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L11n151 L11n153

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DrawMorseLink

PD Presentation:

X8192 X18,9,19,10 X6718 X19,7,20,22 X12,5,13,6 X10,4,11,3 X4,15,5,16 X16,12,17,11 X13,21,14,20 X21,15,22,14 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

q-9/2 - 3q-7/2 + 4q-5/2 - 6q-3/2 + 6q-1/2 - 7q1/2 + 5q3/2 - 4q5/2 + 2q7/2

A2 (sl(3)) Invariant:

- q-14 + q-12 + 3q-6 + q-4 + 3q-2 + 2 + q2 + 2q4 - q6 + q8 - q10 - 2q12

HOMFLY-PT Polynomial:

a-3z-1 + 2a-3z - 3a-1z-1 - 6a-1z - 3a-1z3 + 2az-1 + 4az + 3az3 + az5 - a3z - a3z3

Kauffman Polynomial:

a-4 - 2a-4z2 - a-3z-1 + 3a-3z - 2a-3z3 - a-3z5 + 3a-2 - 3a-2z2 - 4a-2z4 + 3a-2z6 - a-2z8 - 3a-1z-1 + 10a-1z - 11a-1z3 + 2a-1z5 + 2a-1z7 - a-1z9 + 3 - 13z4 + 14z6 - 4z8 - 2az-1 + 9az - 18az3 + 14az5 - az7 - az9 - 6a2z4 + 10a2z6 - 3a2z8 + 2a3z - 9a3z3 + 11a3z5 - 3a3z7 - a4z2 + 3a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 8                 2 j = 6               2   j = 4             3 2   j = 2           4 2     j = 0         3 4       j = -2       3 3         j = -4     2 4           j = -6   1 2             j = -8   2               j = -10 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, 152]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 152]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 9, 19, 10], X[6, 7, 1, 8], X[19, 7, 20, 22], > X[12, 5, 13, 6], X[10, 4, 11, 3], X[4, 15, 5, 16], X[16, 12, 17, 11], > X[13, 21, 14, 20], X[21, 15, 22, 14], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 6, -7, 5, -3}, > {3, -1, 2, -6, 8, -5, -9, 10, 7, -8, 11, -2, -4, 9, -10, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 4 6 6 3/2 5/2 7/2 q - ---- + ---- - ---- + ------- - 7 Sqrt[q] + 5 q - 4 q + 2 q 7/2 5/2 3/2 Sqrt[q] q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 3 -4 3 2 4 6 8 10 12 2 - q + q + -- + q + -- + q + 2 q - q + q - q - 2 q 6 2 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 152]][a, z]
Out[8]=	3 1 3 2 a 2 z 6 z 3 3 z 3 3 3 5 ---- - --- + --- + --- - --- + 4 a z - a z - ---- + 3 a z - a z + a z 3 a z z 3 a a a z a
In[9]:=	Kauffman[Link[11, NonAlternating, 152]][a, z]
Out[9]=	2 2 -4 3 1 3 2 a 3 z 10 z 3 2 z 3 z 3 + a + -- - ---- - --- - --- + --- + ---- + 9 a z + 2 a z - ---- - ---- - 2 3 a z z 3 a 4 2 a a z a a a 3 3 4 4 2 2 z 11 z 3 3 3 4 4 z 2 4 > a z - ---- - ----- - 18 a z - 9 a z - 13 z - ---- - 6 a z + 3 a 2 a a 5 5 6 4 4 z 2 z 5 3 5 6 3 z 2 6 > 3 a z - -- + ---- + 14 a z + 11 a z + 14 z + ---- + 10 a z - 3 a 2 a a 7 8 9 4 6 2 z 7 3 7 8 z 2 8 z 9 > a z + ---- - a z - 3 a z - 4 z - -- - 3 a z - -- - a z a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 2 2 4 3 3 3 4 + 4 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t q t q t 2 4 4 2 6 2 8 3 > 2 q t + 3 q t + 2 q t + 2 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n152

L11n151 L11n153