L11a53

L11a52

L11a54

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Link Presentations

[edit Notes on L11a53's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(u-1) (v-1) \left(v^2+1\right) \left(2 v^2-3 v+2\right)}{\sqrt{u} v^{5/2}}$ (db)
Jones polynomial	$-q^{11/2}+4 q^{9/2}-7 q^{7/2}+11 q^{5/2}-16 q^{3/2}+17 \sqrt{q}-\frac{18}{\sqrt{q}}+\frac{15}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z^7+z^7 a^{-1} -a^3 z^5+4 a z^5+3 z^5 a^{-1} -z^5 a^{-3} -3 a^3 z^3+7 a z^3-2 z^3 a^{-3} -3 a^3 z+8 a z-6 z a^{-1} +z a^{-3} -a^3 z^{-1} +4 a z^{-1} -4 a^{-1} z^{-1} + a^{-3} z^{-1}$ (db)
Kauffman polynomial	$a^6 z^4-a^6 z^2+z^7 a^{-5} +3 a^5 z^5-3 z^5 a^{-5} -2 a^5 z^3+2 z^3 a^{-5} +4 z^8 a^{-4} +6 a^4 z^6-15 z^6 a^{-4} -6 a^4 z^4+15 z^4 a^{-4} +4 a^4 z^2-z^2 a^{-4} -a^4- a^{-4} +5 z^9 a^{-3} +9 a^3 z^7-17 z^7 a^{-3} -15 a^3 z^5+16 z^5 a^{-3} +15 a^3 z^3-3 z^3 a^{-3} -6 a^3 z-2 z a^{-3} +a^3 z^{-1} + a^{-3} z^{-1} +2 z^{10} a^{-2} +8 a^2 z^8+3 z^8 a^{-2} -8 a^2 z^6-23 z^6 a^{-2} -6 a^2 z^4+18 z^4 a^{-2} +12 a^2 z^2+3 z^2 a^{-2} -4 a^2-4 a^{-2} +5 a z^9+10 z^9 a^{-1} +2 a z^7-25 z^7 a^{-1} -26 a z^5+11 z^5 a^{-1} +29 a z^3+7 z^3 a^{-1} -15 a z-11 z a^{-1} +4 a z^{-1} +4 a^{-1} z^{-1} +2 z^{10}+7 z^8-22 z^6+4 z^4+11 z^2-7$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ 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 6 χ 12                       1 1 10                     3   -3 8                   4 1   3 6                 7 3     -4 4               9 4       5 2             8 7         -1 0           10 9           1 -2         7 10             3 -4       5 8               -3 -6     2 7                 5 -8   1 5                   -4 -10   2                     2 -12 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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