L11a467

L11a466

L11a468

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Link Presentations

[edit Notes on L11a467's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X18,12,19,11 X16,8,17,7 X8,16,9,15 X20,13,21,14 X22,20,15,19 X12,21,13,22 X14,18,5,17 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {5, -4, 9, -3, 7, -6, 8, -7}, {10, -1, 4, -5, 11, -2, 3, -8, 6, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(3)-1)\left(-t(2)t(3)^{3}+t(3)^{3}+t(1)t(2)^{2}t(3)^{2}-2t(2)^{2}t(3)^{2}+2t(1)t(3)^{2}-3t(1)t(2)t(3)^{2}+5t(2)t(3)^{2}-2t(3)^{2}-2t(1)t(2)^{2}t(3)+2t(2)^{2}t(3)-2t(1)t(3)+5t(1)t(2)t(3)-3t(2)t(3)+t(3)+t(1)t(2)^{2}-t(1)t(2)\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-3q^{-5}+4q^{4}+8q^{-4}-9q^{3}-13q^{-3}+15q^{2}+19q^{-2}-19q-21q^{-1}+23$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}-3a^{4}z^{2}+a^{4}z^{-2}-z^{2}a^{-4}-a^{4}+3a^{2}z^{4}+2z^{4}a^{-2}+2a^{2}z^{2}-2a^{2}z^{-2}+z^{2}a^{-2}-2a^{2}-z^{6}-z^{4}-z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+3a^{6}z^{2}-a^{6}+3a^{5}z^{7}-7a^{5}z^{5}+z^{5}a^{-5}+5a^{5}z^{3}-z^{3}a^{-5}-a^{5}z+5a^{4}z^{8}-10a^{4}z^{6}+4z^{6}a^{-4}+7a^{4}z^{4}-5z^{4}a^{-4}-5a^{4}z^{2}+2z^{2}a^{-4}-a^{4}z^{-2}+4a^{4}+4a^{3}z^{9}+a^{3}z^{7}+8z^{7}a^{-3}-18a^{3}z^{5}-12z^{5}a^{-3}+20a^{3}z^{3}+7z^{3}a^{-3}-10a^{3}z-2za^{-3}+2a^{3}z^{-1}+a^{2}z^{10}+16a^{2}z^{8}+9z^{8}a^{-2}-44a^{2}z^{6}-11z^{6}a^{-2}+46a^{2}z^{4}+5z^{4}a^{-2}-30a^{2}z^{2}-4z^{2}a^{-2}-2a^{2}z^{-2}+12a^{2}+2a^{-2}+9az^{9}+5z^{9}a^{-1}-az^{7}+9z^{7}a^{-1}-31az^{5}-33z^{5}a^{-1}+34az^{3}+27z^{3}a^{-1}-14az-7za^{-1}+2az^{-1}+z^{10}+20z^{8}-48z^{6}+46z^{4}-28z^{2}-z^{-2}+10$ (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 11                       1 -1 9                     3   3 7                   6 1   -5 5                 9 3     6 3               11 7       -4 1             12 8         4 -1           10 12           2 -3         9 11             -2 -5       5 11               6 -7     3 8                 -5 -9   1 6                   5 -11   2                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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