L11a265

L11a264

L11a266

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Link Presentations

[edit Notes on L11a265's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X22,12,9,11 X18,8,19,7 X20,18,21,17 X16,22,17,21 X2,9,3,10 X4,16,5,15 X14,6,15,5 X6,14,7,13 X8,20,1,19
Gauss code	{1, -7, 2, -8, 9, -10, 4, -11}, {7, -1, 3, -2, 10, -9, 8, -6, 5, -4, 11, -5, 6, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)+t(2)-1)(t(2)t(1)-t(1)-t(2))(2t(2)t(1)-t(1)-t(2)+2)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$17q^{9/2}-16q^{7/2}+11q^{5/2}-7q^{3/2}+q^{21/2}-3q^{19/2}+6q^{17/2}-11q^{15/2}+15q^{13/2}-17q^{11/2}+3{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-3}-2z^{5}a^{-5}-z^{5}a^{-7}+z^{3}a^{-1}-3z^{3}a^{-5}-z^{3}a^{-7}+z^{3}a^{-9}+za^{-1}+2za^{-3}-za^{-5}-za^{-7}+za^{-9}+a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-3z^{9}a^{-5}-6z^{9}a^{-7}-3z^{9}a^{-9}-5z^{8}a^{-4}-8z^{8}a^{-6}-7z^{8}a^{-8}-4z^{8}a^{-10}-5z^{7}a^{-3}-5z^{7}a^{-5}+4z^{7}a^{-7}+z^{7}a^{-9}-3z^{7}a^{-11}-3z^{6}a^{-2}+4z^{6}a^{-4}+14z^{6}a^{-6}+17z^{6}a^{-8}+9z^{6}a^{-10}-z^{6}a^{-12}-z^{5}a^{-1}+8z^{5}a^{-3}+18z^{5}a^{-5}+10z^{5}a^{-7}+10z^{5}a^{-9}+9z^{5}a^{-11}+5z^{4}a^{-2}+2z^{4}a^{-4}-6z^{4}a^{-6}-11z^{4}a^{-8}-5z^{4}a^{-10}+3z^{4}a^{-12}+2z^{3}a^{-1}-6z^{3}a^{-3}-18z^{3}a^{-5}-13z^{3}a^{-7}-11z^{3}a^{-9}-8z^{3}a^{-11}-2z^{2}a^{-2}-3z^{2}a^{-4}+4z^{2}a^{-8}+z^{2}a^{-10}-2z^{2}a^{-12}-za^{-1}+4za^{-3}+7za^{-5}+4za^{-7}+4za^{-9}+2za^{-11}+a^{-4}-a^{-3}z^{-1}-a^{-5}z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 22                       1 -1 20                     2   2 18                   4 1   -3 16                 7 2     5 14               8 4       -4 12             9 7         2 10           9 9           0 8         7 8             -1 6       4 9               5 4     3 7                 -4 2   1 5                   4 0   2                     -2 -2 1                       1

Integral Khovanov Homology (db, data source)

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

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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