L11a110

L11a109

L11a111

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Link Presentations

[edit Notes on L11a110's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(v^{2}+1\right)\left(v^{4}+1\right)}{{\sqrt {u}}v^{7/2}}}$ (db)
Jones polynomial	$q^{25/2}-2q^{23/2}+3q^{21/2}-3q^{19/2}+4q^{17/2}-4q^{15/2}+4q^{13/2}-4q^{11/2}+2q^{9/2}-3q^{7/2}+q^{5/2}-q^{3/2}$ (db)
Signature	7 (db)
HOMFLY-PT polynomial	$-z^{9}a^{-7}+z^{7}a^{-5}-8z^{7}a^{-7}+z^{7}a^{-9}+7z^{5}a^{-5}-23z^{5}a^{-7}+6z^{5}a^{-9}+16z^{3}a^{-5}-31z^{3}a^{-7}+11z^{3}a^{-9}+14za^{-5}-22za^{-7}+8za^{-9}+4a^{-5}z^{-1}-7a^{-7}z^{-1}+3a^{-9}z^{-1}$ (db)
Kauffman polynomial	$z^{2}a^{-16}+2z^{3}a^{-15}+3z^{4}a^{-14}-3z^{2}a^{-14}+a^{-14}+3z^{5}a^{-13}-4z^{3}a^{-13}+3z^{6}a^{-12}-6z^{4}a^{-12}+3z^{7}a^{-11}-9z^{5}a^{-11}+4z^{3}a^{-11}+3z^{8}a^{-10}-12z^{6}a^{-10}+10z^{4}a^{-10}+3z^{9}a^{-9}-17z^{7}a^{-9}+31z^{5}a^{-9}-25z^{3}a^{-9}+12za^{-9}-3a^{-9}z^{-1}+z^{10}a^{-8}-3z^{8}a^{-8}-7z^{6}a^{-8}+27z^{4}a^{-8}-23z^{2}a^{-8}+7a^{-8}+4z^{9}a^{-7}-28z^{7}a^{-7}+66z^{5}a^{-7}-65z^{3}a^{-7}+30za^{-7}-7a^{-7}z^{-1}+z^{10}a^{-6}-6z^{8}a^{-6}+8z^{6}a^{-6}+8z^{4}a^{-6}-19z^{2}a^{-6}+7a^{-6}+z^{9}a^{-5}-8z^{7}a^{-5}+23z^{5}a^{-5}-30z^{3}a^{-5}+18za^{-5}-4a^{-5}z^{-1}$ (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 26                       1 -1 24                     1   1 22                   2 1   -1 20                 1 1     0 18               3 2       -1 16             1 1         0 14           3 3           0 12         1 1             0 10       1 3               2 8     2 1                 1 6   1 3                   2 4                         0 2 1                       1

Integral Khovanov Homology (db, data source)

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

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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