L11a461

L11a460

L11a462

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Link Presentations

[edit Notes on L11a461's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{3}t(2)^{3}-t(3)^{3}t(2)^{3}-2t(1)t(3)^{2}t(2)^{3}+2t(3)^{2}t(2)^{3}-t(3)t(2)^{3}-2t(1)t(3)^{3}t(2)^{2}+2t(3)^{3}t(2)^{2}+4t(1)t(3)^{2}t(2)^{2}-4t(3)^{2}t(2)^{2}-2t(1)t(3)t(2)^{2}+3t(3)t(2)^{2}-t(2)^{2}+t(1)t(3)^{3}t(2)-3t(1)t(3)^{2}t(2)+2t(3)^{2}t(2)-2t(1)t(2)+4t(1)t(3)t(2)-4t(3)t(2)+2t(2)+t(1)t(3)^{2}+t(1)-2t(1)t(3)+2t(3)-1}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{10}-3q^{9}+7q^{8}-10q^{7}+14q^{6}-15q^{5}+16q^{4}-13q^{3}+10q^{2}-6q+4-q^{-1}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{4}a^{-8}+3z^{2}a^{-8}+a^{-8}z^{-2}+2a^{-8}-2z^{6}a^{-6}-8z^{4}a^{-6}-9z^{2}a^{-6}-2a^{-6}z^{-2}-5a^{-6}+z^{8}a^{-4}+5z^{6}a^{-4}+8z^{4}a^{-4}+5z^{2}a^{-4}+a^{-4}z^{-2}+a^{-4}-z^{6}a^{-2}-3z^{4}a^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-12}-z^{2}a^{-12}+3z^{5}a^{-11}-2z^{3}a^{-11}+6z^{6}a^{-10}-8z^{4}a^{-10}+6z^{2}a^{-10}-2a^{-10}+7z^{7}a^{-9}-8z^{5}a^{-9}+2z^{3}a^{-9}+za^{-9}+7z^{8}a^{-8}-11z^{6}a^{-8}+8z^{4}a^{-8}-6z^{2}a^{-8}-a^{-8}z^{-2}+3a^{-8}+5z^{9}a^{-7}-6z^{7}a^{-7}-4z^{5}a^{-7}+9z^{3}a^{-7}-8za^{-7}+2a^{-7}z^{-1}+2z^{10}a^{-6}+5z^{8}a^{-6}-29z^{6}a^{-6}+37z^{4}a^{-6}-24z^{2}a^{-6}-2a^{-6}z^{-2}+9a^{-6}+10z^{9}a^{-5}-33z^{7}a^{-5}+29z^{5}a^{-5}-z^{3}a^{-5}-8za^{-5}+2a^{-5}z^{-1}+2z^{10}a^{-4}+2z^{8}a^{-4}-28z^{6}a^{-4}+36z^{4}a^{-4}-13z^{2}a^{-4}-a^{-4}z^{-2}+3a^{-4}+5z^{9}a^{-3}-19z^{7}a^{-3}+19z^{5}a^{-3}-5z^{3}a^{-3}+za^{-3}+4z^{8}a^{-2}-16z^{6}a^{-2}+16z^{4}a^{-2}-2z^{2}a^{-2}-2a^{-2}+z^{7}a^{-1}-3z^{5}a^{-1}+z^{3}a^{-1}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 21                       1 1 19                     3 1 -2 17                   4     4 15                 6 3     -3 13               8 4       4 11             8 7         -1 9           8 7           1 7         5 8             3 5       5 8               -3 3     3 7                 4 1   1 3                   -2 -1   3                     3 -3 1                       -1

Integral Khovanov Homology (db, data source)

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

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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