L11a14

L11a13

L11a15

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Link Presentations

[edit Notes on L11a14's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X18,10,19,9 X8493 X14,6,15,5 X20,16,21,15 X22,18,5,17 X16,22,17,21 X10,20,11,19 X2,12,3,11
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, 4, -10, 11, -2, 3, -6, 7, -9, 8, -4, 10, -7, 9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(2v^{4}-4v^{3}+5v^{2}-4v+2\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$22q^{9/2}-22q^{7/2}+17q^{5/2}-13q^{3/2}+{\frac {1}{q^{3/2}}}-q^{19/2}+5q^{17/2}-10q^{15/2}+16q^{13/2}-20q^{11/2}+6{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-7}-z^{3}a^{-7}+2za^{-7}+2a^{-7}z^{-1}+z^{7}a^{-5}+2z^{5}a^{-5}-3z^{3}a^{-5}-9za^{-5}-5a^{-5}z^{-1}+z^{7}a^{-3}+4z^{5}a^{-3}+8z^{3}a^{-3}+9za^{-3}+3a^{-3}z^{-1}-z^{5}a^{-1}-3z^{3}a^{-1}-2za^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-11}+5z^{6}a^{-10}-5z^{4}a^{-10}+a^{-10}+10z^{7}a^{-9}-14z^{5}a^{-9}+3z^{3}a^{-9}+11z^{8}a^{-8}-14z^{6}a^{-8}+z^{4}a^{-8}+7z^{9}a^{-7}-z^{7}a^{-7}-12z^{5}a^{-7}+6z^{3}a^{-7}-3za^{-7}+2a^{-7}z^{-1}+2z^{10}a^{-6}+13z^{8}a^{-6}-28z^{6}a^{-6}+12z^{4}a^{-6}+6z^{2}a^{-6}-5a^{-6}+11z^{9}a^{-5}-17z^{7}a^{-5}+2z^{5}a^{-5}+14z^{3}a^{-5}-13za^{-5}+5a^{-5}z^{-1}+2z^{10}a^{-4}+6z^{8}a^{-4}-17z^{6}a^{-4}+9z^{4}a^{-4}+7z^{2}a^{-4}-5a^{-4}+4z^{9}a^{-3}-3z^{7}a^{-3}-10z^{5}a^{-3}+20z^{3}a^{-3}-14za^{-3}+3a^{-3}z^{-1}+4z^{8}a^{-2}-7z^{6}a^{-2}+3z^{2}a^{-2}+3z^{7}a^{-1}-9z^{5}a^{-1}+9z^{3}a^{-1}-4za^{-1}+z^{6}-3z^{4}+2z^{2}$ (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 20                       1 1 18                     4   -4 16                   6 1   5 14                 10 4     -6 12               10 6       4 10             12 10         -2 8           10 10           0 6         7 12             5 4       6 10               -4 2     2 9                 7 0   1 4                   -3 -2   2                     2 -4 1                       -1

Integral Khovanov Homology (db, data source)

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

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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