L11a518

L11a517

L11a519

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Link Presentations

[edit Notes on L11a518's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X22,9,13,10 X20,11,21,12 X12,19,7,20 X10,21,11,22 X18,14,19,13 X16,6,17,5 X2738 X4,16,5,15 X6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 3, -6, 4, -5}, {7, -2, 10, -8, 11, -7, 5, -4, 6, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-2t(1)^{2}t(3)^{3}+t(1)t(3)^{3}+t(1)^{2}t(2)t(3)^{3}-t(1)t(2)t(3)^{3}+t(1)^{2}t(3)^{2}+t(1)^{2}t(2)^{2}t(3)^{2}-t(1)t(2)^{2}t(3)^{2}-2t(1)t(3)^{2}-2t(1)^{2}t(2)t(3)^{2}+2t(1)t(2)t(3)^{2}-t(2)t(3)^{2}+t(3)^{2}-t(1)^{2}t(2)^{2}t(3)+2t(1)t(2)^{2}t(3)-t(2)^{2}t(3)+t(1)t(3)+t(1)^{2}t(2)t(3)-2t(1)t(2)t(3)+2t(2)t(3)-t(3)-t(1)t(2)^{2}+2t(2)^{2}+t(1)t(2)-t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{6}+3q^{5}+q^{-5}-5q^{4}-q^{-4}+8q^{3}+4q^{-3}-9q^{2}-5q^{-2}+10q+8q^{-1}-9$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}-2a^{2}z^{4}+3z^{4}a^{-2}-z^{4}a^{-4}+3z^{4}+a^{4}z^{2}-7a^{2}z^{2}+2z^{2}a^{-2}-2z^{2}a^{-4}+2z^{2}+3a^{4}-6a^{2}+a^{-2}+2+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$z^{3}a^{-7}+3z^{4}a^{-6}-z^{2}a^{-6}+5z^{5}a^{-5}-3z^{3}a^{-5}+a^{4}z^{8}-7a^{4}z^{6}+7z^{6}a^{-4}+17a^{4}z^{4}-10z^{4}a^{-4}-17a^{4}z^{2}+4z^{2}a^{-4}-a^{4}z^{-2}+7a^{4}+a^{3}z^{9}-4a^{3}z^{7}+7z^{7}a^{-3}+a^{3}z^{5}-13z^{5}a^{-3}+10a^{3}z^{3}+5z^{3}a^{-3}-9a^{3}z+2a^{3}z^{-1}+a^{2}z^{10}-3a^{2}z^{8}+5z^{8}a^{-2}-3a^{2}z^{6}-8z^{6}a^{-2}+17a^{2}z^{4}-5z^{4}a^{-2}-20a^{2}z^{2}+7z^{2}a^{-2}-2a^{2}z^{-2}+11a^{2}-2a^{-2}+4az^{9}+3z^{9}a^{-1}-16az^{7}-5z^{7}a^{-1}+15az^{5}-4z^{5}a^{-1}+4az^{3}+3z^{3}a^{-1}-9az+2az^{-1}+z^{10}+z^{8}-11z^{6}+8z^{4}-z^{2}-z^{-2}+3$ (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 13                       1 -1 11                     2   2 9                   3 1   -2 7                 5 2     3 5               5 4       -1 3             5 4         1 1           6 7           1 -1         2 3             -1 -3       3 6               3 -5     1 2                 -1 -7     3                   3 -9 1 1                     0 -11 1                       1

Integral Khovanov Homology (db, data source)

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

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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