L11n366

L11n365

L11n367

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Link Presentations

[edit Notes on L11n366's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X15,17,16,22 X18,10,19,9 X8,18,9,17 X13,21,14,20 X21,15,22,14 X19,5,20,16 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {6, -5, -9, 7, -8, 4}, {10, -1, 3, -6, 5, -2, 11, -3, -7, 8, -4, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(w-1)\left(v^{2}w^{3}-1\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{9}-q^{8}+2q^{7}-2q^{6}+3q^{5}-2q^{4}+3q^{3}-q^{2}+q$ (db)
Signature	6 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-6}+z^{6}a^{-4}-7z^{6}a^{-6}+z^{6}a^{-8}+6z^{4}a^{-4}-17z^{4}a^{-6}+6z^{4}a^{-8}+11z^{2}a^{-4}-20z^{2}a^{-6}+10z^{2}a^{-8}-z^{2}a^{-10}+7a^{-4}-12a^{-6}+6a^{-8}-a^{-10}+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$a^{-12}-za^{-11}+z^{4}a^{-10}-5z^{2}a^{-10}+2a^{-10}+2z^{7}a^{-9}-10z^{5}a^{-9}+11z^{3}a^{-9}-3za^{-9}+3z^{8}a^{-8}-18z^{6}a^{-8}+33z^{4}a^{-8}-26z^{2}a^{-8}-a^{-8}z^{-2}+9a^{-8}+z^{9}a^{-7}-3z^{7}a^{-7}-6z^{5}a^{-7}+19z^{3}a^{-7}-12za^{-7}+2a^{-7}z^{-1}+4z^{8}a^{-6}-25z^{6}a^{-6}+49z^{4}a^{-6}-39z^{2}a^{-6}-2a^{-6}z^{-2}+15a^{-6}+z^{9}a^{-5}-5z^{7}a^{-5}+4z^{5}a^{-5}+8z^{3}a^{-5}-10za^{-5}+2a^{-5}z^{-1}+z^{8}a^{-4}-7z^{6}a^{-4}+17z^{4}a^{-4}-18z^{2}a^{-4}-a^{-4}z^{-2}+8a^{-4}$ (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   \ -2 -1 0 1 2 3 4 5 6 χ 19                 1 1 17               2 2 0 15             1 1 1 1 13           2 2     0 11         1 1 1     1 9       1 2         1 7     2 1           1 5   1 3             2 3                   0 1 1                 1

Integral Khovanov Homology (db, data source)

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

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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