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L11a117 L11a119

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DrawMorseLink

PD Presentation:

X6172 X14,3,15,4 X16,10,17,9 X20,11,21,12 X8,21,9,22 X18,7,19,8 X12,19,13,20 X10,16,11,15 X22,17,5,18 X2536 X4,13,1,14

Gauss Code:

{{1, -10, 2, -11}, {10, -1, 6, -5, 3, -8, 4, -7, 11, -2, 8, -3, 9, -6, 7, -4, 5, -9}}

Jones Polynomial:

q-21/2 - 3q-19/2 + 8q-17/2 - 15q-15/2 + 20q-13/2 - 24q-11/2 + 24q-9/2 - 22q-7/2 + 16q-5/2 - 10q-3/2 + 4q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-34 - 2q-32 + q-30 - 3q-26 + 6q-24 + 5q-18 - 3q-16 + 4q-14 - 3q-12 + q-10 + 4q-8 - 4q-6 + 5q-4 - 2 + q2

HOMFLY-PT Polynomial:

- az3 - a3z-1 - 4a3z - 2a3z3 + a3z5 + 2a5z-1 + 6a5z + 6a5z3 + 3a5z5 - 3a7z-1 - 8a7z - 6a7z3 + 3a9z-1 + 4a9z - a11z-1

Kauffman Polynomial:

az3 - az5 + 4a2z4 - 4a2z6 - a3z-1 + 5a3z - 10a3z3 + 14a3z5 - 9a3z7 + 5a4z2 - 16a4z4 + 21a4z6 - 12a4z8 - 2a5z-1 + 11a5z - 26a5z3 + 22a5z5 + a5z7 - 8a5z9 - 2a6 + 15a6z2 - 44a6z4 + 52a6z6 - 20a6z8 - 2a6z10 - 3a7z-1 + 15a7z - 26a7z3 + 13a7z5 + 15a7z7 - 13a7z9 + 9a8z2 - 28a8z4 + 36a8z6 - 13a8z8 - 2a8z10 - 3a9z-1 + 12a9z - 17a9z3 + 13a9z5 + 2a9z7 - 5a9z9 + 2a10 - 4a10z2 - a10z4 + 8a10z6 - 5a10z8 - a11z-1 + 3a11z - 6a11z3 + 7a11z5 - 3a11z7 + a12 - 3a12z2 + 3a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                       1 j = 0                     3   j = -2                   7 1   j = -4                 10 4     j = -6               12 6       j = -8             12 10         j = -10           12 12           j = -12         9 13             j = -14       6 11               j = -16     2 9                 j = -18   1 6                   j = -20   2                     j = -22 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 118]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 118]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[16, 10, 17, 9], X[20, 11, 21, 12], > X[8, 21, 9, 22], X[18, 7, 19, 8], X[12, 19, 13, 20], X[10, 16, 11, 15], > X[22, 17, 5, 18], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 6, -5, 3, -8, 4, -7, 11, -2, 8, -3, 9, -6, > 7, -4, 5, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 3 8 15 20 24 24 22 16 10 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q 4 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 2 -30 3 6 5 3 4 3 -10 4 4 -2 - q - --- + q - --- + --- + --- - --- + --- - --- + q + -- - -- + 32 26 24 18 16 14 12 8 6 q q q q q q q q q 5 2 > -- + q 4 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 118]][a, z]
Out[8]=	3 5 7 9 11 a 2 a 3 a 3 a a 3 5 7 9 3 -(--) + ---- - ---- + ---- - --- - 4 a z + 6 a z - 8 a z + 4 a z - a z - z z z z z 3 3 5 3 7 3 3 5 5 5 > 2 a z + 6 a z - 6 a z + a z + 3 a z
In[9]:=	Kauffman[Link[11, Alternating, 118]][a, z]
Out[9]=	3 5 7 9 11 6 10 12 a 2 a 3 a 3 a a 3 5 -2 a + 2 a + a - -- - ---- - ---- - ---- - --- + 5 a z + 11 a z + z z z z z 7 9 11 4 2 6 2 8 2 10 2 > 15 a z + 12 a z + 3 a z + 5 a z + 15 a z + 9 a z - 4 a z - 12 2 3 3 3 5 3 7 3 9 3 11 3 > 3 a z + a z - 10 a z - 26 a z - 26 a z - 17 a z - 6 a z + 2 4 4 4 6 4 8 4 10 4 12 4 5 > 4 a z - 16 a z - 44 a z - 28 a z - a z + 3 a z - a z + 3 5 5 5 7 5 9 5 11 5 2 6 4 6 > 14 a z + 22 a z + 13 a z + 13 a z + 7 a z - 4 a z + 21 a z + 6 6 8 6 10 6 12 6 3 7 5 7 7 7 > 52 a z + 36 a z + 8 a z - a z - 9 a z + a z + 15 a z + 9 7 11 7 4 8 6 8 8 8 10 8 5 9 > 2 a z - 3 a z - 12 a z - 20 a z - 13 a z - 5 a z - 8 a z - 7 9 9 9 6 10 8 10 > 13 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 7 1 2 1 6 2 9 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q q t q t q t q t q t q t q t 11 9 13 12 12 12 10 12 6 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 10 t 2 2 > ---- + 3 t + -- + q t 4 2 q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a118

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