L11a395

L11a394

L11a396

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Link Presentations

[edit Notes on L11a395's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{u v w^3-4 u v w^2+4 u v w-2 u v-2 u w^3+5 u w^2-5 u w+2 u-2 v w^3+5 v w^2-5 v w+2 v+2 w^3-4 w^2+4 w-1}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db)
Jones polynomial	$1-4 q^{-1} +8 q^{-2} -11 q^{-3} +15 q^{-4} -15 q^{-5} +17 q^{-6} -12 q^{-7} +9 q^{-8} -5 q^{-9} +2 q^{-10} - q^{-11}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{12} z^{-2} +4 a^{10} z^{-2} +4 a^{10}-6 z^2 a^8-5 a^8 z^{-2} -11 a^8+4 z^4 a^6+9 z^2 a^6+2 a^6 z^{-2} +7 a^6-z^6 a^4-2 z^4 a^4-z^2 a^4+z^4 a^2+z^2 a^2$ (db)
Kauffman polynomial	$z^5 a^{13}-3 z^3 a^{13}+3 z a^{13}-a^{13} z^{-1} +2 z^6 a^{12}-4 z^4 a^{12}+3 z^2 a^{12}+a^{12} z^{-2} -2 a^{12}+2 z^7 a^{11}+2 z^5 a^{11}-12 z^3 a^{11}+13 z a^{11}-5 a^{11} z^{-1} +2 z^8 a^{10}+4 z^6 a^{10}-13 z^4 a^{10}+14 z^2 a^{10}+4 a^{10} z^{-2} -10 a^{10}+2 z^9 a^9+z^7 a^9+3 z^5 a^9-16 z^3 a^9+21 z a^9-9 a^9 z^{-1} +z^{10} a^8+4 z^8 a^8-4 z^6 a^8-8 z^4 a^8+20 z^2 a^8+5 a^8 z^{-2} -14 a^8+6 z^9 a^7-9 z^7 a^7+4 z^5 a^7-6 z^3 a^7+11 z a^7-5 a^7 z^{-1} +z^{10} a^6+8 z^8 a^6-22 z^6 a^6+11 z^4 a^6+6 z^2 a^6+2 a^6 z^{-2} -7 a^6+4 z^9 a^5-4 z^7 a^5-8 z^5 a^5+6 z^3 a^5+6 z^8 a^4-15 z^6 a^4+8 z^4 a^4-2 z^2 a^4+4 z^7 a^3-10 z^5 a^3+5 z^3 a^3+z^6 a^2-2 z^4 a^2+z^2 a^2$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                       1 1 -1                     3   -3 -3                   5 1   4 -5                 7 4     -3 -7               8 4       4 -9             7 7         0 -11           10 8           2 -13         5 10             5 -15       4 7               -3 -17     1 5                 4 -19   1 4                   -3 -21   1                     1 -23 1                       -1

Integral Khovanov Homology (db, data source)

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

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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