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L11a138 L11a140

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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X22,10,7,9 X2738 X4,15,5,16 X12,5,13,6 X16,12,17,11 X6,18,1,17 X14,20,15,19 X20,14,21,13 X18,21,19,22

Gauss Code:

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

Jones Polynomial:

q-9/2 - 5q-7/2 + 10q-5/2 - 17q-3/2 + 22q-1/2 - 26q1/2 + 25q3/2 - 22q5/2 + 16q7/2 - 9q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-14 + 2q-12 + 2q-10 - 2q-8 + 6q-6 - q-4 + 4 - 4q2 + 5q4 - 3q6 + 2q8 + 2q10 - 5q12 + 3q14 - q16 - q18 + q20

HOMFLY-PT Polynomial:

- a-5z - a-5z3 + 3a-3z + 4a-3z3 + 2a-3z5 - 3a-1z - 5a-1z3 - 3a-1z5 - a-1z7 - az-1 + 3az3 + 2az5 + a3z-1 - a3z3

Kauffman Polynomial:

a-7z3 - a-7z5 - 2a-6z2 + 5a-6z4 - 4a-6z6 + 2a-5z - 7a-5z3 + 11a-5z5 - 8a-5z7 - 5a-4z4 + 12a-4z6 - 10a-4z8 + 6a-3z - 22a-3z3 + 25a-3z5 - 4a-3z7 - 7a-3z9 + 8a-2z2 - 34a-2z4 + 49a-2z6 - 21a-2z8 - 2a-2z10 + 6a-1z - 26a-1z3 + 25a-1z5 + 11a-1z7 - 14a-1z9 + 8z2 - 36z4 + 53z6 - 20z8 - 2z10 + az-1 + az - 17az3 + 22az5 + 2az7 - 7az9 - a2 + 2a2z2 - 11a2z4 + 19a2z6 - 9a2z8 + a3z-1 - a3z - 5a3z3 + 10a3z5 - 5a3z7 + a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 14                       1 j = 12                     3   j = 10                   6 1   j = 8                 10 3     j = 6               12 6       j = 4             13 10         j = 2           13 12           j = 0         10 14             j = -2       7 12               j = -4     4 11                 j = -6   1 6                   j = -8   4                     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, Alternating, 139]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 139]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[22, 10, 7, 9], X[2, 7, 3, 8], > X[4, 15, 5, 16], X[12, 5, 13, 6], X[16, 12, 17, 11], X[6, 18, 1, 17], > X[14, 20, 15, 19], X[20, 14, 21, 13], X[18, 21, 19, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -5, 6, -8}, > {4, -1, 3, -2, 7, -6, 10, -9, 5, -7, 8, -11, 9, -10, 11, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 5 10 17 22 3/2 5/2 q - ---- + ---- - ---- + ------- - 26 Sqrt[q] + 25 q - 22 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 16 q - 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 2 2 2 6 -4 2 4 6 8 10 4 - q + --- + --- - -- + -- - q - 4 q + 5 q - 3 q + 2 q + 2 q - 12 10 8 6 q q q q 12 14 16 18 20 > 5 q + 3 q - q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 139]][a, z]
Out[8]=	3 3 3 3 5 5 a a z 3 z 3 z z 4 z 5 z 3 3 3 2 z 3 z -(-) + -- - -- + --- - --- - -- + ---- - ---- + 3 a z - a z + ---- - ---- + z z 5 3 a 5 3 a 3 a a a a a a 7 5 z > 2 a z - -- a
In[9]:=	Kauffman[Link[11, Alternating, 139]][a, z]
Out[9]=	3 2 2 2 a a 2 z 6 z 6 z 3 2 2 z 8 z 2 2 -a + - + -- + --- + --- + --- + a z - a z + 8 z - ---- + ---- + 2 a z + z z 5 3 a 6 2 a a a a 3 3 3 3 4 4 z 7 z 22 z 26 z 3 3 3 4 5 z 5 z > -- - ---- - ----- - ----- - 17 a z - 5 a z - 36 z + ---- - ---- - 7 5 3 a 6 4 a a a a a 4 5 5 5 5 34 z 2 4 4 4 z 11 z 25 z 25 z 5 > ----- - 11 a z + a z - -- + ----- + ----- + ----- + 22 a z + 2 7 5 3 a a a a a 6 6 6 7 7 3 5 6 4 z 12 z 49 z 2 6 4 6 8 z 4 z > 10 a z + 53 z - ---- + ----- + ----- + 19 a z - a z - ---- - ---- + 6 4 2 5 3 a a a a a 7 8 8 9 9 11 z 7 3 7 8 10 z 21 z 2 8 7 z 14 z > ----- + 2 a z - 5 a z - 20 z - ----- - ----- - 9 a z - ---- - ----- - a 4 2 3 a a a a 10 9 10 2 z > 7 a z - 2 z - ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 4 1 6 4 11 7 10 14 + 13 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + -- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t q t q t q t q t q t q t q t 12 2 4 4 2 6 2 6 3 8 3 > ---- + 12 q t + 13 q t + 10 q t + 12 q t + 6 q t + 10 q t + 2 q t 8 4 10 4 10 5 12 5 14 6 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a139

L11a138 L11a140