L11a493

L11a492

L11a494

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Link Presentations

[edit Notes on L11a493's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X10,4,11,3 X18,12,5,11 X20,10,21,9 X22,16,19,15 X14,22,15,21 X2,14,3,13 X8,20,9,19
Gauss code	{1, -10, 5, -3}, {11, -7, 9, -8}, {4, -1, 2, -11, 7, -5, 6, -4, 10, -9, 8, -2, 3, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(3)-1)\left(t(3)^{2}-t(3)+1\right)\left(t(2)t(3)^{2}-2t(2)t(3)+2t(3)-1\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$q^{10}-4q^{9}+10q^{8}-16q^{7}+20q^{6}-23q^{5}+24q^{4}-18q^{3}+15q^{2}-8q+4-q^{-1}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{8}a^{-4}-z^{6}a^{-2}+5z^{6}a^{-4}-2z^{6}a^{-6}-3z^{4}a^{-2}+10z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}-2z^{2}a^{-2}+9z^{2}a^{-4}-9z^{2}a^{-6}+2z^{2}a^{-8}+a^{-2}+2a^{-4}-5a^{-6}+2a^{-8}+a^{-2}z^{-2}-2a^{-4}z^{-2}+a^{-6}z^{-2}$ (db)
Kauffman polynomial	$2z^{10}a^{-4}+2z^{10}a^{-6}+5z^{9}a^{-3}+14z^{9}a^{-5}+9z^{9}a^{-7}+4z^{8}a^{-2}+12z^{8}a^{-4}+24z^{8}a^{-6}+16z^{8}a^{-8}+z^{7}a^{-1}-11z^{7}a^{-3}-26z^{7}a^{-5}+2z^{7}a^{-7}+16z^{7}a^{-9}-14z^{6}a^{-2}-56z^{6}a^{-4}-76z^{6}a^{-6}-24z^{6}a^{-8}+10z^{6}a^{-10}-3z^{5}a^{-1}-2z^{5}a^{-3}-11z^{5}a^{-5}-39z^{5}a^{-7}-23z^{5}a^{-9}+4z^{5}a^{-11}+17z^{4}a^{-2}+66z^{4}a^{-4}+66z^{4}a^{-6}+8z^{4}a^{-8}-8z^{4}a^{-10}+z^{4}a^{-12}+3z^{3}a^{-1}+14z^{3}a^{-3}+32z^{3}a^{-5}+34z^{3}a^{-7}+13z^{3}a^{-9}-8z^{2}a^{-2}-29z^{2}a^{-4}-25z^{2}a^{-6}+4z^{2}a^{-10}-za^{-1}-3za^{-3}-11za^{-5}-13za^{-7}-4za^{-9}+4a^{-4}+5a^{-6}+a^{-8}-a^{-10}-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-2}z^{-2}+2a^{-4}z^{-2}+a^{-6}z^{-2}$ (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 21                       1 1 19                     3   -3 17                   7 1   6 15                 9 3     -6 13               11 7       4 11             14 11         -3 9           10 9           1 7         8 14             6 5       7 10               -3 3     3 10                 7 1   1 5                   -4 -1   3                     3 -3 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=4$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{11}$ $r=5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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