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L11n103 L11n105

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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X7,16,8,17 X22,18,5,17 X18,14,19,13 X21,14,22,15 X9,20,10,21 X15,8,16,9 X19,10,20,11 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

q-17/2 - 3q-15/2 + 6q-13/2 - 9q-11/2 + 10q-9/2 - 11q-7/2 + 8q-5/2 - 7q-3/2 + 4q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-28 - 2q-26 + q-24 - q-20 + 4q-18 + q-16 + 3q-14 + 2q-12 + 3q-8 - 2q-6 + 2q-4 - 2 + q2

HOMFLY-PT Polynomial:

- az3 + a3z3 + a3z5 - 2a5z-1 - 6a5z - 4a5z3 + 3a7z-1 + 4a7z - a9z-1

Kauffman Polynomial:

az3 - az5 + 7a2z4 - 4a2z6 - 5a3z3 + 13a3z5 - 6a3z7 + 2a4z2 - 2a4z4 + 6a4z6 - 4a4z8 + 2a5z-1 - 6a5z - 4a5z3 + 12a5z5 - 5a5z7 - a5z9 - 3a6 + 10a6z2 - 17a6z4 + 12a6z6 - 5a6z8 + 3a7z-1 - 7a7z + 6a7z3 - 5a7z5 + a7z7 - a7z9 - 3a8 + 10a8z2 - 9a8z4 + 2a8z6 - a8z8 + a9z-1 - a9z + 4a9z3 - 3a9z5 - a10 + 2a10z2 - a10z4

Khovanov Homology:

trqj 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               4 1   j = -4             5 4     j = -6           6 3       j = -8         4 5         j = -10       5 6           j = -12     2 5             j = -14   1 4               j = -16   2                 j = -18 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, NonAlternating, 104]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 104]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[7, 16, 8, 17], X[22, 18, 5, 17], > X[18, 14, 19, 13], X[21, 14, 22, 15], X[9, 20, 10, 21], X[15, 8, 16, 9], > X[19, 10, 20, 11], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 8, -7, 9, 11, -2, 5, 6, -8, 3, 4, -5, > -9, 7, -6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 6 9 10 11 8 7 4 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - Sqrt[q] 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 2 -24 -20 4 -16 3 2 3 2 2 2 -2 - q - --- + q - q + --- + q + --- + --- + -- - -- + -- + q 26 18 14 12 8 6 4 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 104]][a, z]
Out[8]=	5 7 9 -2 a 3 a a 5 7 3 3 3 5 3 3 5 ----- + ---- - -- - 6 a z + 4 a z - a z + a z - 4 a z + a z z z z
In[9]:=	Kauffman[Link[11, NonAlternating, 104]][a, z]
Out[9]=	5 7 9 6 8 10 2 a 3 a a 5 7 9 4 2 -3 a - 3 a - a + ---- + ---- + -- - 6 a z - 7 a z - a z + 2 a z + z z z 6 2 8 2 10 2 3 3 3 5 3 7 3 > 10 a z + 10 a z + 2 a z + a z - 5 a z - 4 a z + 6 a z + 9 3 2 4 4 4 6 4 8 4 10 4 5 > 4 a z + 7 a z - 2 a z - 17 a z - 9 a z - a z - a z + 3 5 5 5 7 5 9 5 2 6 4 6 6 6 > 13 a z + 12 a z - 5 a z - 3 a z - 4 a z + 6 a z + 12 a z + 8 6 3 7 5 7 7 7 4 8 6 8 8 8 5 9 > 2 a z - 6 a z - 5 a z + a z - 4 a z - 5 a z - a z - a z - 7 9 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 4 1 2 1 4 2 5 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 6 4 5 6 3 5 t 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 3 t + -- + q t 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n104

L11n103 L11n105