11z^{4}a^{-8}-2z^{4}a^{-10}-4z^{4}+3z^{3}a^{-1}+15z^{3}a^{-3}+29z^{3}a^{-5}+23z^{3}a^{-7}+6z^{3}a^{-9}-9z^{2}a^{-2}-27z^{2}a^{-4}-25z^{2}a^{-6}-6z^{2}a^{-8}+z^{2}a^{-10}-za^{-1}-za^{-3}-5za^{-5}-7za^{-7}-2za^{-9}+a^{-2}+2a^{-4}+3a^{-6}+2a^{-8}+1-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-2}z^{-2}+2a^{-4}z^{-2}+a^{-6}z^{-2}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 19                       1 1 17                     3   -3 15                   6 1   5 13                 9 3     -6 11               13 6       7 9             14 11         -3 7           13 11           2 5         9 14             5 3       9 13               -4 1     4 11                 7 -1   1 7                   -6 -3   4                     4 -5 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=4$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{13}$ $r=5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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