L10a39

L10a38

L10a40

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Link Presentations

[edit Notes on L10a39's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(v^{2}-v+1\right)^{2}}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-3q^{9/2}+{\frac {1}{q^{9/2}}}+7q^{7/2}-{\frac {2}{q^{7/2}}}-10q^{5/2}+{\frac {5}{q^{5/2}}}+11q^{3/2}-{\frac {9}{q^{3/2}}}+q^{11/2}-13{\sqrt {q}}+{\frac {10}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-3}-a^{3}z^{3}+3z^{3}a^{-3}-3a^{3}z+4za^{-3}-2a^{3}z^{-1}+2a^{-3}z^{-1}-z^{7}a^{-1}+2az^{5}-5z^{5}a^{-1}+8az^{3}-11z^{3}a^{-1}+12az-13za^{-1}+7az^{-1}-7a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-2a^{2}z^{8}-4z^{8}a^{-2}-6z^{8}-2a^{3}z^{7}-5az^{7}-10z^{7}a^{-1}-7z^{7}a^{-3}-a^{4}z^{6}+a^{2}z^{6}-2z^{6}a^{-2}-6z^{6}a^{-4}+6z^{6}+6a^{3}z^{5}+19az^{5}+27z^{5}a^{-1}+11z^{5}a^{-3}-3z^{5}a^{-5}+4a^{4}z^{4}+11a^{2}z^{4}+17z^{4}a^{-2}+8z^{4}a^{-4}-z^{4}a^{-6}+15z^{4}-6a^{3}z^{3}-21az^{3}-27z^{3}a^{-1}-10z^{3}a^{-3}+2z^{3}a^{-5}-5a^{4}z^{2}-17a^{2}z^{2}-20z^{2}a^{-2}-6z^{2}a^{-4}+z^{2}a^{-6}-25z^{2}+4a^{3}z+16az+18za^{-1}+6za^{-3}+2a^{4}+8a^{2}+8a^{-2}+2a^{-4}+13-2a^{3}z^{-1}-7az^{-1}-7a^{-1}z^{-1}-2a^{-3}z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                     1 -1 10                   2   2 8                 5 1   -4 6               5 2     3 4             6 5       -1 2           7 5         2 0         5 8           3 -2       4 5             -1 -4     1 5               4 -6   1 4                 -3 -8   1                   1 -10 1                     -1

Integral Khovanov Homology (db, data source)

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

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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